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    \textbf{SYN000$\wedge$0.ax} A simple include file for THF






$\mbox{ia}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ia1\_type},\mbox{type})$

$\mbox{ia}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ia2\_type},\mbox{type})$

$\mbox{ia}_3{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ia3\_type},\mbox{type})$

$\mbox{ia}_1\qquad\mbox{thf}(\mbox{ia}_1,\mbox{axiom})$

$\mbox{ia}_2\qquad\mbox{thf}(\mbox{ia}_2,\mbox{axiom})$

$\mbox{ia}_3\qquad\mbox{thf}(\mbox{ia}_3,\mbox{axiom})$

\medskip\textbf{SYN000-0.ax} A simple include file for CNF






$\mbox{ia}_1\qquad\mbox{cnf}(\mbox{ia}_1,\mbox{axiom})$

$\mbox{ia}_2\qquad\mbox{cnf}(\mbox{ia}_2,\mbox{axiom})$

$\mbox{ia}_3\qquad\mbox{cnf}(\mbox{ia}_3,\mbox{axiom})$

\medskip\textbf{SYN002+0.ax} Orevkov formula

r(x,y,z)=y+2$\wedge$x=z




$\forall y{:}\;r(y,0,\mbox{succ}(y))\qquad\mbox{fof}(\mbox{hyp}_1,\mbox{axiom})$

$\forall y,x,z,z_1{:}\;( r(y,x,z) \ \Rightarrow \ ( r(z,x,z_1) \ \Rightarrow \ r(y,\mbox{succ}(x),z_1) ) )\qquad\mbox{fof}(\mbox{hyp}_2,\mbox{axiom})$

\medskip\textbf{SYN000+0.ax} A simple include file for FOF






$\mbox{ia}_1\qquad\mbox{fof}(\mbox{ia}_1,\mbox{axiom})$

$\mbox{ia}_2\qquad\mbox{fof}(\mbox{ia}_2,\mbox{axiom})$

$\mbox{ia}_3\qquad\mbox{fof}(\mbox{ia}_3,\mbox{axiom})$

\medskip\textbf{SYN000\_0.ax} A simple include file for TFF






$\mbox{ia}_1{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{ia1\_type},\mbox{type})$

$\mbox{ia}_2{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{ia2\_type},\mbox{type})$

$\mbox{ia}_3{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{ia3\_type},\mbox{type})$

$\mbox{ia}_1\qquad\mbox{tff}(\mbox{ia}_1,\mbox{axiom})$

$\mbox{ia}_2\qquad\mbox{tff}(\mbox{ia}_2,\mbox{axiom})$

$\mbox{ia}_3\qquad\mbox{tff}(\mbox{ia}_3,\mbox{axiom})$

\medskip\textbf{SYN356+1.p} Peter Andrews Problem X2107






$( \mbox{big\_r}(a,b)\mbox{ and }\forall x,y{:}\;( \mbox{big\_r}(x,y) \ \Rightarrow \ ( \mbox{big\_r}(y,x)\mbox{ and }\mbox{big\_q}(x,y) ) )\mbox{ and }\forall u,v{:}\;( \mbox{big\_q}(u,v) \ \Rightarrow \ \mbox{big\_q}(u,u) ) ) \ \Rightarrow \ ( \mbox{big\_q}(a,a)\mbox{ and }\mbox{big\_q}(b,b) )\qquad\mbox{fof}(x_{2107},\mbox{conjecture})$

\medskip\textbf{SYN623-1.p} Harrison problem 4027






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_8(x_{58},x_{58})\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_7(x_{51},x_{51})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{48})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_3(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_2(x_{30},x_{30})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{17}(x_{23},x_{23})\qquad\mbox{cnf}(\mbox{p17}_7,\mbox{negated\_conjecture})$

$p_{14}(x_{10},x_{10})\qquad\mbox{cnf}(\mbox{p14}_8,\mbox{negated\_conjecture})$

$p_{12}(x_3,x_3)\qquad\mbox{cnf}(\mbox{p12}_9,\mbox{negated\_conjecture})$

$p_{18}(c_{26},c_{25})\qquad\mbox{cnf}(\mbox{p18}_{10},\mbox{negated\_conjecture})$

$p_3(c_{22},c_{23})\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_2(c_{24},c_{25})\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$\neg\,p_{16}(c_{22},c_{21})\qquad\mbox{cnf}(\mbox{not\_p16}_{13},\mbox{negated\_conjecture})$

$p_3(c_{23},f_4(c_{24},c_{20}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_{16}(f_4(c_{19},c_{20}),c_{21})\qquad\mbox{cnf}(\mbox{p16}_{15},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{16},\mbox{negated\_conjecture})$

$( p_8(x_{58},x_{59})\mbox{ and }p_8(x_{58},x_{60}) ) \ \Rightarrow \ p_8(x_{59},x_{60})\qquad\mbox{cnf}(\mbox{p8}_{17},\mbox{negated\_conjecture})$

$( p_7(x_{51},x_{52})\mbox{ and }p_7(x_{51},x_{53}) ) \ \Rightarrow \ p_7(x_{52},x_{53})\qquad\mbox{cnf}(\mbox{p7}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{48},x_{49})\mbox{ and }p_5(x_{48},x_{50}) ) \ \Rightarrow \ p_5(x_{49},x_{50})\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$( p_3(x_{37},x_{38})\mbox{ and }p_3(x_{37},x_{39}) ) \ \Rightarrow \ p_3(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p3}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{31})\mbox{ and }p_2(x_{30},x_{32}) ) \ \Rightarrow \ p_2(x_{31},x_{32})\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_{17}(x_{23},x_{24})\mbox{ and }p_{17}(x_{23},x_{25}) ) \ \Rightarrow \ p_{17}(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p17}_{22},\mbox{negated\_conjecture})$

$( p_{14}(x_{10},x_{11})\mbox{ and }p_{14}(x_{10},x_{12}) ) \ \Rightarrow \ p_{14}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p14}_{23},\mbox{negated\_conjecture})$

$( p_{12}(x_3,x_4)\mbox{ and }p_{12}(x_3,x_5) ) \ \Rightarrow \ p_{12}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p12}_{24},\mbox{negated\_conjecture})$

$p_{16}(f_4(f_6(c_{27},c_{25}),x_{21}),x_{22}) \ \Rightarrow \ p_{16}(f_4(c_{25},x_{21}),x_{22})\qquad\mbox{cnf}(\mbox{p16}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{15},x_{13})\mbox{ and }p_8(x_{16},x_{14})\mbox{ and }p_{16}(x_{15},x_{16}) ) \ \Rightarrow \ p_{16}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p16}_{26},\mbox{negated\_conjecture})$

$( p_{18}(x_{28},x_{29})\mbox{ and }p_2(x_{29},x_{27})\mbox{ and }p_{17}(x_{28},x_{26}) ) \ \Rightarrow \ p_{18}(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p18}_{27},\mbox{negated\_conjecture})$

$( p_2(x_{62},x_{64})\mbox{ and }p_7(x_{61},x_{63}) ) \ \Rightarrow \ p_8(f_9(x_{61},x_{62}),f_9(x_{63},x_{64}))\qquad\mbox{cnf}(\mbox{p8}_{28},\mbox{negated\_conjecture})$

$( p_{14}(x_6,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_{12}(f_{15}(x_6,x_7),f_{15}(x_8,x_9))\qquad\mbox{cnf}(\mbox{p12}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{34},x_{36})\mbox{ and }p_5(x_{33},x_{35}) ) \ \Rightarrow \ p_2(f_6(x_{33},x_{34}),f_6(x_{35},x_{36}))\qquad\mbox{cnf}(\mbox{p2}_{30},\mbox{negated\_conjecture})$

$( p_{12}(x_{40},x_{42})\mbox{ and }p_2(x_{41},x_{43}) ) \ \Rightarrow \ p_3(f_{13}(x_{40},x_{41}),f_{13}(x_{42},x_{43}))\qquad\mbox{cnf}(\mbox{p3}_{31},\mbox{negated\_conjecture})$

$( p_2(x_{44},x_{46})\mbox{ and }p_3(x_{45},x_{47}) ) \ \Rightarrow \ p_3(f_4(x_{44},x_{45}),f_4(x_{46},x_{47}))\qquad\mbox{cnf}(\mbox{p3}_{32},\mbox{negated\_conjecture})$

$( p_{10}(x_{54},x_{56})\mbox{ and }p_2(x_{55},x_{57}) ) \ \Rightarrow \ p_7(f_{11}(x_{54},x_{55}),f_{11}(x_{56},x_{57}))\qquad\mbox{cnf}(\mbox{p7}_{33},\mbox{negated\_conjecture})$

$( p_2(x_{17},x_{20})\mbox{ and }p_{18}(c_{26},x_{20})\mbox{ and }p_{16}(f_4(c_{19},x_{18}),x_{19})\mbox{ and }p_{16}(f_4(x_{20},f_{13}(f_{15}(c_{28},x_{17}),x_{20})),f_9(f_{11}(c_{29},x_{17}),x_{20})) ) \ \Rightarrow \ p_{16}(f_4(x_{17},x_{18}),x_{19})\qquad\mbox{cnf}(\mbox{p16}_{34},\mbox{negated\_conjecture})$

$( p_2(x_{17},x_{20})\mbox{ and }p_{18}(c_{26},x_{20})\mbox{ and }p_{16}(f_4(c_{19},x_{18}),x_{19}) ) \ \Rightarrow \ ( p_{16}(f_4(x_{17},x_{18}),x_{19})\mbox{ or }p_{16}(f_4(f_6(c_{27},x_{20}),f_{13}(f_{15}(c_{28},x_{17}),x_{20})),f_9(f_{11}(c_{29},x_{17}),x_{20})) )\qquad\mbox{cnf}(\mbox{p16}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{SYN031-1.p} MQW






$g(a,a)\mbox{ or }g(f(a),a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$g(a,a)\mbox{ or }g(a,f(a))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$g(a,b) \ \Rightarrow \ g(f(b),b)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$g(a,b) \ \Rightarrow \ g(b,f(b))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$g(a,b) \ \Rightarrow \ \neg\,g(b,a)\qquad\mbox{cnf}(\mbox{theorem},\mbox{negated\_conjecture})$

\medskip\textbf{SYN004-1.007.p} Implications that form a contradiction

P1 \& Q1 $\to$ P2     P1 \& Q1 $\to$ Q2     :
 P2 \& Q2 $\to$ P3     P2 \& Q2 $\to$ Q3     :
 ......           ......           :
 Pk-1 \& Qk-1 $\to$Pk  Pk-1 \& Qk-1 $\to$ Qk :
 P1     Q1         ~Pk $\vee$ ~Qk         :
 : The size is k, in the above.




$( p_1\mbox{ and }q_1 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{pqp}_1,\mbox{negated\_conjecture})$

$( p_2\mbox{ and }q_2 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{pqp}_2,\mbox{negated\_conjecture})$

$( p_3\mbox{ and }q_3 ) \ \Rightarrow \ p_4\qquad\mbox{cnf}(\mbox{pqp}_3,\mbox{negated\_conjecture})$

$( p_4\mbox{ and }q_4 ) \ \Rightarrow \ p_5\qquad\mbox{cnf}(\mbox{pqp}_4,\mbox{negated\_conjecture})$

$( p_5\mbox{ and }q_5 ) \ \Rightarrow \ p_6\qquad\mbox{cnf}(\mbox{pqp}_5,\mbox{negated\_conjecture})$

$( p_6\mbox{ and }q_6 ) \ \Rightarrow \ p_7\qquad\mbox{cnf}(\mbox{pqp}_6,\mbox{negated\_conjecture})$

$( p_1\mbox{ and }q_1 ) \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{pqq}_1,\mbox{negated\_conjecture})$

$( p_2\mbox{ and }q_2 ) \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{pqq}_2,\mbox{negated\_conjecture})$

$( p_3\mbox{ and }q_3 ) \ \Rightarrow \ q_4\qquad\mbox{cnf}(\mbox{pqq}_3,\mbox{negated\_conjecture})$

$( p_4\mbox{ and }q_4 ) \ \Rightarrow \ q_5\qquad\mbox{cnf}(\mbox{pqq}_4,\mbox{negated\_conjecture})$

$( p_5\mbox{ and }q_5 ) \ \Rightarrow \ q_6\qquad\mbox{cnf}(\mbox{pqq}_5,\mbox{negated\_conjecture})$

$( p_6\mbox{ and }q_6 ) \ \Rightarrow \ q_7\qquad\mbox{cnf}(\mbox{pqq}_6,\mbox{negated\_conjecture})$

$p_1\qquad\mbox{cnf}(\mbox{base}_1,\mbox{negated\_conjecture})$

$q_1\qquad\mbox{cnf}(\mbox{base}_2,\mbox{negated\_conjecture})$

$p_7 \ \Rightarrow \ \neg\,q_7\qquad\mbox{cnf}(\mbox{base}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN320+1.p} Church problem 46.3 (1)






$\exists z,x{:}\;\forall y_1,y_2{:}\;( ( \neg\,\mbox{big\_f}(x,z) \ \Rightarrow \ \mbox{big\_f}(z,y_1) ) \ \Rightarrow \ ( \mbox{big\_f}(y_2,z) \ \Rightarrow \ \mbox{big\_f}(z,y_2) ) )\qquad\mbox{fof}(\mbox{church\_46\_3}_1,\mbox{conjecture})$

\medskip\textbf{SYN064-1.p} Pelletier Problem 35






$\mbox{big\_p}(x,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_p}(f(x,y),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN380+1.p} Peter Andrews Problem X2132






$\forall w{:}\;\neg\,\mbox{big\_r}(w,w) \ \Rightarrow \ \exists x,y{:}\;( \neg\,\mbox{big\_r}(x,y)\mbox{ and }( \mbox{big\_q}(y,x) \ \Rightarrow \ \forall z{:}\;\mbox{big\_q}(z,z) ) )\qquad\mbox{fof}(x_{2132},\mbox{conjecture})$

\medskip\textbf{SYN615-1.p} Harrison problem 3879






$p_{13}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p13}_1,\mbox{negated\_conjecture})$

$p_7(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_{15}(x_7,x_7)\qquad\mbox{cnf}(\mbox{p15}_4,\mbox{negated\_conjecture})$

$p_2(c_{26},f_3(c_{22}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{21},f_3(c_{22}))\qquad\mbox{cnf}(\mbox{not\_p2}_6,\mbox{negated\_conjecture})$

$p_{19}(c_{23},f_3(f_4(f_5(c_{22}))))\qquad\mbox{cnf}(\mbox{p19}_7,\mbox{negated\_conjecture})$

$p_{20}(c_{24},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\qquad\mbox{cnf}(\mbox{p20}_8,\mbox{negated\_conjecture})$

$p_7(x_{52},x_{53}) \ \Rightarrow \ p_7(f_{12}(x_{52}),f_{12}(x_{53}))\qquad\mbox{cnf}(\mbox{p7}_9,\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_2(f_3(x_{29}),f_3(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{31},x_{32}) \ \Rightarrow \ p_2(f_4(x_{31}),f_4(x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{33},x_{34}) \ \Rightarrow \ p_2(f_5(x_{33}),f_5(x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_7(f_{11}(x_{50}),f_{11}(x_{51}))\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$p_2(c_{26},f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$( p_{13}(x_0,x_1)\mbox{ and }p_{13}(x_0,x_2) ) \ \Rightarrow \ p_{13}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p13}_{15},\mbox{negated\_conjecture})$

$( p_7(x_{43},x_{44})\mbox{ and }p_7(x_{43},x_{45}) ) \ \Rightarrow \ p_7(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_{15}(x_7,x_8)\mbox{ and }p_{15}(x_7,x_9) ) \ \Rightarrow \ p_{15}(x_8,x_9)\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{42},x_{40})\mbox{ and }p_{20}(x_{41},x_{42})\mbox{ and }p_2(x_{41},x_{39}) ) \ \Rightarrow \ p_{20}(x_{39},x_{40})\qquad\mbox{cnf}(\mbox{p20}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{14})\mbox{ and }p_2(x_{17},x_{15})\mbox{ and }p_{19}(x_{16},x_{17}) ) \ \Rightarrow \ p_{19}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p19}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_4,x_6)\mbox{ and }p_2(x_3,x_5) ) \ \Rightarrow \ p_{13}(f_{16}(x_3,x_4),f_{16}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p13}_{21},\mbox{negated\_conjecture})$

$( p_7(x_{58},x_{60})\mbox{ and }p_7(x_{59},x_{61}) ) \ \Rightarrow \ p_7(f_9(x_{58},x_{59}),f_9(x_{60},x_{61}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_7(x_{54},x_{56})\mbox{ and }p_7(x_{55},x_{57}) ) \ \Rightarrow \ p_7(f_8(x_{54},x_{55}),f_8(x_{56},x_{57}))\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{47},x_{49})\mbox{ and }p_7(x_{46},x_{48}) ) \ \Rightarrow \ p_7(f_{10}(x_{46},x_{47}),f_{10}(x_{48},x_{49}))\qquad\mbox{cnf}(\mbox{p7}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{35},x_{37})\mbox{ and }p_2(x_{36},x_{38}) ) \ \Rightarrow \ p_2(f_6(x_{35},x_{36}),f_6(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_2(x_{26},x_{28}) ) \ \Rightarrow \ p_2(f_{18}(x_{25},x_{26}),f_{18}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$( p_{13}(x_{21},x_{23})\mbox{ and }p_7(x_{22},x_{24}) ) \ \Rightarrow \ p_2(f_{14}(x_{21},x_{22}),f_{14}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_{15}(f_{17}(x_{10},x_{11}),f_{17}(x_{12},x_{13}))\qquad\mbox{cnf}(\mbox{p15}_{28},\mbox{negated\_conjecture})$

$p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{26}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{p7}_{29},\mbox{negated\_conjecture})$

$( p_{19}(x_{62},f_3(f_4(f_5(c_{22}))))\mbox{ and }p_{20}(x_{63},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\mbox{ and }p_2(f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25}),f_3(c_{22})) ) \ \Rightarrow \ \neg\,p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),x_{62}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25})),f_9(f_{11}(x_{63}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{not\_p19}_{30},\mbox{negated\_conjecture})$

$( p_{19}(x_{62},f_3(f_4(f_5(c_{22}))))\mbox{ and }p_{20}(x_{63},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\mbox{ and }p_{19}(f_6(f_3(f_4(f_5(c_{22}))),f_{18}(c_{21},f_3(f_5(c_{22})))),x_{63}) ) \ \Rightarrow \ \neg\,p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),x_{62}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25})),f_9(f_{11}(x_{63}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{not\_p19}_{31},\mbox{negated\_conjecture})$

\medskip\textbf{SYN055$\wedge$5.p} TPS problem PELL25






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ ( \neg\,\mbox{cG}@ \mbox{xx}\mbox{ and }\mbox{cR}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cG}@ \mbox{xx}\mbox{ and }\mbox{cF}@ \mbox{xx} ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{xx} ) )\mbox{ or }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}\mbox{ and }\mbox{cR}@ \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}\mbox{ and }\mbox{cP}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPELL}_{25},\mbox{conjecture})$

\medskip\textbf{SYN355+1.p} Peter Andrews Problem X2106






$( \forall x{:}\;( \mbox{big\_r}(x) \ \Rightarrow \ \mbox{big\_p}(x) )\mbox{ and }\forall x{:}\;( \neg\,\mbox{big\_q}(x) \ \Rightarrow \ \mbox{big\_r}(x) ) ) \ \Rightarrow \ \forall x{:}\;( \mbox{big\_p}(x)\mbox{ or }\mbox{big\_q}(x) )\qquad\mbox{fof}(x_{2106},\mbox{conjecture})$

\medskip\textbf{SYN041$\wedge$4.p} Pelletier Problem 3






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ p ) ) )\qquad\mbox{thf}(\mbox{pel}_3,\mbox{conjecture})$

\medskip\textbf{SYN340-1.p} Church problem 46.15 (5)






$f(x,y(x),z_1,z_2,z_1)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,f(z_1,x,y(x),z_1,z_2)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN000\_2.p} Advanced TPTP TF0 syntax without arithmetic

Advanced TPTP TF0 syntax that you will encounter some time.




$\mbox{"An Apple"}\ne \mbox{"A \"Microsoft \\ escape\""}\qquad\mbox{tff}(\mbox{distinct\_object},\mbox{axiom})$

$a{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{b\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$g{:}\;( \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{tff}(\mbox{g\_type},\mbox{type})$

$h{:}\;( \mbox{\$i}\times \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{tff}(\mbox{h\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{p\_type},\mbox{type})$

$q{:}\;( \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{q\_type},\mbox{type})$

$\forall z{:}\;\mbox{\$i}{:}\;\mbox{\$ite\_f}(\exists x{:}\;\mbox{\$i}{:}\;p(x),\forall x{:}\;\mbox{\$i}{:}\;q(x,x),q(z,\mbox{\$ite\_t}(\forall x{:}\;\mbox{\$i}{:}\;p(x),f(a),f(z))))\qquad\mbox{tff}(\mbox{conditionals},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;\mbox{\$let\_ff}(\forall y_1{:}\;\mbox{\$i},y_2{:}\;\mbox{\$i}{:}\;( q(y_1,y_2)\iff p(y_1) ),q(\mbox{\$let\_tt}(\forall z_1{:}\;\mbox{\$i}{:}\;f(z_1) = g(z_1,b),f(a)),x)\mbox{ and }p(\mbox{\$let\_ft}(\forall y_3{:}\;\mbox{\$i},y_4{:}\;\mbox{\$i}{:}\;( q(y_3,y_4)\iff \mbox{\$ite\_f}(y_3 = y_4,q(a,a),q(y_3,y_4)) ),\mbox{\$ite\_t}(q(b,b),f(a),f(x)))))\qquad\mbox{tff}(\mbox{let\_binders},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( p(x)\mbox{ nor }\neg\,q(x,a) )\mbox{ nand }p(x)\qquad\mbox{tff}(\mbox{never\_used\_connectives},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;f(a) = f(x)\qquad\mbox{tff}(\mbox{role\_definition},\mbox{definition})$

$p(a)\qquad\mbox{tff}(\mbox{role\_assumption},\mbox{assumption})$

$p(a)\qquad\mbox{tff}(\mbox{role\_lemma},\mbox{lemma})$

$p(a)\qquad\mbox{tff}(\mbox{role\_theorem},\mbox{theorem})$

$p(a)\qquad\mbox{tff}(\mbox{role\_unknown},\mbox{unknown})$

$\mbox{include}(\mbox{'Axioms/SYN000\_0.ax'},[\mbox{ia}_1,\mbox{ia}_3])$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{source\_unknown},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{source},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{source\_name},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{source\_copy},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{source\_introduced\_assumption},\mbox{axiom})$

$p(a)\qquad\mbox{tff}(\mbox{source\_inference},\mbox{axiom})$

$p(a)\qquad\mbox{tff}(\mbox{source\_inference\_with\_bind},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{useful\_info},\mbox{axiom})$

\medskip\textbf{SYN405+1.p} Kalish and Montague Problem 239

Laws of alphabetic variance.




$( \forall x{:}\;f(x)\mbox{ and }\exists y{:}\;g(y) ) \ \Rightarrow \ \exists z{:}\;( f(z)\mbox{ and }g(z) )\qquad\mbox{fof}(\mbox{kalish}_{239},\mbox{conjecture})$

\medskip\textbf{SYN083-1.p} Pelletier Problem 61






$f(x,f(y,z)) = f(f(x,y),z)\qquad\mbox{cnf}(\mbox{f\_is\_associative},\mbox{axiom})$

$f(a,f(b,f(c,d)))\ne f(f(f(a,b),c),d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN391$\wedge$4.p} Pelletier 9






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_2 ) ) ) ) )@ ( \mbox{inot}@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_2 ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_9,\mbox{conjecture})$

\medskip\textbf{SYN001+1.p} Pelletier 2

2: A biconditional version of the 'most difficult' theorem
 proved by the new logic theorist.




$\neg\,\neg\,p\iff p\qquad\mbox{fof}(\mbox{pel}_2,\mbox{conjecture})$

\medskip\textbf{SYN337-1.p} Church problem 46.15 (2)






$f(a,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(z(y),a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(z(y),y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,f(b,y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,f(b,z(y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN080-1.p} Pelletier Problem 58






$f(x) = g(y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$f(f(a))\ne f(g(b))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN128-1.p} RPT63 synthetic problem 26 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_5(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN180-1.p} RPT63 synthetic problem 78 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_4(a,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN348-1.p} Church problem 46.17 (4)






$( f(x,g(x,y))\mbox{ and }f(y,g(x,y))\mbox{ and }f(w(x),g(x,y)) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(g(x,y),x)\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(y,g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(g(x,y),x)\mbox{ or }f(w(x),g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(y,g(x,y)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),x)\mbox{ or }f(w(x),g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(y,g(x,y))\mbox{ and }f(w(x),g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),x) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(x,g(x,y))\mbox{ or }f(g(x,y),y)\mbox{ or }f(y,g(x,y))\mbox{ or }f(g(x,y),x)\mbox{ or }f(w(x),g(x,y))\mbox{ or }f(g(x,y),w(x))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(w(x),g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),y)\mbox{ or }f(y,g(x,y))\mbox{ or }f(g(x,y),x) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y)\mbox{ and }f(w(x),g(x,y)) ) \ \Rightarrow \ ( f(y,g(x,y))\mbox{ or }f(g(x,y),x)\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y)\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(y,g(x,y))\mbox{ or }f(g(x,y),x)\mbox{ or }f(w(x),g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),x) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(y,g(x,y))\mbox{ or }f(w(x),g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),x)\mbox{ and }f(w(x),g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(y,g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(g(x,y),x)\mbox{ and }f(w(x),g(x,y)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(y,g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(g(x,y),x)\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(y,g(x,y))\mbox{ or }f(w(x),g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( f(y,g(x,y))\mbox{ and }f(g(x,y),x)\mbox{ and }f(w(x),g(x,y)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),y)\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( f(y,g(x,y))\mbox{ and }f(g(x,y),x)\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),y)\mbox{ or }f(w(x),g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y)\mbox{ and }f(y,g(x,y))\mbox{ and }f(g(x,y),x) ) \ \Rightarrow \ ( f(w(x),g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y)\mbox{ and }f(y,g(x,y))\mbox{ and }f(g(x,y),x)\mbox{ and }f(w(x),g(x,y)) ) \ \Rightarrow \ \neg\,f(g(x,y),w(x))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

\medskip\textbf{SYN968+1.p} Syntactic from Shults






$\exists x{:}\;\forall y{:}\;( p(x) \ \Rightarrow \ p(y) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN047+1.p} Pelletier Problem 17

A problem which appears to not be provable by [BBH72].
 For details of why not, see [Pel82] p.135f.




$( ( p\mbox{ and }( q \ \Rightarrow \ r ) ) \ \Rightarrow \ s )\iff ( ( \neg\,p\mbox{ or }q\mbox{ or }s )\mbox{ and }( \neg\,p\mbox{ or }\neg\,r\mbox{ or }s ) )\qquad\mbox{fof}(\mbox{pel}_{17},\mbox{conjecture})$

\medskip\textbf{SYN246-1.p} RPT63 synthetic problem 31 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(c,b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN416+1.p} Pelletier Problem 16

16:A surprising theorem of propositional logic.




$( p \ \Rightarrow \ q )\mbox{ or }( q \ \Rightarrow \ p )\qquad\mbox{fof}(\mbox{pel}_{16},\mbox{conjecture})$

\medskip\textbf{SYN985$\wedge$1.p} Factoring application over conjunction with abstraction

Variation of BB-1 with functional extensionality




$\neg\,\forall p{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o},a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ \lambda x{:}\;\mbox{\$i}{:}\;( a@ x )\mbox{ and }p@ \lambda x{:}\;\mbox{\$i}{:}\;( b@ x ) ) \ \Rightarrow \ ( p@ \lambda x{:}\;\mbox{\$i}{:}\;( a@ x\mbox{ and }b@ x ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN109-1.p} RPT63 synthetic problem 7 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN119-1.p} RPT63 synthetic problem 17 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_0(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN009-2.p} A problem to demonstrate the usefulness of relevancy testing






$p(x,y,z) \ \Rightarrow \ \neg\,t(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q(x,y,z) \ \Rightarrow \ \neg\,t(y,z,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(x,y,z) \ \Rightarrow \ \neg\,t(z,x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$s(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$s(c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$t(c,c,c)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( s(x)\mbox{ and }s(y)\mbox{ and }s(z) ) \ \Rightarrow \ ( p(x,y,z)\mbox{ or }q(y,z,x)\mbox{ or }r(z,x,y) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN352-1.p} Church problem 46.18 (4)






$f(a,b)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ ( f(b,z(x,y))\mbox{ or }f(y,z(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(x,y)\mbox{ or }f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(b,z(x,y))\mbox{ or }f(x,z(x,y))\mbox{ or }f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( f(b,z(x,y))\mbox{ and }f(x,z(x,y)) ) \ \Rightarrow \ f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(x,y)\mbox{ and }f(x,z(x,y)) ) \ \Rightarrow \ \neg\,f(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ ( f(x,z(x,y))\mbox{ or }f(y,z(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN293-1.p} RPT63 synthetic problem 78 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(d,d,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN964+1.p} Syntactic from Shults






$\exists x{:}\;p(x) \ \Rightarrow \ \exists z{:}\;p(z)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN629-1.p} Harrison problem 4093






$p_2(x_4,x_4)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_9(x_{40},x_{40})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_6(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_5(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_3(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_9(c_{28},c_{26})\qquad\mbox{cnf}(\mbox{p9}_6,\mbox{negated\_conjecture})$

$p_{20}(c_{22},c_{28})\qquad\mbox{cnf}(\mbox{p20}_7,\mbox{negated\_conjecture})$

$p_{20}(c_{28},c_{26})\qquad\mbox{cnf}(\mbox{p20}_8,\mbox{negated\_conjecture})$

$p_{21}(c_{26},c_{27})\qquad\mbox{cnf}(\mbox{p21}_9,\mbox{negated\_conjecture})$

$p_{19}(c_{22},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{10},\mbox{negated\_conjecture})$

$p_{19}(c_{27},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{11},\mbox{negated\_conjecture})$

$p_{19}(c_{26},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{12},\mbox{negated\_conjecture})$

$p_9(x_{53},x_{54}) \ \Rightarrow \ p_9(f_{16}(x_{53}),f_{16}(x_{54}))\qquad\mbox{cnf}(\mbox{p9}_{13},\mbox{negated\_conjecture})$

$p_2(x_{22},x_{23}) \ \Rightarrow \ p_3(f_4(x_{22}),f_4(x_{23}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_5(x_{27},x_{28}) \ \Rightarrow \ p_5(f_{12}(x_{27}),f_{12}(x_{28}))\qquad\mbox{cnf}(\mbox{p5}_{15},\mbox{negated\_conjecture})$

$p_5(x_{29},x_{30}) \ \Rightarrow \ p_5(f_{13}(x_{29}),f_{13}(x_{30}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$p_5(x_{31},x_{32}) \ \Rightarrow \ p_5(f_{14}(x_{31}),f_{14}(x_{32}))\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$p_5(x_{47},x_{48}) \ \Rightarrow \ p_9(f_{11}(x_{47}),f_{11}(x_{48}))\qquad\mbox{cnf}(\mbox{p9}_{18},\mbox{negated\_conjecture})$

$( p_2(x_4,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(x_5,x_6)\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_9(x_{40},x_{41})\mbox{ and }p_9(x_{40},x_{42}) ) \ \Rightarrow \ p_9(x_{41},x_{42})\qquad\mbox{cnf}(\mbox{p9}_{20},\mbox{negated\_conjecture})$

$( p_6(x_{33},x_{34})\mbox{ and }p_6(x_{33},x_{35}) ) \ \Rightarrow \ p_6(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p6}_{21},\mbox{negated\_conjecture})$

$( p_5(x_{24},x_{25})\mbox{ and }p_5(x_{24},x_{26}) ) \ \Rightarrow \ p_5(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p5}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{19},x_{20})\mbox{ and }p_3(x_{19},x_{21}) ) \ \Rightarrow \ p_3(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_3,x_1)\mbox{ and }p_9(x_2,x_0)\mbox{ and }p_{19}(x_2,x_3) ) \ \Rightarrow \ p_{19}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p19}_{24},\mbox{negated\_conjecture})$

$( p_9(x_{17},x_{15})\mbox{ and }p_9(x_{18},x_{16})\mbox{ and }p_{21}(x_{17},x_{18}) ) \ \Rightarrow \ p_{21}(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p21}_{25},\mbox{negated\_conjecture})$

$( p_9(x_{13},x_{11})\mbox{ and }p_9(x_{14},x_{12})\mbox{ and }p_{20}(x_{13},x_{14}) ) \ \Rightarrow \ p_{20}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p20}_{26},\mbox{negated\_conjecture})$

$( p_5(x_7,x_9)\mbox{ and }p_6(x_8,x_{10}) ) \ \Rightarrow \ p_2(f_7(x_7,x_8),f_7(x_9,x_{10}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_9(x_{59},x_{61})\mbox{ and }p_9(x_{60},x_{62}) ) \ \Rightarrow \ p_9(f_{18}(x_{59},x_{60}),f_{18}(x_{61},x_{62}))\qquad\mbox{cnf}(\mbox{p9}_{28},\mbox{negated\_conjecture})$

$( p_9(x_{55},x_{57})\mbox{ and }p_9(x_{56},x_{58}) ) \ \Rightarrow \ p_9(f_{17}(x_{55},x_{56}),f_{17}(x_{57},x_{58}))\qquad\mbox{cnf}(\mbox{p9}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{49},x_{51})\mbox{ and }p_9(x_{50},x_{52}) ) \ \Rightarrow \ p_9(f_{15}(x_{49},x_{50}),f_{15}(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p9}_{30},\mbox{negated\_conjecture})$

$( p_9(x_{43},x_{45})\mbox{ and }p_9(x_{44},x_{46}) ) \ \Rightarrow \ p_9(f_{10}(x_{43},x_{44}),f_{10}(x_{45},x_{46}))\qquad\mbox{cnf}(\mbox{p9}_{31},\mbox{negated\_conjecture})$

$( p_5(x_{36},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_6(f_8(x_{36},x_{37}),f_8(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p6}_{32},\mbox{negated\_conjecture})$

$( p_{21}(x_{63},c_{26})\mbox{ and }p_{21}(c_{22},x_{63}) ) \ \Rightarrow \ \neg\,p_{19}(x_{63},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{not\_p21}_{33},\mbox{negated\_conjecture})$

$p_{20}(f_{16}(f_{17}(c_{28},c_{27})),f_{10}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{28}),f_{11}(f_{12}(f_{13}(f_{14}(c_{29}))))))\qquad\mbox{cnf}(\mbox{p20}_{34},\mbox{negated\_conjecture})$

$\neg\,p_{20}(f_{16}(f_{17}(c_{26},c_{27})),f_{10}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{26}),f_{11}(f_{12}(f_{13}(f_{14}(c_{29}))))))\qquad\mbox{cnf}(\mbox{not\_p20}_{35},\mbox{negated\_conjecture})$

$p_{20}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{28}),f_{18}(f_{11}(f_{12}(f_{13}(f_{14}(c_{29})))),f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{26})))\qquad\mbox{cnf}(\mbox{p20}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{SYN299-1.p} RPT63 synthetic problem 84 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(b,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN143-1.p} RPT63 synthetic problem 41 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_5(d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN936+1.p} From Smullyan






$\forall x{:}\;( c \ \Rightarrow \ p(x) )\iff ( c \ \Rightarrow \ \forall x{:}\;p(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN278-1.p} RPT63 synthetic problem 63 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,y,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN967+1.p} Syntactic from Shults






$\forall a,b{:}\;( \forall y{:}\;( q(y) \ \Rightarrow \ p(y) ) \ \Rightarrow \ \exists x{:}\;( ( p(x) \ \Rightarrow \ p(a) )\mbox{ and }( q(x) \ \Rightarrow \ p(b) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN177-1.p} RPT63 synthetic problem 75 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_3(x,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN326-1.p} Church problem 46.12 (1)






$f(y(x),z(x))\mbox{ or }f(x,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$g(y(x))\mbox{ or }f(x,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$h(x) \ \Rightarrow \ f(x,x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(z(x),x)\mbox{ or }h(z(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$g(x) \ \Rightarrow \ h(z(x))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$f(x,y(x))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,f(z(x),z(x))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN346-1.p} Church problem 46.17 (2)






$f(b,z_1(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y_1,z_2(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y_1,z_1(y_1,y_2)) \ \Rightarrow \ \neg\,f(y_2,z_1(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(b,z_2(y_1,y_2)) \ \Rightarrow \ \neg\,f(y_2,z_2(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN202-1.p} RPT63 synthetic problem 100 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_2(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN060+1.p} Pelletier Problem 30






$\forall x{:}\;( ( \mbox{big\_f}(x)\mbox{ or }\mbox{big\_g}(x) ) \ \Rightarrow \ \neg\,\mbox{big\_h}(x) )\qquad\mbox{fof}(\mbox{pel30}_1,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_g}(x) \ \Rightarrow \ \neg\,\mbox{big\_i}(x) ) \ \Rightarrow \ ( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_h}(x) ) )\qquad\mbox{fof}(\mbox{pel30}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{big\_i}(x)\qquad\mbox{fof}(\mbox{pel}_{30},\mbox{conjecture})$

\medskip\textbf{SYN000\_1.p} Basic TPTP TF0 syntax without arithmetic

Basic TPTP TF0 syntax that you can't survive without parsing.




$p_0{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{p0\_type},\mbox{type})$

$q_0{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{q0\_type},\mbox{type})$

$r_0{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{r0\_type},\mbox{type})$

$s_0{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{s0\_type},\mbox{type})$

$( p_0\mbox{ and }\neg\,q_0 ) \ \Rightarrow \ ( r_0\mbox{ or }\neg\,s_0 )\qquad\mbox{tff}(\mbox{propositional},\mbox{axiom})$

$a{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{b\_type},\mbox{type})$

$h{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{h\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$g{:}\;( \mbox{\$i}\times \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{tff}(\mbox{g\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{p\_type},\mbox{type})$

$q{:}\;( \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{q\_type},\mbox{type})$

$r{:}\;( \mbox{\$i}\times \mbox{\$i}\times \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{r\_type},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{s\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( p(x)\mbox{ or }\neg\,q(x,a) ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r(x,f(y),g(x,f(y),z))\mbox{ and }\neg\,s(f(f(f(b)))) ) )\qquad\mbox{tff}(\mbox{first\_order},\mbox{axiom})$

$\exists y{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( f(y) = g(x,f(y),z)\mbox{ or }f(f(f(b)))\ne a\mbox{ or }x = f(y) )\qquad\mbox{tff}(\mbox{equality},\mbox{axiom})$

$\mbox{\$true}\mbox{ or }\mbox{\$false}\qquad\mbox{tff}(\mbox{true\_false},\mbox{axiom})$

$\mbox{'A proposition'}{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{quoted\_proposition\_type},\mbox{type})$

$\mbox{'A predicate'}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{quoted\_predicate\_type},\mbox{type})$

$\mbox{'A constant'}{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{quoted\_constant\_type},\mbox{type})$

$\mbox{'A function'}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{tff}(\mbox{quoted\_function\_type},\mbox{type})$

$\mbox{'A \'quoted \\ escape\''}{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{quoted\_escape\_type},\mbox{type})$

$\mbox{'A proposition'}\mbox{ or }\mbox{'A predicate'}(a)\mbox{ or }p(\mbox{'A constant'})\mbox{ or }p(\mbox{'A function'}(a))\mbox{ or }p(\mbox{'A \'quoted \\ escape\''})\qquad\mbox{tff}(\mbox{single\_quoted},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( p(x) \ \Leftarrow \ \neg\,q(x,a) )\iff \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;r(x,f(y),g(x,f(y),z)) <~> \neg\,s(f(f(f(b)))) )\qquad\mbox{tff}(\mbox{useful\_connectives},\mbox{axiom})$

$\mbox{new}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{new\_type},\mbox{type})$

$\mbox{newc}{:}\;\mbox{new}\qquad\mbox{tff}(\mbox{newc\_type},\mbox{type})$

$\mbox{newf}{:}\;( \mbox{new}\times \mbox{\$i} )\to \mbox{new}\qquad\mbox{tff}(\mbox{newf\_type},\mbox{type})$

$\mbox{newp}{:}\;( \mbox{new}\times \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{newp\_type},\mbox{type})$

$\forall x{:}\;\mbox{new}{:}\;\mbox{newp}(\mbox{newf}(\mbox{newc},a),a)\qquad\mbox{tff}(\mbox{new\_axiom},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( p(x)\mbox{ or }\neg\,q(x,a) ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r(x,f(y),g(x,f(y),z))\mbox{ and }\neg\,s(f(f(f(b)))) ) )\qquad\mbox{tff}(123,\mbox{axiom})$

$p(h)\qquad\mbox{tff}(\mbox{role\_hypothesis},\mbox{hypothesis})$

$\exists x{:}\;\mbox{\$i}{:}\;p(x)\qquad\mbox{tff}(\mbox{role\_conjecture},\mbox{conjecture})$

$\mbox{include}(\mbox{'Axioms/SYN000\_0.ax'})$

\medskip\textbf{SYN339+1.p} Church problem 46.15 (4)






$\exists x{:}\;\forall y{:}\;\exists z{:}\;( \mbox{big\_f}(x,y,z) \ \Rightarrow \ \mbox{big\_f}(y,z,z) )\qquad\mbox{fof}(\mbox{church\_46\_15}_4,\mbox{conjecture})$

\medskip\textbf{SYN590-1.p} Harrison problem 3582






$p_{11}(c_{14})\qquad\mbox{cnf}(\mbox{c14\_is\_p11}_1,\mbox{negated\_conjecture})$

$p_{11}(c_{17})\qquad\mbox{cnf}(\mbox{c17\_is\_p11}_2,\mbox{negated\_conjecture})$

$p_{11}(c_{16})\qquad\mbox{cnf}(\mbox{c16\_is\_p11}_3,\mbox{negated\_conjecture})$

$p_5(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_2(x_{14},x_{14})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_3(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_3(f_4(c_{17}),f_6(c_{18}))\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_3(f_4(c_{16}),f_6(c_{15}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_{11}(x_4) \ \Rightarrow \ p_{11}(f_9(x_4))\qquad\mbox{cnf}(\mbox{p11}_9,\mbox{negated\_conjecture})$

$( p_{11}(x_9)\mbox{ and }p_2(x_9,x_8) ) \ \Rightarrow \ p_{11}(x_8)\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_{10}(f_6(x_5),f_6(x_6)) \ \Rightarrow \ p_{12}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p12}_{11},\mbox{negated\_conjecture})$

$p_2(x_{32},x_{33}) \ \Rightarrow \ p_5(f_8(x_{32}),f_8(x_{33}))\qquad\mbox{cnf}(\mbox{p5}_{12},\mbox{negated\_conjecture})$

$p_{12}(x_5,x_6) \ \Rightarrow \ p_{10}(f_6(x_5),f_6(x_6))\qquad\mbox{cnf}(\mbox{p10}_{13},\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_2(f_9(x_{17}),f_9(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_2(x_{22},x_{23}) \ \Rightarrow \ p_3(f_4(x_{22}),f_4(x_{23}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{25},x_{26}) \ \Rightarrow \ p_3(f_6(x_{25}),f_6(x_{26}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_5(x_{30},x_{31}) \ \Rightarrow \ p_5(f_7(x_{30}),f_7(x_{31}))\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$p_{11}(x_{24}) \ \Rightarrow \ p_3(f_4(x_{24}),f_6(f_8(x_{24})))\qquad\mbox{cnf}(\mbox{p3}_{18},\mbox{negated\_conjecture})$

$p_{11}(x_4) \ \Rightarrow \ p_{10}(f_4(f_9(x_4)),f_4(x_4))\qquad\mbox{cnf}(\mbox{p10}_{19},\mbox{negated\_conjecture})$

$p_{11}(x_7) \ \Rightarrow \ p_{10}(f_6(f_7(c_{13})),f_4(x_7))\qquad\mbox{cnf}(\mbox{p10}_{20},\mbox{negated\_conjecture})$

$p_{11}(x_4) \ \Rightarrow \ p_{10}(f_6(f_7(c_{13})),f_4(x_4))\qquad\mbox{cnf}(\mbox{p10}_{21},\mbox{negated\_conjecture})$

$( p_5(x_{27},x_{28})\mbox{ and }p_5(x_{27},x_{29}) ) \ \Rightarrow \ p_5(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p5}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{14},x_{15})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_2(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{19},x_{20})\mbox{ and }p_3(x_{19},x_{21}) ) \ \Rightarrow \ p_3(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_{11}(x_{34})\mbox{ and }p_{12}(x_{35},c_{15}) ) \ \Rightarrow \ \neg\,p_3(f_4(x_{34}),f_6(x_{35}))\qquad\mbox{cnf}(\mbox{not\_p11}_{25},\mbox{negated\_conjecture})$

$( p_3(x_2,x_0)\mbox{ and }p_3(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{26},\mbox{negated\_conjecture})$

$( p_5(x_{12},x_{10})\mbox{ and }p_5(x_{13},x_{11})\mbox{ and }p_{12}(x_{12},x_{13}) ) \ \Rightarrow \ p_{12}(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN392+1.p} Pelletier 14






$( p_1\iff p_2 )\iff ( ( p_2\mbox{ or }\neg\,p_1 )\mbox{ and }( \neg\,p_2\mbox{ or }p_1 ) )\qquad\mbox{fof}(\mbox{pel}_{14},\mbox{conjecture})$

\medskip\textbf{SYN048+1.p} Pelletier Problem 18






$\exists y{:}\;\forall x{:}\;( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel}_{18},\mbox{conjecture})$

\medskip\textbf{SYN598-1.p} Harrison problem 3756






$p_{17}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p17}_1,\mbox{negated\_conjecture})$

$p_{18}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p18}_2,\mbox{negated\_conjecture})$

$p_7(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(x_9,x_9)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_6,\mbox{negated\_conjecture})$

$p_3(f_4(f_6(c_{21})),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$( p_{17}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{17}(x_0)\qquad\mbox{cnf}(\mbox{p17}_8,\mbox{negated\_conjecture})$

$( p_{18}(x_3)\mbox{ and }p_5(x_3,x_2) ) \ \Rightarrow \ p_{18}(x_2)\qquad\mbox{cnf}(\mbox{p18}_9,\mbox{negated\_conjecture})$

$p_5(x_{49},x_{50}) \ \Rightarrow \ p_7(f_9(x_{49}),f_9(x_{50}))\qquad\mbox{cnf}(\mbox{p7}_{10},\mbox{negated\_conjecture})$

$p_5(x_{12},x_{13}) \ \Rightarrow \ p_2(f_6(x_{12}),f_6(x_{13}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{21},x_{22}) \ \Rightarrow \ p_3(f_{12}(x_{21}),f_{12}(x_{22}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_3(x_{23},x_{24}) \ \Rightarrow \ p_3(f_{13}(x_{23}),f_{13}(x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{25},x_{26}) \ \Rightarrow \ p_3(f_{15}(x_{25}),f_{15}(x_{26}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{27},x_{28}) \ \Rightarrow \ p_3(f_{16}(x_{27}),f_{16}(x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_3(f_4(x_{29}),f_4(x_{30}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_3(x_{45},x_{46}) \ \Rightarrow \ p_7(f_{14}(x_{45}),f_{14}(x_{46}))\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$p_7(x_{47},x_{48}) \ \Rightarrow \ p_7(f_8(x_{47}),f_8(x_{48}))\qquad\mbox{cnf}(\mbox{p7}_{18},\mbox{negated\_conjecture})$

$( p_2(x_9,x_{10})\mbox{ and }p_2(x_9,x_{11}) ) \ \Rightarrow \ p_2(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{34},x_{35})\mbox{ and }p_7(x_{34},x_{36}) ) \ \Rightarrow \ p_7(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{21},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_7(x_6,x_4)\mbox{ and }p_7(x_7,x_5)\mbox{ and }p_{19}(x_6,x_7) ) \ \Rightarrow \ p_{19}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p19}_{23},\mbox{negated\_conjecture})$

$( p_7(x_{16},x_{14})\mbox{ and }p_7(x_{17},x_{15})\mbox{ and }p_{20}(x_{16},x_{17}) ) \ \Rightarrow \ p_{20}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p20}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{42},x_{44})\mbox{ and }p_7(x_{41},x_{43}) ) \ \Rightarrow \ p_7(f_{11}(x_{41},x_{42}),f_{11}(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$( p_7(x_{38},x_{40})\mbox{ and }p_7(x_{37},x_{39}) ) \ \Rightarrow \ p_7(f_{10}(x_{37},x_{38}),f_{10}(x_{39},x_{40}))\qquad\mbox{cnf}(\mbox{p7}_{26},\mbox{negated\_conjecture})$

$\neg\,p_{19}(f_{10}(f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),c_{22}),f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),f_{12}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(c_{23})))))))))),f_8(f_9(c_{21})))\qquad\mbox{cnf}(\mbox{not\_p19}_{27},\mbox{negated\_conjecture})$

$p_{18}(x_8) \ \Rightarrow \ p_{19}(f_{10}(f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),f_4(f_6(x_8))),f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),f_{12}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(c_{23})))))))))),f_8(f_9(x_8)))\qquad\mbox{cnf}(\mbox{p19}_{28},\mbox{negated\_conjecture})$

$p_{18}(x_8) \ \Rightarrow \ p_{20}(f_8(f_9(x_8)),f_{10}(f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),f_{16}(f_4(f_6(x_8)))),f_{11}(f_{14}(f_{12}(f_{15}(f_{13}(c_{23})))),f_{12}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(f_{13}(c_{23})))))))))))\qquad\mbox{cnf}(\mbox{p20}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{SYN981+1.p} Syntactic from Shults






$( \forall x{:}\;( p(x) \ \Rightarrow \ q(x) )\mbox{ and }\exists y{:}\;( q(y) \ \Rightarrow \ r(y) ) ) \ \Rightarrow \ \exists z{:}\;( p(z) \ \Rightarrow \ r(z) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN274-1.p} RPT63 synthetic problem 59 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_0(x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN034-1.p} QW






$p(a,a)\mbox{ or }p(a,f(a))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$p(a,a)\mbox{ or }p(f(a),a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( p(a,b)\mbox{ and }p(b,a) ) \ \Rightarrow \ \neg\,p(b,a)\qquad\mbox{cnf}(\mbox{theorem},\mbox{negated\_conjecture})$

\medskip\textbf{SYN330-1.p} Church problem 46.14 (2)






$f(z(x,y),y) \ \Rightarrow \ f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,z(x,y)) \ \Rightarrow \ f(z(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(z(x,y),z(x,y)) \ \Rightarrow \ f(z(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(z(x,y),y) \ \Rightarrow \ f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(y,x) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$f(y,x) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$f(x,y)\mbox{ or }f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ \neg\,f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN001$\wedge$4.004.p} ILTP Problem SYJ212+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$a_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ a_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_4 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_3 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_2 )@ ( \mbox{iatom}@ a_1 ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYN954+1.p} Syntactic from Shults






$\forall a,b{:}\;( \forall z{:}\;( q(z) \ \Rightarrow \ p(z) ) \ \Rightarrow \ \exists x{:}\;( ( p(x) \ \Rightarrow \ p(a) )\mbox{ and }( q(x) \ \Rightarrow \ p(b) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN350-1.p} Church problem 46.18 (2)






$f(a,z(x,y)) \ \Rightarrow \ f(z(x,y),a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(z(x,y),a) \ \Rightarrow \ f(a,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,z(x,y))\mbox{ or }f(y,z(x,y))\mbox{ or }f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(a,z(x,y))\mbox{ and }f(y,z(x,y)) ) \ \Rightarrow \ f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ ( f(a,z(x,y))\mbox{ or }f(y,z(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(a,z(x,y))\mbox{ and }f(y,z(x,y)) ) \ \Rightarrow \ \neg\,f(x,y)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN958+1.p} Syntactic from Shults






$( \exists x{:}\;a(x)\mbox{ and }\forall x{:}\;b(x) ) \ \Rightarrow \ \exists x{:}\;( a(x)\mbox{ and }b(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN235-1.p} RPT63 synthetic problem 20 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(e,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN984$\wedge$1.p} Factoring application over conjunction with lambda

Variation of BB-1 with functional extensionality




$a{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{a\_const},\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{b\_const},\mbox{type})$

$p{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_const},\mbox{type})$

$( p@ \lambda x{:}\;\mbox{\$i}{:}\;a\mbox{ and }p@ \lambda x{:}\;\mbox{\$i}{:}\;b ) \ \Rightarrow \ ( p@ \lambda x{:}\;\mbox{\$i}{:}\;( a\mbox{ and }b ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN342-1.p} Church problem 46.15 (7)






$f(a,y_2,y_3(y_2))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,f(y_4,y_4,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN147-1.p} RPT63 synthetic problem 45 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_1(x,d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN732+1.p} Peter Andrews Problem X3411






$\forall y{:}\;( \forall x{:}\;p(x,y) \ \Rightarrow \ \forall u{:}\;q(u,y) ) \ \Rightarrow \ \exists z{:}\;( \exists v{:}\;p(v,z) \ \Rightarrow \ \exists w{:}\;( p(z,w)\mbox{ or }q(w,z) ) )\qquad\mbox{fof}(x_{3411},\mbox{conjecture})$

\medskip\textbf{SYN036+2.p} Andrews Challenge Problem






$( \exists x{:}\;\forall y{:}\;( \mbox{big\_p}(x)\iff \mbox{big\_p}(y) )\iff ( \exists u{:}\;\mbox{big\_q}(u)\iff \forall w{:}\;\mbox{big\_p}(w) ) )\iff ( \exists x_1{:}\;\forall y_1{:}\;( \mbox{big\_q}(x_1)\iff \mbox{big\_q}(y_1) )\iff ( \exists u_1{:}\;\mbox{big\_p}(u_1)\iff \forall w_1{:}\;\mbox{big\_q}(w_1) ) )\qquad\mbox{fof}(\mbox{pel}_{34},\mbox{conjecture})$

\medskip\textbf{SYN996$\wedge$1.p} Test for naive Skolemization






$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_decl},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ \exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( r@ x@ ( f@ x ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN052$\wedge$7.p} Pelletier Problem 22






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{big\_f}{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{big\_f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_f}@ x ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_f}@ x ) )@ ( \mbox{mbox\_s}_4@ p ) ) ) ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x_1{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_f}@ x_1 ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x_1{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_f}@ x_1 ) ) ) )@ ( \mbox{mbox\_s}_4@ p ) ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{22},\mbox{conjecture})$

\medskip\textbf{SYN132-1.p} RPT63 synthetic problem 30 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_0(x,e,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN977+1.p} Syntactic from Shults






$( a\iff b )\mbox{ or }a\mbox{ or }b\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN948+1.p} Syntactic from Shults






$\forall x{:}\;\exists y{:}\;( a(x,y)\mbox{ and }a(y,y) ) \ \Rightarrow \ \exists z{:}\;a(z,z)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN183-1.p} RPT63 synthetic problem 81 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_5(e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN213-1.p} RPT63 synthetic problem 111 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_5(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN000-1.p} Basic TPTP CNF syntax

Basic TPTP CNF syntax that you can't survive without parsing.




$( q_0\mbox{ and }s_0 ) \ \Rightarrow \ ( p_0\mbox{ or }r_0 )\qquad\mbox{cnf}(\mbox{propositional},\mbox{axiom})$

$( q(x,a)\mbox{ and }s(f(f(f(b)))) ) \ \Rightarrow \ ( p(x)\mbox{ or }r(x,f(y),g(x,f(y),z)) )\qquad\mbox{cnf}(\mbox{first\_order},\mbox{axiom})$

$f(f(f(b))) = a \ \Rightarrow \ ( f(y) = g(x,f(y),z)\mbox{ or }x = f(y) )\qquad\mbox{cnf}(\mbox{equality},\mbox{axiom})$

$\mbox{\$true}\mbox{ or }\mbox{\$false}\qquad\mbox{cnf}(\mbox{true\_false},\mbox{axiom})$

$\mbox{'A proposition'}\mbox{ or }\mbox{'A predicate'}(y)\mbox{ or }p(\mbox{'A constant'})\mbox{ or }p(\mbox{'A function'}(a))\mbox{ or }p(\mbox{'A \'quoted \\ escape\''})\qquad\mbox{cnf}(\mbox{single\_quoted},\mbox{axiom})$

$( q(x,a)\mbox{ and }s(f(f(f(b)))) ) \ \Rightarrow \ ( p(x)\mbox{ or }r(x,f(y),g(x,f(y),z)) )\qquad\mbox{cnf}(123,\mbox{axiom})$

$p(h)\qquad\mbox{cnf}(\mbox{role\_hypothesis},\mbox{hypothesis})$

$\neg\,p(x)\qquad\mbox{cnf}(\mbox{role\_negated\_conjecture},\mbox{negated\_conjecture})$

$\mbox{include}(\mbox{'Axioms/SYN000-0.ax'})$

\medskip\textbf{SYN057+1.p} Pelletier Problem 27






$\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }\neg\,\mbox{big\_g}(x) )\qquad\mbox{fof}(\mbox{pel27}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_h}(x) )\qquad\mbox{fof}(\mbox{pel27}_2,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_j}(x)\mbox{ and }\mbox{big\_i}(x) ) \ \Rightarrow \ \mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel27}_3,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_h}(x)\mbox{ and }\neg\,\mbox{big\_g}(x) ) \ \Rightarrow \ \forall x_1{:}\;( \mbox{big\_i}(x_1) \ \Rightarrow \ \neg\,\mbox{big\_h}(x_1) )\qquad\mbox{fof}(\mbox{pel27}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_j}(x) \ \Rightarrow \ \neg\,\mbox{big\_i}(x) )\qquad\mbox{fof}(\mbox{pel}_{27},\mbox{conjecture})$

\medskip\textbf{SYN552-1.p} The E Killer






$g(x_1) = g(x_2)\qquad\mbox{cnf}(c_1,\mbox{axiom})$

$f(g(b),b)\ne f(g(a),a)\qquad\mbox{cnf}(c_2,\mbox{axiom})$

\medskip\textbf{SYN006-1.p} A problem to demonstrate controlling splits






$p_1(x,f(y)) \ \Rightarrow \ ( q_1(f(x),y)\mbox{ or }q_2(x,f(y)) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p_1(a,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,q_2(x,f(y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p_2(x,f(y)) \ \Rightarrow \ ( q_3(f(x),y)\mbox{ or }q_4(x,f(y)) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,q_3(x,g(y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\neg\,q_4(f(f(x)),f(g(f(y))))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$q_1(x,f(y)) \ \Rightarrow \ p_2(f(x),y)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN327-1.p} Church problem 46.12 (2)






$f(y,a)\mbox{ or }f(a,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(a,z(y)) \ \Rightarrow \ f(y,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,z(y))\mbox{ or }f(a,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(y,a)\mbox{ and }f(z(y),y) ) \ \Rightarrow \ f(a,z(y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(a,y) \ \Rightarrow \ \neg\,f(a,z(y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(a,y)\mbox{ and }f(y,a) ) \ \Rightarrow \ \neg\,f(z(y),y)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN082-1.p} Pelletier Problem 60






$\mbox{big\_f}(y,b) \ \Rightarrow \ ( \mbox{big\_f}(a,f(a))\mbox{ or }\mbox{big\_f}(y,f(a)) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{big\_f}(a,f(a))\mbox{ or }\mbox{big\_f}(a,b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(a,x)\mbox{ and }\mbox{big\_f}(y,b) ) \ \Rightarrow \ ( \mbox{big\_f}(g(x),x)\mbox{ or }\mbox{big\_f}(y,f(a)) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{big\_f}(a,x) \ \Rightarrow \ ( \mbox{big\_f}(g(x),x)\mbox{ or }\mbox{big\_f}(a,b) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(a,x)\mbox{ and }\mbox{big\_f}(a,f(a)) ) \ \Rightarrow \ \mbox{big\_f}(g(x),x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(g(x),x)\mbox{ and }\mbox{big\_f}(a,x)\mbox{ and }\mbox{big\_f}(y,b) ) \ \Rightarrow \ \mbox{big\_f}(y,f(a))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(g(x),x)\mbox{ and }\mbox{big\_f}(a,x) ) \ \Rightarrow \ \mbox{big\_f}(a,b)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(g(x),x)\mbox{ and }\mbox{big\_f}(a,x) ) \ \Rightarrow \ \neg\,\mbox{big\_f}(a,f(a))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN067+1.p} Pelletier Problem 38






$\forall x{:}\;( ( \mbox{big\_p}(a)\mbox{ and }( \mbox{big\_p}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{big\_p}(y)\mbox{ and }\mbox{big\_r}(x,y) ) ) ) \ \Rightarrow \ \exists z,w{:}\;( \mbox{big\_p}(z)\mbox{ and }\mbox{big\_r}(x,w)\mbox{ and }\mbox{big\_r}(w,z) ) )\iff \forall x_1{:}\;( ( \neg\,\mbox{big\_p}(a)\mbox{ or }\mbox{big\_p}(x_1)\mbox{ or }\exists z_1,w_1{:}\;( \mbox{big\_p}(z_1)\mbox{ and }\mbox{big\_r}(x_1,w_1)\mbox{ and }\mbox{big\_r}(w_1,z_1) ) )\mbox{ and }( \neg\,\mbox{big\_p}(a)\mbox{ or }\neg\,\exists y_1{:}\;( \mbox{big\_p}(y_1)\mbox{ and }\mbox{big\_r}(x_1,y_1) )\mbox{ or }\exists z_2,w_2{:}\;( \mbox{big\_p}(z_2)\mbox{ and }\mbox{big\_r}(x_1,w_2)\mbox{ and }\mbox{big\_r}(w_2,z_2) ) ) )\qquad\mbox{fof}(\mbox{pel}_{38},\mbox{conjecture})$

\medskip\textbf{SYN160-1.p} RPT63 synthetic problem 58 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN266-1.p} RPT63 synthetic problem 51 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_3(b,e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN250-1.p} RPT63 synthetic problem 35 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(e,a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN232-1.p} RPT63 synthetic problem 17 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN055-1.p} Pelletier Problem 25






$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(x) ) \ \Rightarrow \ \neg\,\mbox{big\_r}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_p}(b) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_r}(b) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\mbox{big\_q}(x) \ \Rightarrow \ \neg\,\mbox{big\_p}(x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN099-1.003.p} Plaisted problem sym(m(t(2,3)))






$\neg\,\mbox{p\_1}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2(x)\mbox{ and }\mbox{p\_1}_1(x) ) \ \Rightarrow \ \mbox{p\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x)\mbox{ and }\mbox{q\_1}_1(x) ) \ \Rightarrow \ \mbox{p\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_3(x)\mbox{ and }\mbox{p\_1}_2(x) ) \ \Rightarrow \ \mbox{p\_1}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_3,\mbox{axiom})$

$( \mbox{q\_2}_3(x)\mbox{ and }\mbox{q\_1}_2(x) ) \ \Rightarrow \ \mbox{p\_1}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_4,\mbox{axiom})$

$( \mbox{p\_3}_3(x)\mbox{ and }\mbox{p\_2}_2(x) ) \ \Rightarrow \ \mbox{p\_2}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_5,\mbox{axiom})$

$( \mbox{q\_3}_3(x)\mbox{ and }\mbox{q\_2}_2(x) ) \ \Rightarrow \ \mbox{p\_2}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_6,\mbox{axiom})$

$( \mbox{p\_2}_2(x)\mbox{ and }\mbox{q\_1}_1(x) ) \ \Rightarrow \ \mbox{q\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x)\mbox{ and }\mbox{p\_1}_1(x) ) \ \Rightarrow \ \mbox{q\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_2,\mbox{axiom})$

$( \mbox{p\_2}_3(x)\mbox{ and }\mbox{q\_1}_2(x) ) \ \Rightarrow \ \mbox{q\_1}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_3,\mbox{axiom})$

$( \mbox{q\_2}_3(x)\mbox{ and }\mbox{p\_1}_2(x) ) \ \Rightarrow \ \mbox{q\_1}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_4,\mbox{axiom})$

$( \mbox{p\_3}_3(x)\mbox{ and }\mbox{q\_2}_2(x) ) \ \Rightarrow \ \mbox{q\_2}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_5,\mbox{axiom})$

$( \mbox{q\_3}_3(x)\mbox{ and }\mbox{p\_2}_2(x) ) \ \Rightarrow \ \mbox{q\_2}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_6,\mbox{axiom})$

$\mbox{p\_1}_2(x) \ \Rightarrow \ \mbox{p\_1}_1(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2(x) \ \Rightarrow \ \mbox{q\_1}_1(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2(x) \ \Rightarrow \ \mbox{p\_2}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_1,\mbox{axiom})$

$\mbox{p\_2}_3(x) \ \Rightarrow \ \mbox{p\_3}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_2,\mbox{axiom})$

$\mbox{q\_1}_2(x) \ \Rightarrow \ \mbox{q\_2}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_3,\mbox{axiom})$

$\mbox{q\_2}_3(x) \ \Rightarrow \ \mbox{q\_3}_3(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_4,\mbox{axiom})$

$\mbox{p\_1}_1(a)\qquad\mbox{cnf}(\mbox{m\_t2}_1,\mbox{axiom})$

$\mbox{q\_1}_1(a)\qquad\mbox{cnf}(\mbox{m\_t2}_2,\mbox{axiom})$

$\mbox{p\_2}_2(a)\qquad\mbox{cnf}(\mbox{m\_t2}_3,\mbox{axiom})$

$\mbox{q\_2}_2(a)\qquad\mbox{cnf}(\mbox{m\_t2}_4,\mbox{axiom})$

$\mbox{p\_3}_3(a)\qquad\mbox{cnf}(\mbox{m\_t2}_5,\mbox{axiom})$

$\mbox{q\_3}_3(a)\qquad\mbox{cnf}(\mbox{m\_t2}_6,\mbox{axiom})$

$\mbox{sym\_p\_1}_3(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_goal}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_2(x) \ \Rightarrow \ ( \mbox{sym\_p\_2}_2(x)\mbox{ or }\mbox{sym\_p\_1}_1(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_2(x) \ \Rightarrow \ ( \mbox{sym\_q\_2}_2(x)\mbox{ or }\mbox{sym\_q\_1}_1(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_2,\mbox{axiom})$

$\mbox{sym\_p\_1}_3(x) \ \Rightarrow \ ( \mbox{sym\_p\_2}_3(x)\mbox{ or }\mbox{sym\_p\_1}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_3,\mbox{axiom})$

$\mbox{sym\_p\_1}_3(x) \ \Rightarrow \ ( \mbox{sym\_q\_2}_3(x)\mbox{ or }\mbox{sym\_q\_1}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_4,\mbox{axiom})$

$\mbox{sym\_p\_2}_3(x) \ \Rightarrow \ ( \mbox{sym\_p\_3}_3(x)\mbox{ or }\mbox{sym\_p\_2}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_5,\mbox{axiom})$

$\mbox{sym\_p\_2}_3(x) \ \Rightarrow \ ( \mbox{sym\_q\_3}_3(x)\mbox{ or }\mbox{sym\_q\_2}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type11}_6,\mbox{axiom})$

$\mbox{sym\_q\_1}_2(x) \ \Rightarrow \ ( \mbox{sym\_p\_2}_2(x)\mbox{ or }\mbox{sym\_q\_1}_1(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_1,\mbox{axiom})$

$\mbox{sym\_q\_1}_2(x) \ \Rightarrow \ ( \mbox{sym\_q\_2}_2(x)\mbox{ or }\mbox{sym\_p\_1}_1(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_2,\mbox{axiom})$

$\mbox{sym\_q\_1}_3(x) \ \Rightarrow \ ( \mbox{sym\_p\_2}_3(x)\mbox{ or }\mbox{sym\_q\_1}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_3,\mbox{axiom})$

$\mbox{sym\_q\_1}_3(x) \ \Rightarrow \ ( \mbox{sym\_q\_2}_3(x)\mbox{ or }\mbox{sym\_p\_1}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_4,\mbox{axiom})$

$\mbox{sym\_q\_2}_3(x) \ \Rightarrow \ ( \mbox{sym\_p\_3}_3(x)\mbox{ or }\mbox{sym\_q\_2}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_5,\mbox{axiom})$

$\mbox{sym\_q\_2}_3(x) \ \Rightarrow \ ( \mbox{sym\_q\_3}_3(x)\mbox{ or }\mbox{sym\_p\_2}_2(x) )\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type12}_6,\mbox{axiom})$

$\mbox{sym\_p\_1}_1(x) \ \Rightarrow \ \mbox{sym\_p\_1}_2(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type21}_1,\mbox{axiom})$

$\mbox{sym\_q\_1}_1(x) \ \Rightarrow \ \mbox{sym\_q\_1}_2(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type21}_2,\mbox{axiom})$

$\mbox{sym\_p\_2}_2(x) \ \Rightarrow \ \mbox{sym\_p\_1}_2(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type22}_1,\mbox{axiom})$

$\mbox{sym\_p\_3}_3(x) \ \Rightarrow \ \mbox{sym\_p\_2}_3(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type22}_2,\mbox{axiom})$

$\mbox{sym\_q\_2}_2(x) \ \Rightarrow \ \mbox{sym\_q\_1}_2(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type22}_3,\mbox{axiom})$

$\mbox{sym\_q\_3}_3(x) \ \Rightarrow \ \mbox{sym\_q\_2}_3(x)\qquad\mbox{cnf}(\mbox{sym\_m\_s2\_type22}_4,\mbox{axiom})$

$\neg\,\mbox{sym\_p\_1}_1(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_1,\mbox{axiom})$

$\neg\,\mbox{sym\_q\_1}_1(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_2,\mbox{axiom})$

$\neg\,\mbox{sym\_p\_2}_2(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_3,\mbox{axiom})$

$\neg\,\mbox{sym\_q\_2}_2(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_4,\mbox{axiom})$

$\neg\,\mbox{sym\_p\_3}_3(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_5,\mbox{axiom})$

$\neg\,\mbox{sym\_q\_3}_3(a)\qquad\mbox{cnf}(\mbox{sym\_m\_t2}_6,\mbox{axiom})$

\medskip\textbf{SYN087-1.003.p} Plaisted problem s(3,3)






$p_0 \ \Rightarrow \ \neg\,q_0\qquad\mbox{cnf}(\mbox{s3\_goal}_1,\mbox{negated\_conjecture})$

$( p_1\mbox{ and }p_2 ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{s3\_type11}_1,\mbox{axiom})$

$( q_1\mbox{ and }q_2 ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{s3\_type11}_2,\mbox{axiom})$

$( p_2\mbox{ and }p_3 ) \ \Rightarrow \ p_1\qquad\mbox{cnf}(\mbox{s3\_type11}_3,\mbox{axiom})$

$( q_2\mbox{ and }q_3 ) \ \Rightarrow \ p_1\qquad\mbox{cnf}(\mbox{s3\_type11}_4,\mbox{axiom})$

$( p_3\mbox{ and }p_4 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{s3\_type11}_5,\mbox{axiom})$

$( q_3\mbox{ and }q_4 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{s3\_type11}_6,\mbox{axiom})$

$( p_4\mbox{ and }p_5 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{s3\_type11}_7,\mbox{axiom})$

$( q_4\mbox{ and }q_5 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{s3\_type11}_8,\mbox{axiom})$

$( p_1\mbox{ and }q_2 ) \ \Rightarrow \ q_0\qquad\mbox{cnf}(\mbox{s3\_type12}_1,\mbox{axiom})$

$( q_1\mbox{ and }p_2 ) \ \Rightarrow \ q_0\qquad\mbox{cnf}(\mbox{s3\_type12}_2,\mbox{axiom})$

$( p_2\mbox{ and }q_3 ) \ \Rightarrow \ q_1\qquad\mbox{cnf}(\mbox{s3\_type12}_3,\mbox{axiom})$

$( q_2\mbox{ and }p_3 ) \ \Rightarrow \ q_1\qquad\mbox{cnf}(\mbox{s3\_type12}_4,\mbox{axiom})$

$( p_3\mbox{ and }q_4 ) \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{s3\_type12}_5,\mbox{axiom})$

$( q_3\mbox{ and }p_4 ) \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{s3\_type12}_6,\mbox{axiom})$

$( p_4\mbox{ and }q_5 ) \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{s3\_type12}_7,\mbox{axiom})$

$( q_4\mbox{ and }p_5 ) \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{s3\_type12}_8,\mbox{axiom})$

$p_2 \ \Rightarrow \ p_5\qquad\mbox{cnf}(\mbox{s3\_type2}_1,\mbox{axiom})$

$p_3 \ \Rightarrow \ p_6\qquad\mbox{cnf}(\mbox{s3\_type2}_2,\mbox{axiom})$

$q_2 \ \Rightarrow \ q_5\qquad\mbox{cnf}(\mbox{s3\_type2}_3,\mbox{axiom})$

$q_3 \ \Rightarrow \ q_6\qquad\mbox{cnf}(\mbox{s3\_type2}_4,\mbox{axiom})$

\medskip\textbf{SYN170-1.p} RPT63 synthetic problem 68 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(x,c,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN373+1.p} Peter Andrews Problem X2124






$\exists x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_q}(x) )\iff ( \forall x{:}\;\mbox{big\_p}(x) \ \Rightarrow \ \exists x{:}\;\mbox{big\_q}(x) )\qquad\mbox{fof}(x_{2124},\mbox{conjecture})$

\medskip\textbf{SYN942+1.p} Syntactic from Shults






$( \forall x{:}\;( a(x) \ \Rightarrow \ ( b(x)\mbox{ or }c(x) ) )\mbox{ and }\neg\,\forall x{:}\;( a(x) \ \Rightarrow \ b(x) ) ) \ \Rightarrow \ \exists x{:}\;( a(x)\mbox{ and }c(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN171-1.p} RPT63 synthetic problem 69 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(d,a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN121-1.p} RPT63 synthetic problem 19 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(a,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN935+1.p} From Smullyan






$\exists x{:}\;( p(x) \ \Rightarrow \ c )\iff ( \forall x{:}\;p(x) \ \Rightarrow \ c )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN326+1.p} Church problem 46.12 (1)






$\exists x{:}\;\forall y,z{:}\;( ( ( \mbox{big\_f}(y,z) \ \Rightarrow \ ( \mbox{big\_g}(y) \ \Rightarrow \ \mbox{big\_h}(x) ) ) \ \Rightarrow \ \mbox{big\_f}(x,x) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(z,x) \ \Rightarrow \ \mbox{big\_g}(x) ) \ \Rightarrow \ \mbox{big\_h}(z) ) \ \Rightarrow \ ( \mbox{big\_f}(x,y) \ \Rightarrow \ \mbox{big\_f}(z,z) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_12}_1,\mbox{conjecture})$

\medskip\textbf{SYN722+1.p} Peter Andrews Problem THM119






$\neg\,\forall z{:}\;( ( p(z)\mbox{ or }r(z) )\mbox{ and }q(z) )\mbox{ and }\forall x{:}\;\exists y{:}\;( p(x)\mbox{ or }\neg\,q(x)\mbox{ or }\neg\,q(y)\mbox{ or }\neg\,q(c)\mbox{ or }\neg\,q(d) )\mbox{ and }( \neg\,p(a)\mbox{ or }\neg\,p(b) )\qquad\mbox{fof}(\mbox{thm}_{119},\mbox{conjecture})$

\medskip\textbf{SYN931+1.p} From Smullyan






$\exists x{:}\;( p(x)\mbox{ and }c )\iff ( \exists x{:}\;p(x)\mbox{ and }c )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN916+1.p} FALSE






$\mbox{\$false}\qquad\mbox{fof}(\mbox{falsity},\mbox{conjecture})$

\medskip\textbf{SYN374$\wedge$5.p} TPS problem X2125






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} )\iff ( \mbox{cP}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cX}_{2125},\mbox{conjecture})$

\medskip\textbf{SYN129-1.p} RPT63 synthetic problem 27 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_5(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN600-1.p} Harrison problem 3759






$p_{19}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p19}_1,\mbox{negated\_conjecture})$

$p_{19}(c_{22})\qquad\mbox{cnf}(\mbox{c22\_is\_p19}_2,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_7(x_{32},x_{32})\qquad\mbox{cnf}(\mbox{p7}_4,\mbox{negated\_conjecture})$

$p_5(x_{29},x_{29})\qquad\mbox{cnf}(\mbox{p5}_5,\mbox{negated\_conjecture})$

$p_3(x_{16},x_{16})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_7(f_{16}(c_{21}),f_{16}(c_{22}))\qquad\mbox{cnf}(\mbox{p7}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(f_4(f_6(c_{21})),f_4(f_6(c_{22})))\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$( p_{19}(x_{10})\mbox{ and }p_5(x_{10},x_9) ) \ \Rightarrow \ p_{19}(x_9)\qquad\mbox{cnf}(\mbox{p19}_9,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{44}) \ \Rightarrow \ p_7(f_{16}(x_{43}),f_{16}(x_{44}))\qquad\mbox{cnf}(\mbox{p7}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{19},x_{20}) \ \Rightarrow \ p_3(f_{11}(x_{19}),f_{11}(x_{20}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_3(x_{21},x_{22}) \ \Rightarrow \ p_3(f_{12}(x_{21}),f_{12}(x_{22}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{23},x_{24}) \ \Rightarrow \ p_3(f_{14}(x_{23}),f_{14}(x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{25},x_{26}) \ \Rightarrow \ p_3(f_4(x_{25}),f_4(x_{26}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_7(f_{13}(x_{39}),f_{13}(x_{40}))\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$p_7(x_{41},x_{42}) \ \Rightarrow \ p_7(f_{15}(x_{41}),f_{15}(x_{42}))\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{32},x_{33})\mbox{ and }p_7(x_{32},x_{34}) ) \ \Rightarrow \ p_7(x_{33},x_{34})\qquad\mbox{cnf}(\mbox{p7}_{19},\mbox{negated\_conjecture})$

$( p_5(x_{29},x_{30})\mbox{ and }p_5(x_{29},x_{31}) ) \ \Rightarrow \ p_5(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p5}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{16},x_{17})\mbox{ and }p_3(x_{16},x_{18}) ) \ \Rightarrow \ p_3(x_{17},x_{18})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$( p_7(x_2,x_0)\mbox{ and }p_7(x_3,x_1)\mbox{ and }p_{17}(x_2,x_3) ) \ \Rightarrow \ p_{17}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p17}_{22},\mbox{negated\_conjecture})$

$( p_7(x_7,x_5)\mbox{ and }p_7(x_8,x_6)\mbox{ and }p_{18}(x_7,x_8) ) \ \Rightarrow \ p_{18}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p18}_{23},\mbox{negated\_conjecture})$

$( p_7(x_{50},x_{52})\mbox{ and }p_7(x_{49},x_{51}) ) \ \Rightarrow \ p_7(f_9(x_{49},x_{50}),f_9(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p7}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{36},x_{38})\mbox{ and }p_7(x_{35},x_{37}) ) \ \Rightarrow \ p_7(f_{10}(x_{35},x_{36}),f_{10}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$( p_7(x_{45},x_{47})\mbox{ and }p_7(x_{46},x_{48}) ) \ \Rightarrow \ p_7(f_8(x_{45},x_{46}),f_8(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p7}_{26},\mbox{negated\_conjecture})$

$p_{19}(x_4) \ \Rightarrow \ p_{18}(f_9(f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_4(f_6(x_4))),f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_{11}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(c_{20})))))))))),f_{15}(f_{16}(x_4)))\qquad\mbox{cnf}(\mbox{p18}_{27},\mbox{negated\_conjecture})$

$p_{19}(x_4) \ \Rightarrow \ p_{17}(f_{15}(f_{16}(x_4)),f_8(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_9(f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_4(f_6(x_4))),f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_{11}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(c_{20}))))))))))))\qquad\mbox{cnf}(\mbox{p17}_{28},\mbox{negated\_conjecture})$

$( p_{19}(x_{27})\mbox{ and }p_{18}(f_9(f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),x_{28}),f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_{11}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(c_{20})))))))))),f_{15}(f_{16}(x_{27})))\mbox{ and }p_{17}(f_{15}(f_{16}(x_{27})),f_8(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_9(f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),x_{28}),f_{10}(f_{13}(f_{11}(f_{14}(f_{12}(c_{20})))),f_{11}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(f_{12}(c_{20})))))))))))) ) \ \Rightarrow \ p_3(f_4(f_6(x_{27})),x_{28})\qquad\mbox{cnf}(\mbox{p3}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{SYN989$\wedge$1.p} All things are true or false






$\forall x{:}\;\mbox{\$o}{:}\;( x\mbox{ or }\neg\,x )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN548+1.p} dia box (box (p or box q)$\leq$$>$box p or box q)






$\forall x{:}\;\mbox{reachable}(x,x)\qquad\mbox{fof}(\mbox{reflexivity\_of\_reachable},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{reachable}(x,y)\mbox{ and }\mbox{reachable}(y,z) ) \ \Rightarrow \ \mbox{reachable}(x,z) )\qquad\mbox{fof}(\mbox{transitivity\_of\_reachable},\mbox{axiom})$

$\exists x{:}\;( \mbox{reachable}(\mbox{initial\_world},x)\mbox{ and }\forall y{:}\;( \mbox{reachable}(x,y) \ \Rightarrow \ ( \forall z{:}\;( \mbox{reachable}(y,z) \ \Rightarrow \ ( p(z)\mbox{ or }\forall v{:}\;( \mbox{reachable}(z,v) \ \Rightarrow \ q(v) ) ) )\iff ( \forall w{:}\;( \mbox{reachable}(y,w) \ \Rightarrow \ p(w) )\mbox{ or }\forall w{:}\;( \mbox{reachable}(y,w) \ \Rightarrow \ q(w) ) ) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN122-1.p} RPT63 synthetic problem 20 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(b,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN148-1.p} RPT63 synthetic problem 46 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_1(c,d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN359+1.p} Peter Andrews Problem X2110






$( \exists x{:}\;\mbox{big\_r}(x)\mbox{ and }\forall y{:}\;( \mbox{big\_r}(y) \ \Rightarrow \ \exists z{:}\;\mbox{big\_q}(y,z) )\mbox{ and }\forall x,y{:}\;( \mbox{big\_q}(x,y) \ \Rightarrow \ \mbox{big\_q}(x,x) ) ) \ \Rightarrow \ \exists x,y{:}\;( \mbox{big\_q}(x,y)\mbox{ and }\mbox{big\_r}(y) )\qquad\mbox{fof}(x_{2110},\mbox{conjecture})$

\medskip\textbf{SYN110-1.p} RPT63 synthetic problem 8 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(c,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN330+1.p} Church problem 46.14 (2)






$\exists x,y{:}\;\forall z{:}\;( ( \mbox{big\_f}(x,z)\iff \mbox{big\_f}(z,y) ) \ \Rightarrow \ ( ( \mbox{big\_f}(z,y)\iff \mbox{big\_f}(z,z) ) \ \Rightarrow \ ( ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(y,x) ) \ \Rightarrow \ ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(x,z) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_2,\mbox{conjecture})$

\medskip\textbf{SYN597-1.p} Harrison problem 3670






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_9(x_{50},x_{50})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{35})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{28},x_{28})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{12}(x_9,x_9)\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{negated\_conjecture})$

$p_{16}(c_{21},c_{22})\qquad\mbox{cnf}(\mbox{p16}_6,\mbox{negated\_conjecture})$

$p_3(c_{21},c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_{17}(c_{21},f_8(c_{20}))\qquad\mbox{cnf}(\mbox{p17}_8,\mbox{negated\_conjecture})$

$p_{16}(f_5(f_6(f_7(c_{19}))),f_4(c_{20}))\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_3(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_2(x_{38},x_{39}) \ \Rightarrow \ p_3(f_{15}(x_{38}),f_{15}(x_{39}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_2(x_{40},x_{41}) \ \Rightarrow \ p_3(f_4(x_{40}),f_4(x_{41}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_3(x_{42},x_{43}) \ \Rightarrow \ p_3(f_5(x_{42}),f_5(x_{43}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{44},x_{45}) \ \Rightarrow \ p_3(f_6(x_{44}),f_6(x_{45}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{46},x_{47}) \ \Rightarrow \ p_3(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{16},\mbox{negated\_conjecture})$

$( p_9(x_{50},x_{51})\mbox{ and }p_9(x_{50},x_{52}) ) \ \Rightarrow \ p_9(x_{51},x_{52})\qquad\mbox{cnf}(\mbox{p9}_{17},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{36})\mbox{ and }p_3(x_{35},x_{37}) ) \ \Rightarrow \ p_3(x_{36},x_{37})\qquad\mbox{cnf}(\mbox{p3}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{28},x_{29})\mbox{ and }p_2(x_{28},x_{30}) ) \ \Rightarrow \ p_2(x_{29},x_{30})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_{12}(x_9,x_{10})\mbox{ and }p_{12}(x_9,x_{11}) ) \ \Rightarrow \ p_{12}(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p12}_{20},\mbox{negated\_conjecture})$

$p_{18}(c_{20},f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{21})),c_{23},c_{24}))\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$\neg\,p_{18}(c_{20},f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{22})),c_{23},c_{24}))\qquad\mbox{cnf}(\mbox{not\_p18}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{16})\mbox{ and }p_3(x_{19},x_{17})\mbox{ and }p_{16}(x_{18},x_{19}) ) \ \Rightarrow \ p_{16}(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p16}_{23},\mbox{negated\_conjecture})$

$( p_{18}(x_{27},x_{26})\mbox{ and }p_2(x_{27},x_{24})\mbox{ and }p_{10}(x_{26},x_{25}) ) \ \Rightarrow \ p_{18}(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p18}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{22},x_{20})\mbox{ and }p_3(x_{23},x_{21})\mbox{ and }p_{17}(x_{22},x_{23}) ) \ \Rightarrow \ p_{17}(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p17}_{25},\mbox{negated\_conjecture})$

$( p_{12}(x_{32},x_{34})\mbox{ and }p_3(x_{31},x_{33}) ) \ \Rightarrow \ p_2(f_{13}(x_{31},x_{32}),f_{13}(x_{33},x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{14})\mbox{ and }p_3(x_{13},x_{15}) ) \ \Rightarrow \ p_{12}(f_{14}(x_{12},x_{13}),f_{14}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

$( p_2(x_3,x_6)\mbox{ and }p_9(x_4,x_7)\mbox{ and }p_{10}(x_5,x_8) ) \ \Rightarrow \ p_{10}(f_{11}(x_3,x_4,x_5),f_{11}(x_6,x_7,x_8))\qquad\mbox{cnf}(\mbox{p10}_{28},\mbox{negated\_conjecture})$

\medskip\textbf{SYN363+1.p} Peter Andrews Problem X2114






$( \forall x{:}\;\mbox{big\_r}(x,b)\mbox{ and }\forall y{:}\;( \exists z{:}\;\mbox{big\_r}(y,z) \ \Rightarrow \ \mbox{big\_r}(a,y) ) ) \ \Rightarrow \ \exists u{:}\;\forall v{:}\;\mbox{big\_r}(u,v)\qquad\mbox{fof}(x_{2114},\mbox{conjecture})$

\medskip\textbf{SYN218-1.p} RPT63 synthetic problem 3 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN329+1.p} Church problem 46.14 (1)






$\exists x,y{:}\;\forall z{:}\;( \mbox{big\_f}(x,x) \ \Rightarrow \ ( \mbox{big\_f}(y,y) \ \Rightarrow \ ( \mbox{big\_f}(x,z) \ \Rightarrow \ \mbox{big\_f}(z,y) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_1,\mbox{conjecture})$

\medskip\textbf{SYN282-1.p} RPT63 synthetic problem 67 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,d,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN195-1.p} RPT63 synthetic problem 93 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_5(d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN310-1.p} Problem for testing satisfiability






$p(x_2,x_1,x) \ \Rightarrow \ p(x,x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(x_1,x,x_2) \ \Rightarrow \ p(x,x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(x,x_1,g(x_2)) \ \Rightarrow \ p(x,x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(f(x),x_1,x_2) \ \Rightarrow \ p(x,x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,p(a,b,c)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$p(f(g(a)),f(g(b)),f(g(c)))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN107-1.p} RPT63 synthetic problem 5 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN950+1.p} Syntactic from Shults






$\forall a,b{:}\;( \forall y{:}\;( q(y) \ \Rightarrow \ p(y) ) \ \Rightarrow \ \exists x{:}\;( ( p(x) \ \Rightarrow \ p(a) )\mbox{ and }( q(x) \ \Rightarrow \ p(b) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN995$\wedge$1.p} Every function has a fixed point






$\mbox{leibeq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x )\iff ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}@ ( f@ x )@ x )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN973+1.p} Syntactic from Shults






$p(z) \ \Rightarrow \ p(z)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN641-1.p} Harrison problem 4296






$p_{15}(f_3(c_{19}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_1,\mbox{negated\_conjecture})$

$p_{15}(f_3(c_{18}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_2,\mbox{negated\_conjecture})$

$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_3,\mbox{negated\_conjecture})$

$p_6(x_{45},x_{45})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_4(x_{36},x_{36})\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{16}(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p16}_7,\mbox{negated\_conjecture})$

$p_{12}(x_3,x_3)\qquad\mbox{cnf}(\mbox{p12}_8,\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{19})),f_8(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_9,\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{18})),f_5(c_{18}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{19})),f_5(c_{19}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{18})),f_7(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{12},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{19})),f_7(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_{13},\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{18})),f_8(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{14},\mbox{negated\_conjecture})$

$( p_{15}(x_{17})\mbox{ and }p_2(x_{17},x_{16}) ) \ \Rightarrow \ p_{15}(x_{16})\qquad\mbox{cnf}(\mbox{p15}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_6,x_7) \ \Rightarrow \ p_{12}(f_{13}(x_6),f_{13}(x_7))\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_6(f_8(x_{50}),f_8(x_{51}))\qquad\mbox{cnf}(\mbox{p6}_{17},\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_6(f_7(x_{48}),f_7(x_{49}))\qquad\mbox{cnf}(\mbox{p6}_{18},\mbox{negated\_conjecture})$

$p_2(x_{43},x_{44}) \ \Rightarrow \ p_4(f_5(x_{43}),f_5(x_{44}))\qquad\mbox{cnf}(\mbox{p4}_{19},\mbox{negated\_conjecture})$

$p_2(x_{34},x_{35}) \ \Rightarrow \ p_2(f_3(x_{34}),f_3(x_{35}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_{15}(f_{10}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{22},\mbox{negated\_conjecture})$

$( p_6(x_{45},x_{46})\mbox{ and }p_6(x_{45},x_{47}) ) \ \Rightarrow \ p_6(x_{46},x_{47})\qquad\mbox{cnf}(\mbox{p6}_{23},\mbox{negated\_conjecture})$

$( p_4(x_{36},x_{37})\mbox{ and }p_4(x_{36},x_{38}) ) \ \Rightarrow \ p_4(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p4}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_2(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_{16}(x_{18},x_{19})\mbox{ and }p_{16}(x_{18},x_{20}) ) \ \Rightarrow \ p_{16}(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p16}_{26},\mbox{negated\_conjecture})$

$( p_{12}(x_3,x_4)\mbox{ and }p_{12}(x_3,x_5) ) \ \Rightarrow \ p_{12}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

$\neg\,p_6(f_7(f_{10}(f_3(c_{19}),f_3(c_{18}))),f_9(f_7(c_{19}),f_7(c_{18})))\qquad\mbox{cnf}(\mbox{not\_p6}_{28},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,f_3(c_{19})) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,c_{19}) )\qquad\mbox{cnf}(\mbox{p12}_{29},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},f_3(c_{18})) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},c_{18}) )\qquad\mbox{cnf}(\mbox{p12}_{30},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,c_{19}) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,f_3(c_{19})) )\qquad\mbox{cnf}(\mbox{p12}_{31},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},c_{18}) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},f_3(c_{18})) )\qquad\mbox{cnf}(\mbox{p12}_{32},\mbox{negated\_conjecture})$

$( p_6(x_{53},x_{55})\mbox{ and }p_6(x_{52},x_{54}) ) \ \Rightarrow \ p_6(f_9(x_{52},x_{53}),f_9(x_{54},x_{55}))\qquad\mbox{cnf}(\mbox{p6}_{33},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{32})\mbox{ and }p_2(x_{31},x_{33}) ) \ \Rightarrow \ p_2(f_{10}(x_{30},x_{31}),f_{10}(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p2}_{34},\mbox{negated\_conjecture})$

$( p_4(x_{39},x_{41})\mbox{ and }p_4(x_{40},x_{42}) ) \ \Rightarrow \ p_4(f_{14}(x_{39},x_{40}),f_{14}(x_{41},x_{42}))\qquad\mbox{cnf}(\mbox{p4}_{35},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_8(f_{10}(x_{14},x_{15})),f_9(f_8(x_{14}),f_8(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{36},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_4(f_5(f_{10}(x_{14},x_{15})),f_{14}(f_5(x_{14}),f_5(x_{15})))\qquad\mbox{cnf}(\mbox{p4}_{37},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_7(f_{10}(x_{14},x_{15})),f_9(f_7(x_{14}),f_7(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{38},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{23})\mbox{ and }p_{16}(x_{25},x_{22})\mbox{ and }p_{11}(x_{24},x_{21})\mbox{ and }p_{17}(x_{24},x_{25},x_{26}) ) \ \Rightarrow \ p_{17}(x_{21},x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p17}_{39},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{14})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{40},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{41},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{14})\mbox{ and }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{15}) )\qquad\mbox{cnf}(\mbox{p12}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{SYN731+1.p} Peter Andrews Problem X2150






$\exists w{:}\;( \forall x{:}\;\exists y{:}\;p(w,x,y) \ \Rightarrow \ \exists z{:}\;p(z,z,w) )\qquad\mbox{fof}(x_{2150},\mbox{conjecture})$

\medskip\textbf{SYN045-1.p} Pelletier Problem 13






$p\mbox{ or }q\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p\mbox{ or }r\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ \neg\,r\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN060-1.p} Pelletier Problem 30






$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,\mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_g}(x)\mbox{ or }\mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_g}(x) \ \Rightarrow \ \neg\,\mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_g}(x)\mbox{ or }\mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_i}(x)\mbox{ or }\mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_i}(x)\mbox{ or }\mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\neg\,\mbox{big\_i}(a)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN561-1.p} Harrison problem 2539






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_9(c_{14},c_{10})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_8(c_{11},c_{14})\qquad\mbox{cnf}(\mbox{p8}_3,\mbox{negated\_conjecture})$

$p_2(f_7(c_{10},c_{14}),c_{15})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(f_6(c_{15},c_{11}),c_{16})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(f_4(f_5(c_{12})),c_{13}),c_{16})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_2(x_9,x_{10}) \ \Rightarrow \ p_2(f_5(x_9),f_5(x_{10}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_4(x_7),f_4(x_8))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{24})\mbox{ and }p_9(x_{25},x_{26})\mbox{ and }p_2(x_{25},x_{23}) ) \ \Rightarrow \ p_9(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p9}_{10},\mbox{negated\_conjecture})$

$( p_2(x_{22},x_{20})\mbox{ and }p_8(x_{21},x_{22})\mbox{ and }p_2(x_{21},x_{19}) ) \ \Rightarrow \ p_8(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p8}_{11},\mbox{negated\_conjecture})$

$( p_2(x_3,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(f_3(x_3,x_4),f_3(x_5,x_6))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{17})\mbox{ and }p_2(x_{16},x_{18}) ) \ \Rightarrow \ p_2(f_7(x_{15},x_{16}),f_7(x_{17},x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{14})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(f_6(x_{11},x_{12}),f_6(x_{13},x_{14}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$( p_9(x_{28},c_{10})\mbox{ and }p_8(c_{11},x_{28})\mbox{ and }p_2(f_7(c_{10},x_{28}),x_{27}) ) \ \Rightarrow \ \neg\,p_2(f_6(x_{27},c_{11}),f_3(f_4(f_5(c_{12})),c_{13}))\qquad\mbox{cnf}(\mbox{not\_p9}_{15},\mbox{negated\_conjecture})$

\medskip\textbf{SYN142-1.p} RPT63 synthetic problem 40 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_5(d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN151-1.p} RPT63 synthetic problem 49 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_2(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN377+1.p} Peter Andrews Problem X2128






$\forall x{:}\;( \mbox{big\_p}(x)\iff \forall y{:}\;\mbox{big\_p}(y) )\iff ( \exists x{:}\;\mbox{big\_p}(x)\iff \forall y{:}\;\mbox{big\_p}(y) )\qquad\mbox{fof}(x_{2128},\mbox{conjecture})$

\medskip\textbf{SYN069-1.p} Pelletier Problem 45






$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ ( \mbox{big\_g}(f(x))\mbox{ or }\mbox{big\_k}(y) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ ( \mbox{big\_h}(x,f(x))\mbox{ or }\mbox{big\_k}(y) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_j}(x,f(x))\mbox{ and }\mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ \mbox{big\_k}(y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_l}(x) \ \Rightarrow \ \neg\,\mbox{big\_k}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_h}(a,x) \ \Rightarrow \ \mbox{big\_l}(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_h}(a,x) ) \ \Rightarrow \ \mbox{big\_j}(a,x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(g(x))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_h}(x,g(x))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN163-1.p} RPT63 synthetic problem 61 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(e,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN569-1.p} Harrison problem 3088






$p_2(x_6,x_6)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_6(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$p_8(c_{14},c_{12})\qquad\mbox{cnf}(\mbox{p8}_3,\mbox{negated\_conjecture})$

$p_8(c_{11},c_{12})\qquad\mbox{cnf}(\mbox{p8}_4,\mbox{negated\_conjecture})$

$p_8(c_{11},c_{14})\qquad\mbox{cnf}(\mbox{p8}_5,\mbox{negated\_conjecture})$

$p_9(c_{13},c_{15},c_{14})\qquad\mbox{cnf}(\mbox{p9}_6,\mbox{negated\_conjecture})$

$p_8(x_4,x_5) \ \Rightarrow \ p_{10}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p10}_7,\mbox{negated\_conjecture})$

$p_2(x_{13},x_{14}) \ \Rightarrow \ p_2(f_3(x_{13}),f_3(x_{14}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_6(x_{27},x_{28})\mbox{ and }p_6(x_{27},x_{29}) ) \ \Rightarrow \ p_6(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p6}_9,\mbox{negated\_conjecture})$

$( p_2(x_6,x_7)\mbox{ and }p_2(x_6,x_8) ) \ \Rightarrow \ p_2(x_7,x_8)\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$( p_9(x_{11},x_9,x_{12})\mbox{ and }p_9(x_{11},x_{10},x_{12}) ) \ \Rightarrow \ p_2(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_{10}(x_{40},c_{12})\mbox{ and }p_{10}(c_{11},x_{40}) ) \ \Rightarrow \ p_9(c_{13},f_3(x_{40}),x_{40})\qquad\mbox{cnf}(\mbox{p9}_{12},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{31})\mbox{ and }p_8(x_{32},x_{33})\mbox{ and }p_2(x_{32},x_{30}) ) \ \Rightarrow \ p_8(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p8}_{14},\mbox{negated\_conjecture})$

$p_2(f_5(f_7(c_{13},c_{12}),f_7(c_{13},c_{11})),f_4(f_5(c_{12},c_{11}),c_{15}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{17})\mbox{ and }p_2(x_{16},x_{18}) ) \ \Rightarrow \ p_2(f_4(x_{15},x_{16}),f_4(x_{17},x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{24},x_{26})\mbox{ and }p_6(x_{23},x_{25}) ) \ \Rightarrow \ p_2(f_7(x_{23},x_{24}),f_7(x_{25},x_{26}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{21})\mbox{ and }p_2(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_5(x_{19},x_{20}),f_5(x_{21},x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{37},x_{35})\mbox{ and }p_2(x_{38},x_{36})\mbox{ and }p_6(x_{39},x_{34})\mbox{ and }p_9(x_{39},x_{37},x_{38}) ) \ \Rightarrow \ p_9(x_{34},x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_8(x_{41},c_{12})\mbox{ and }p_8(c_{11},x_{41}) ) \ \Rightarrow \ \neg\,p_2(f_5(f_7(c_{13},c_{12}),f_7(c_{13},c_{11})),f_4(f_5(c_{12},c_{11}),f_3(x_{41})))\qquad\mbox{cnf}(\mbox{not\_p8}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{SYN079+1.p} Pelletier Problem 57






$\mbox{big\_f}(f(a,b),f(b,c))\qquad\mbox{fof}(\mbox{pel57}_1,\mbox{axiom})$

$\mbox{big\_f}(f(b,c),f(a,c))\qquad\mbox{fof}(\mbox{pel57}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(y,z) ) \ \Rightarrow \ \mbox{big\_f}(x,z) )\qquad\mbox{fof}(\mbox{pel57}_3,\mbox{axiom})$

$\mbox{big\_f}(f(a,b),f(a,c))\qquad\mbox{fof}(\mbox{pel}_{57},\mbox{conjecture})$

\medskip\textbf{SYN560-1.p} Harrison problem 2529






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$\neg\,p_2(c_9,f_3(c_{10}))\qquad\mbox{cnf}(\mbox{not\_p2}_2,\mbox{negated\_conjecture})$

$p_2(c_{11},f_4(f_5(c_{12},c_{13})))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_2(f_4(x_9),f_5(x_9,f_3(f_7(c_{10}))))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_2(f_7(x_{18}),f_7(x_{19}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(x_5,x_6) \ \Rightarrow \ p_2(f_3(x_5),f_3(x_6))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_4(x_7),f_4(x_8))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_8(x_{24},x_{26})\mbox{ and }p_8(x_{26},x_{25}) ) \ \Rightarrow \ p_8(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p8}_9,\mbox{negated\_conjecture})$

$p_8(f_6(c_9,f_4(c_{12})),f_6(c_9,f_4(f_5(c_{12},c_{14}))))\qquad\mbox{cnf}(\mbox{p8}_{10},\mbox{negated\_conjecture})$

$\neg\,p_8(f_6(c_9,f_4(c_{12})),f_6(c_9,f_4(f_4(f_5(c_{12},c_{14})))))\qquad\mbox{cnf}(\mbox{not\_p8}_{11},\mbox{negated\_conjecture})$

$( p_2(x_{23},x_{21})\mbox{ and }p_8(x_{22},x_{23})\mbox{ and }p_2(x_{22},x_{20}) ) \ \Rightarrow \ p_8(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p8}_{12},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(f_5(x_{10},x_{11}),f_5(x_{12},x_{13}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{17})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_2(f_6(x_{14},x_{15}),f_6(x_{16},x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_2(x_3,f_3(c_{10}))\mbox{ or }p_8(f_6(x_3,x_4),f_6(x_3,f_5(x_4,f_3(f_7(c_{10})))))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

\medskip\textbf{SYN956+1.p} Syntactic from Shults






$\exists x{:}\;( a(x) \ \Rightarrow \ b(x) ) \ \Rightarrow \ ( \forall x{:}\;a(x) \ \Rightarrow \ \exists x{:}\;b(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN354+1.p} Church problem 46.20 (1)






$\forall x_1,x_2{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( \mbox{big\_f}(x_1,x_2) \ \Rightarrow \ ( \mbox{big\_g}(x_1,x_2) \ \Rightarrow \ ( ( ( \mbox{big\_g}(x_2,z)\iff \mbox{big\_g}(y_2,z) ) \ \Rightarrow \ ( \mbox{big\_f}(y_1,y_2) \ \Rightarrow \ \mbox{big\_f}(x_2,y_2) ) ) \ \Rightarrow \ ( ( \mbox{big\_g}(x_2,z)\iff \mbox{big\_g}(y_1,z) ) \ \Rightarrow \ ( \mbox{big\_f}(x_1,y_1)\mbox{ and }\mbox{big\_f}(x_2,y_1)\mbox{ and }\mbox{big\_f}(y_1,y_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_20}_1,\mbox{conjecture})$

\medskip\textbf{SYN414+1.p} Kalish and Montague Problem 265






$\forall x{:}\;( \exists y{:}\;( h(x,y)\mbox{ and }f(y) ) \ \Rightarrow \ \exists z{:}\;( h(x,z)\mbox{ and }g(z) ) )\iff \forall u,v,w{:}\;( ( h(u,v)\mbox{ and }f(v) ) \ \Rightarrow \ ( h(u,w)\mbox{ and }g(w) ) )\qquad\mbox{fof}(\mbox{kalish}_{265},\mbox{conjecture})$

\medskip\textbf{SYN642-1.p} Harrison problem 4297






$p_{15}(f_3(c_{19}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_1,\mbox{negated\_conjecture})$

$p_{15}(f_3(c_{18}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_2,\mbox{negated\_conjecture})$

$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_3,\mbox{negated\_conjecture})$

$p_6(x_{45},x_{45})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_4(x_{36},x_{36})\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{16}(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p16}_7,\mbox{negated\_conjecture})$

$p_{12}(x_3,x_3)\qquad\mbox{cnf}(\mbox{p12}_8,\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{19})),f_8(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_9,\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{18})),f_5(c_{18}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{19})),f_5(c_{19}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{18})),f_7(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{12},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{19})),f_7(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_{13},\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{18})),f_8(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{14},\mbox{negated\_conjecture})$

$( p_{15}(x_{17})\mbox{ and }p_2(x_{17},x_{16}) ) \ \Rightarrow \ p_{15}(x_{16})\qquad\mbox{cnf}(\mbox{p15}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_6,x_7) \ \Rightarrow \ p_{12}(f_{13}(x_6),f_{13}(x_7))\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_6(f_8(x_{50}),f_8(x_{51}))\qquad\mbox{cnf}(\mbox{p6}_{17},\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_6(f_7(x_{48}),f_7(x_{49}))\qquad\mbox{cnf}(\mbox{p6}_{18},\mbox{negated\_conjecture})$

$p_2(x_{43},x_{44}) \ \Rightarrow \ p_4(f_5(x_{43}),f_5(x_{44}))\qquad\mbox{cnf}(\mbox{p4}_{19},\mbox{negated\_conjecture})$

$p_2(x_{34},x_{35}) \ \Rightarrow \ p_2(f_3(x_{34}),f_3(x_{35}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_{15}(f_{10}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{22},\mbox{negated\_conjecture})$

$( p_6(x_{45},x_{46})\mbox{ and }p_6(x_{45},x_{47}) ) \ \Rightarrow \ p_6(x_{46},x_{47})\qquad\mbox{cnf}(\mbox{p6}_{23},\mbox{negated\_conjecture})$

$( p_4(x_{36},x_{37})\mbox{ and }p_4(x_{36},x_{38}) ) \ \Rightarrow \ p_4(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p4}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_2(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_{16}(x_{18},x_{19})\mbox{ and }p_{16}(x_{18},x_{20}) ) \ \Rightarrow \ p_{16}(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p16}_{26},\mbox{negated\_conjecture})$

$( p_{12}(x_3,x_4)\mbox{ and }p_{12}(x_3,x_5) ) \ \Rightarrow \ p_{12}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

$\neg\,p_6(f_8(f_{10}(f_3(c_{19}),f_3(c_{18}))),f_9(f_8(c_{19}),f_8(c_{18})))\qquad\mbox{cnf}(\mbox{not\_p6}_{28},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,f_3(c_{19})) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,c_{19}) )\qquad\mbox{cnf}(\mbox{p12}_{29},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},f_3(c_{18})) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},c_{18}) )\qquad\mbox{cnf}(\mbox{p12}_{30},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,c_{19}) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,f_3(c_{19})) )\qquad\mbox{cnf}(\mbox{p12}_{31},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},c_{18}) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},f_3(c_{18})) )\qquad\mbox{cnf}(\mbox{p12}_{32},\mbox{negated\_conjecture})$

$( p_6(x_{53},x_{55})\mbox{ and }p_6(x_{52},x_{54}) ) \ \Rightarrow \ p_6(f_9(x_{52},x_{53}),f_9(x_{54},x_{55}))\qquad\mbox{cnf}(\mbox{p6}_{33},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{32})\mbox{ and }p_2(x_{31},x_{33}) ) \ \Rightarrow \ p_2(f_{10}(x_{30},x_{31}),f_{10}(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p2}_{34},\mbox{negated\_conjecture})$

$( p_4(x_{39},x_{41})\mbox{ and }p_4(x_{40},x_{42}) ) \ \Rightarrow \ p_4(f_{14}(x_{39},x_{40}),f_{14}(x_{41},x_{42}))\qquad\mbox{cnf}(\mbox{p4}_{35},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_8(f_{10}(x_{14},x_{15})),f_9(f_8(x_{14}),f_8(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{36},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_4(f_5(f_{10}(x_{14},x_{15})),f_{14}(f_5(x_{14}),f_5(x_{15})))\qquad\mbox{cnf}(\mbox{p4}_{37},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_7(f_{10}(x_{14},x_{15})),f_9(f_7(x_{14}),f_7(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{38},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{23})\mbox{ and }p_{16}(x_{25},x_{22})\mbox{ and }p_{11}(x_{24},x_{21})\mbox{ and }p_{17}(x_{24},x_{25},x_{26}) ) \ \Rightarrow \ p_{17}(x_{21},x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p17}_{39},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{14})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{40},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{41},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{14})\mbox{ and }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{15}) )\qquad\mbox{cnf}(\mbox{p12}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{SYN588-1.p} Harrison problem 3577






$p_2(x_8,x_8)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_7(x_{25},x_{25})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_2(c_{25},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{16},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{not\_p2}_4,\mbox{negated\_conjecture})$

$p_{14}(c_{18},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_5,\mbox{negated\_conjecture})$

$p_{14}(c_{23},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_6,\mbox{negated\_conjecture})$

$p_{14}(c_{19},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_{14}(c_{24},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_8,\mbox{negated\_conjecture})$

$p_7(x_{34},x_{35}) \ \Rightarrow \ p_7(f_{12}(x_{34}),f_{12}(x_{35}))\qquad\mbox{cnf}(\mbox{p7}_9,\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_2(f_3(x_{15}),f_3(x_{16}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_2(f_4(x_{17}),f_4(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{19},x_{20}) \ \Rightarrow \ p_2(f_5(x_{19}),f_5(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{32},x_{33}) \ \Rightarrow \ p_7(f_{11}(x_{32}),f_{11}(x_{33}))\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$( p_7(x_{25},x_{26})\mbox{ and }p_7(x_{25},x_{27}) ) \ \Rightarrow \ p_7(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p7}_{14},\mbox{negated\_conjecture})$

$( p_2(x_8,x_9)\mbox{ and }p_2(x_8,x_{10}) ) \ \Rightarrow \ p_2(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{15}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))),c_{19})\qquad\mbox{cnf}(\mbox{p15}_{16},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{14}(x_2,x_3) ) \ \Rightarrow \ p_{14}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p14}_{17},\mbox{negated\_conjecture})$

$( p_2(x_6,x_4)\mbox{ and }p_2(x_7,x_5)\mbox{ and }p_{15}(x_6,x_7) ) \ \Rightarrow \ p_{15}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{13})\mbox{ and }p_2(x_{12},x_{14}) ) \ \Rightarrow \ p_2(f_{13}(x_{11},x_{12}),f_{13}(x_{13},x_{14}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{40},x_{42})\mbox{ and }p_7(x_{41},x_{43}) ) \ \Rightarrow \ p_7(f_9(x_{40},x_{41}),f_9(x_{42},x_{43}))\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_7(x_{36},x_{38})\mbox{ and }p_7(x_{37},x_{39}) ) \ \Rightarrow \ p_7(f_8(x_{36},x_{37}),f_8(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{31})\mbox{ and }p_7(x_{28},x_{30}) ) \ \Rightarrow \ p_7(f_{10}(x_{28},x_{29}),f_{10}(x_{30},x_{31}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{23})\mbox{ and }p_2(x_{22},x_{24}) ) \ \Rightarrow \ p_2(f_6(x_{21},x_{22}),f_6(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{25}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{24},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$\neg\,p_7(f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))),f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{25}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p7}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{SYN335-1.p} Church problem 46.14 (7)






$f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(z(x,y),z(x,y))\mbox{ and }f(x,y) ) \ \Rightarrow \ g(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(z(x,y),z(x,y))\mbox{ or }f(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$g(z(x,y),z(x,y)) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( g(z(x,y),z(x,y))\mbox{ and }g(x,y) ) \ \Rightarrow \ f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$g(z(x,y),z(x,y))\mbox{ or }g(x,y)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$f(z(x,y),z(x,y)) \ \Rightarrow \ g(x,y)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( g(x,y)\mbox{ and }g(y,z(x,y)) ) \ \Rightarrow \ f(y,x)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$g(x,y)\mbox{ or }g(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$f(y,x) \ \Rightarrow \ g(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$f(z(x,y),y)\mbox{ or }f(y,x)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$f(z(x,y),y) \ \Rightarrow \ \neg\,f(y,x)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{SYN565-1.p} Harrison problem 2889






$p_2(x_9,x_9)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_4(x_{23},x_{23})\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$p_2(f_5(f_7(x_{18})),f_5(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_4(f_8(x_{38},x_{39}),f_6(x_{38},f_7(x_{39})))\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_4(f_7(f_8(x_{32},x_{33})),f_8(x_{33},x_{32}))\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_4(x_{16},x_{17}) \ \Rightarrow \ p_2(f_5(x_{16}),f_5(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_4(x_{30},x_{31}) \ \Rightarrow \ p_4(f_7(x_{30}),f_7(x_{31}))\qquad\mbox{cnf}(\mbox{p4}_7,\mbox{negated\_conjecture})$

$( p_2(x_9,x_{10})\mbox{ and }p_2(x_9,x_{11}) ) \ \Rightarrow \ p_2(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_4(x_{23},x_{24})\mbox{ and }p_4(x_{23},x_{25}) ) \ \Rightarrow \ p_4(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p4}_9,\mbox{negated\_conjecture})$

$p_{10}(f_5(x_7),f_3(f_5(f_6(x_7,x_8)),f_5(x_8)))\qquad\mbox{cnf}(\mbox{p10}_{10},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_9(f_5(c_{11}),f_5(c_{12})),f_5(f_8(c_{12},c_{11})))\qquad\mbox{cnf}(\mbox{not\_p10}_{11},\mbox{negated\_conjecture})$

$p_{10}(f_9(x_4,x_6),x_5) \ \Rightarrow \ p_{10}(x_4,f_3(x_5,x_6))\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$p_{10}(x_4,f_3(x_5,x_6)) \ \Rightarrow \ p_{10}(f_9(x_4,x_6),x_5)\qquad\mbox{cnf}(\mbox{p10}_{13},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_4(x_{34},x_{36})\mbox{ and }p_4(x_{35},x_{37}) ) \ \Rightarrow \ p_4(f_8(x_{34},x_{35}),f_8(x_{36},x_{37}))\qquad\mbox{cnf}(\mbox{p4}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(f_3(x_{12},x_{13}),f_3(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{21})\mbox{ and }p_2(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_9(x_{19},x_{20}),f_9(x_{21},x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_4(x_{26},x_{28})\mbox{ and }p_4(x_{27},x_{29}) ) \ \Rightarrow \ p_4(f_6(x_{26},x_{27}),f_6(x_{28},x_{29}))\qquad\mbox{cnf}(\mbox{p4}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{SYN314-1.002.001.p} Problem for testing satisfiability






$p(x_0,f_0(x_0),x_1,g_0(x_3))\mbox{ or }p(x_0,f_1(x_1),x_2,g_1(x_3))\mbox{ or }p(x_0,f_2(x_2),x_3,g_2(x_3))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(h_0,y_0,j_0(y_1),y_2) \ \Rightarrow \ \neg\,p(h_1(y_0),y_1,j_1(y_2),y_3)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN389+1.p} Pierce's Law

Pierce's Law. Unprovable by Logic Theorist, and tricky
 for 'natural' systems.




$( ( p \ \Rightarrow \ q ) \ \Rightarrow \ p ) \ \Rightarrow \ p\qquad\mbox{fof}(\mbox{pel}_8,\mbox{conjecture})$

\medskip\textbf{SYN268-1.p} RPT63 synthetic problem 53 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(a,x,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN001$\wedge$4.001.p} Pelletier 2






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) ) )@ ( \mbox{iatom}@ p ) )\qquad\mbox{thf}(\mbox{pel}_2,\mbox{conjecture})$

\medskip\textbf{SYN617-1.p} Harrison problem 3908






$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{negated\_conjecture})$

$p_9(x_{53},x_{53})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_7(x_{42},x_{42})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{13}(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p13}_5,\mbox{negated\_conjecture})$

$p_{19}(c_{21},c_{22})\qquad\mbox{cnf}(\mbox{p19}_6,\mbox{negated\_conjecture})$

$p_{11}(f_{12}(f_{14}(x_5,x_6)),x_6)\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_7(f_8(f_{10}(x_{51},x_{52})),x_{52})\qquad\mbox{cnf}(\mbox{p7}_8,\mbox{negated\_conjecture})$

$p_9(f_{18}(f_5(c_{20}),x_{64}),f_{17}(x_{64}))\qquad\mbox{cnf}(\mbox{p9}_9,\mbox{negated\_conjecture})$

$p_{13}(f_{15}(f_5(c_{20}),x_{20}),f_{16}(x_{20}))\qquad\mbox{cnf}(\mbox{p13}_{10},\mbox{negated\_conjecture})$

$p_{11}(x_3,x_4) \ \Rightarrow \ p_{11}(f_{12}(x_3),f_{12}(x_4))\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_7(x_{56},x_{57}) \ \Rightarrow \ p_9(f_{17}(x_{56}),f_{17}(x_{57}))\qquad\mbox{cnf}(\mbox{p9}_{12},\mbox{negated\_conjecture})$

$p_7(x_{49},x_{50}) \ \Rightarrow \ p_7(f_8(x_{49}),f_8(x_{50}))\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$p_2(x_{40},x_{41}) \ \Rightarrow \ p_2(f_6(x_{40}),f_6(x_{41}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_2(x_{38},x_{39}) \ \Rightarrow \ p_2(f_5(x_{38}),f_5(x_{39}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_2(x_{30},x_{31}) \ \Rightarrow \ p_2(f_3(x_{30}),f_3(x_{31}))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$p_{11}(x_{21},x_{22}) \ \Rightarrow \ p_{13}(f_{16}(x_{21}),f_{16}(x_{22}))\qquad\mbox{cnf}(\mbox{p13}_{17},\mbox{negated\_conjecture})$

$p_9(f_{18}(f_3(x_{62}),x_{63}),f_{18}(x_{62},f_8(x_{63})))\qquad\mbox{cnf}(\mbox{p9}_{18},\mbox{negated\_conjecture})$

$p_{13}(f_{15}(f_3(x_{18}),x_{19}),f_{15}(x_{18},f_{12}(x_{19})))\qquad\mbox{cnf}(\mbox{p13}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{20},\mbox{negated\_conjecture})$

$( p_9(x_{53},x_{54})\mbox{ and }p_9(x_{53},x_{55}) ) \ \Rightarrow \ p_9(x_{54},x_{55})\qquad\mbox{cnf}(\mbox{p9}_{21},\mbox{negated\_conjecture})$

$( p_7(x_{42},x_{43})\mbox{ and }p_7(x_{42},x_{44}) ) \ \Rightarrow \ p_7(x_{43},x_{44})\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_2(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_{13}(x_{11},x_{12})\mbox{ and }p_{13}(x_{11},x_{13}) ) \ \Rightarrow \ p_{13}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p13}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{23})\mbox{ and }p_2(x_{26},x_{24})\mbox{ and }p_{19}(x_{25},x_{26}) ) \ \Rightarrow \ p_{19}(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p19}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{58},x_{60})\mbox{ and }p_7(x_{59},x_{61}) ) \ \Rightarrow \ p_9(f_{18}(x_{58},x_{59}),f_{18}(x_{60},x_{61}))\qquad\mbox{cnf}(\mbox{p9}_{26},\mbox{negated\_conjecture})$

$( p_{11}(x_8,x_{10})\mbox{ and }p_{13}(x_7,x_9) ) \ \Rightarrow \ p_{11}(f_{14}(x_7,x_8),f_{14}(x_9,x_{10}))\qquad\mbox{cnf}(\mbox{p11}_{27},\mbox{negated\_conjecture})$

$( p_{11}(x_{15},x_{17})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_{13}(f_{15}(x_{14},x_{15}),f_{15}(x_{16},x_{17}))\qquad\mbox{cnf}(\mbox{p13}_{28},\mbox{negated\_conjecture})$

$( p_2(x_{32},x_{34})\mbox{ and }p_2(x_{33},x_{35}) ) \ \Rightarrow \ p_2(f_4(x_{32},x_{33}),f_4(x_{34},x_{35}))\qquad\mbox{cnf}(\mbox{p2}_{29},\mbox{negated\_conjecture})$

$( p_7(x_{46},x_{48})\mbox{ and }p_9(x_{45},x_{47}) ) \ \Rightarrow \ p_7(f_{10}(x_{45},x_{46}),f_{10}(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p7}_{30},\mbox{negated\_conjecture})$

$p_{19}(x_{37},x_{36}) \ \Rightarrow \ p_2(f_4(f_4(f_3(x_{36}),f_5(f_6(c_{20}))),x_{37}),f_3(f_4(f_4(x_{36},f_5(f_6(c_{20}))),x_{37})))\qquad\mbox{cnf}(\mbox{p2}_{31},\mbox{negated\_conjecture})$

$\neg\,p_{13}(f_{15}(f_4(f_4(f_3(c_{22}),f_5(f_6(c_{20}))),c_{21}),f_{14}(c_{23},c_{24})),f_{15}(f_4(f_4(c_{22},f_5(f_6(c_{20}))),c_{21}),c_{24}))\qquad\mbox{cnf}(\mbox{not\_p13}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{SYN731-1.p} Peter Andrews Problem X2150






$p(a,b,\mbox{sk}_1(b,a))\qquad\mbox{cnf}(\mbox{x2150}_1,\mbox{negated\_conjecture})$

$\neg\,p(a,a,b)\qquad\mbox{cnf}(\mbox{x2150}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN080+1.p} Pelletier Problem 58






$\forall x,y{:}\;f(x) = g(y)\qquad\mbox{fof}(\mbox{pel58}_1,\mbox{axiom})$

$\forall x,y{:}\;f(f(x)) = f(g(y))\qquad\mbox{fof}(\mbox{pel}_{58},\mbox{conjecture})$

\medskip\textbf{SYN217-1.p} RPT63 synthetic problem 2 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN730+1.p} Peter Andrews Problem THM75






$\exists v{:}\;\forall j{:}\;\exists q{:}\;( ( p(a,j,h(j))\mbox{ or }p(v,j,k(j)) ) \ \Rightarrow \ p(v,j,q) )\qquad\mbox{fof}(\mbox{thm}_{75},\mbox{conjecture})$

\medskip\textbf{SYN130-1.p} RPT63 synthetic problem 28 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_0(x,y,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN390+1.p} Pelletier 11

A simple problem designed to see whether 'natural' systems can
 do it efficiently (or whether they incorrectly try to prove
 the $\to$ each way).




$p\iff p\qquad\mbox{fof}(\mbox{pel}_{11},\mbox{conjecture})$

\medskip\textbf{SYN315-1.p} Church problem 46.2 (1)






$f(x) \ \Rightarrow \ p(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(a) \ \Rightarrow \ f(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y(x)) \ \Rightarrow \ \neg\,p(a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(a)\mbox{ or }f(y(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN259-1.p} RPT63 synthetic problem 44 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(x,d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN000$\wedge$1.p} Basic TPTP 

Basic TPTP  that you can't survive without parsing.




$p_0{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$q_0{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{q0\_type},\mbox{type})$

$r_0{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{r0\_type},\mbox{type})$

$s_0{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{s0\_type},\mbox{type})$

$( p_0\mbox{ and }\neg\,q_0 ) \ \Rightarrow \ ( r_0\mbox{ or }\neg\,s_0 )\qquad\mbox{thf}(\mbox{propositional},\mbox{axiom})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$h{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{h\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{s\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( p@ x\mbox{ or }\neg\,q@ x@ a ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r@ x@ ( f@ y )@ ( g@ x@ ( f@ y )@ z )\mbox{ and }\neg\,s@ ( f@ ( f@ ( f@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{first\_order},\mbox{axiom})$

$\exists y{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( f@ y ) = ( g@ x@ ( f@ y )@ z )\mbox{ or }( f@ ( f@ ( f@ b ) ) )\ne a\mbox{ or }x = ( f@ y ) )\qquad\mbox{thf}(\mbox{equality},\mbox{axiom})$

$\mbox{\$true}\mbox{ or }\mbox{\$false}\qquad\mbox{thf}(\mbox{true\_false},\mbox{axiom})$

$\mbox{'A proposition'}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{quoted\_proposition\_type},\mbox{type})$

$\mbox{'A predicate'}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{quoted\_predicate\_type},\mbox{type})$

$\mbox{'A constant'}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{quoted\_constant\_type},\mbox{type})$

$\mbox{'A function'}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{quoted\_function\_type},\mbox{type})$

$\mbox{'A \'quoted \\ escape\''}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{quoted\_escape\_type},\mbox{type})$

$\mbox{'A proposition'}\mbox{ or }\mbox{'A predicate'}@ a\mbox{ or }p@ \mbox{'A constant'}\mbox{ or }p@ ( \mbox{'A function'}@ a )\mbox{ or }p@ \mbox{'A \'quoted \\ escape\''}\qquad\mbox{thf}(\mbox{single\_quoted},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( ( p@ x ) \ \Leftarrow \ \neg\,q@ x@ a )\iff \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r@ x@ ( f@ y )@ ( g@ x@ ( f@ y )@ z ) ) <~> \neg\,s@ ( f@ ( f@ ( f@ b ) ) ) )\qquad\mbox{thf}(\mbox{useful\_connectives},\mbox{axiom})$

$l_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{l1\_type},\mbox{type})$

$l_2{:}\;( \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{l2\_type},\mbox{type})$

$l_1 = ( \lambda c{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( p@ c ) )\qquad\mbox{thf}(\mbox{lambda\_defn},\mbox{axiom})$

$l_2@ \lambda c{:}\;\mbox{\$i},f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( f@ c )\qquad\mbox{thf}(\mbox{lambda\_use},\mbox{axiom})$

$\mbox{new}{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{new\_type},\mbox{type})$

$\mbox{newc}{:}\;\mbox{new}\qquad\mbox{thf}(\mbox{newc\_type},\mbox{type})$

$\mbox{newf}{:}\;\mbox{new}\to \mbox{\$i}\to \mbox{new}\qquad\mbox{thf}(\mbox{newf\_type},\mbox{type})$

$\mbox{newp}{:}\;\mbox{new}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{newp\_type},\mbox{type})$

$\forall x{:}\;\mbox{new}{:}\;( \mbox{newp}@ ( \mbox{newf}@ \mbox{newc}@ a )@ a )\qquad\mbox{thf}(\mbox{new\_axiom},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( p@ x\mbox{ or }\neg\,q@ x@ a ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r@ x@ ( f@ y )@ ( g@ x@ ( f@ y )@ z )\mbox{ and }\neg\,s@ ( f@ ( f@ ( f@ b ) ) ) ) )\qquad\mbox{thf}(123,\mbox{axiom})$

$p@ h\qquad\mbox{thf}(\mbox{role\_hypothesis},\mbox{hypothesis})$

$\exists x{:}\;\mbox{\$i}{:}\;( p@ x )\qquad\mbox{thf}(\mbox{role\_conjecture},\mbox{conjecture})$

$\mbox{include}(\mbox{'Axioms/SYN000\string^0.ax'})$

\medskip\textbf{SYN724-1.p} Peter Andrews Problem THM31






$r(a) \ \Rightarrow \ s(a)\qquad\mbox{cnf}(\mbox{thm31}_3,\mbox{negated\_conjecture})$

$r(\mbox{sk}_1)\mbox{ or }r(\mbox{sk}_2)\qquad\mbox{cnf}(\mbox{thm31}_4,\mbox{negated\_conjecture})$

$( r(\mbox{sk}_2)\mbox{ and }s(\mbox{sk}_2) ) \ \Rightarrow \ r(\mbox{sk}_1)\qquad\mbox{cnf}(\mbox{thm31}_9,\mbox{negated\_conjecture})$

$s(\mbox{sk}_1) \ \Rightarrow \ r(\mbox{sk}_2)\qquad\mbox{cnf}(\mbox{thm31}_{10},\mbox{negated\_conjecture})$

$( r(\mbox{sk}_2)\mbox{ and }s(\mbox{sk}_1) ) \ \Rightarrow \ \neg\,s(\mbox{sk}_2)\qquad\mbox{cnf}(\mbox{thm31}_{15},\mbox{negated\_conjecture})$

\medskip\textbf{SYN335+1.p} Church problem 46.14 (7)






$\exists x,y{:}\;\forall z{:}\;( \mbox{big\_f}(x,z) \ \Rightarrow \ ( ( ( \mbox{big\_f}(z,z) \ \Rightarrow \ \mbox{big\_g}(z,z) )\iff \mbox{big\_f}(x,y) ) \ \Rightarrow \ ( ( ( \mbox{big\_g}(z,z) \ \Rightarrow \ \mbox{big\_f}(z,z) )\iff \mbox{big\_g}(x,y) ) \ \Rightarrow \ ( ( ( \mbox{big\_g}(x,y) \ \Rightarrow \ \mbox{big\_f}(y,x) )\iff \mbox{big\_g}(y,z) ) \ \Rightarrow \ ( \mbox{big\_f}(z,y)\iff \mbox{big\_f}(y,x) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_7,\mbox{conjecture})$

\medskip\textbf{SYN140-1.p} RPT63 synthetic problem 38 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_4(b,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN015-2.p} A problem in quantification theory






$x{=}x\qquad\mbox{cnf}(\mbox{reflexivityish},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetryish},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivityish},\mbox{axiom})$

$( a{=}b\mbox{ and }\mbox{element}(c,a) ) \ \Rightarrow \ \mbox{element}(c,b)\qquad\mbox{cnf}(\mbox{element\_substitutionish}_1,\mbox{axiom})$

$( \mbox{element}(a,b)\mbox{ and }a{=}c ) \ \Rightarrow \ \mbox{element}(c,b)\qquad\mbox{cnf}(\mbox{element\_substitutionish}_2,\mbox{axiom})$

$\mbox{element}(a,a) \ \Rightarrow \ ( a{=}k\mbox{ or }a{=}a )\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$a{=}k \ \Rightarrow \ ( \mbox{element}(a,a)\mbox{ or }a{=}a )\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$( f(a){=}m\mbox{ and }\mbox{element}(a,m) ) \ \Rightarrow \ a{=}m\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$( f(a){=}a\mbox{ and }\mbox{element}(a,m) ) \ \Rightarrow \ a{=}m\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$\mbox{element}(a,m) \ \Rightarrow \ ( \mbox{element}(a,f(a))\mbox{ or }a{=}m )\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$\mbox{element}(a,m) \ \Rightarrow \ ( \mbox{element}(f(a),a)\mbox{ or }a{=}m )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$( \mbox{element}(a,b)\mbox{ and }\mbox{element}(b,a) ) \ \Rightarrow \ ( a{=}b\mbox{ or }a{=}m\mbox{ or }\mbox{element}(b,m)\mbox{ or }b{=}m )\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$g(a){=}n \ \Rightarrow \ ( \mbox{element}(a,n)\mbox{ or }a{=}n )\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$g(a){=}a \ \Rightarrow \ ( \mbox{element}(a,n)\mbox{ or }a{=}n )\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$\mbox{element}(a,g(a))\mbox{ or }\mbox{element}(a,n)\mbox{ or }a{=}n\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$\mbox{element}(g(a),a)\mbox{ or }\mbox{element}(a,n)\mbox{ or }a{=}n\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$( \mbox{element}(a,b)\mbox{ and }\mbox{element}(b,a)\mbox{ and }\mbox{element}(b,n) ) \ \Rightarrow \ ( a{=}b\mbox{ or }a{=}n\mbox{ or }b{=}n )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$a{=}m \ \Rightarrow \ ( \mbox{element}(a,k)\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$a{=}n \ \Rightarrow \ ( \mbox{element}(a,k)\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

$\mbox{element}(a,k) \ \Rightarrow \ ( a{=}n\mbox{ or }a{=}m\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{17},\mbox{negated\_conjecture})$

$\neg\,n{=}a\qquad\mbox{cnf}(c_{18},\mbox{negated\_conjecture})$

$\neg\,m{=}n\qquad\mbox{cnf}(c_{19},\mbox{negated\_conjecture})$

$\neg\,n{=}k\qquad\mbox{cnf}(c_{20},\mbox{negated\_conjecture})$

$\neg\,m{=}a\qquad\mbox{cnf}(c_{21},\mbox{negated\_conjecture})$

$\neg\,k{=}a\qquad\mbox{cnf}(c_{22},\mbox{negated\_conjecture})$

$m{=}k\qquad\mbox{cnf}(c_{23},\mbox{negated\_conjecture})$

\medskip\textbf{SYN200-1.p} RPT63 synthetic problem 98 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_1(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN091-1.003.p} Plaisted problem sym(s(2,3))






$\neg\,\mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_3\mbox{ and }\mbox{p\_1}_2 ) \ \Rightarrow \ \mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_3,\mbox{axiom})$

$( \mbox{q\_2}_3\mbox{ and }\mbox{q\_1}_2 ) \ \Rightarrow \ \mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_4,\mbox{axiom})$

$( \mbox{p\_3}_3\mbox{ and }\mbox{p\_2}_2 ) \ \Rightarrow \ \mbox{p\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_5,\mbox{axiom})$

$( \mbox{q\_3}_3\mbox{ and }\mbox{q\_2}_2 ) \ \Rightarrow \ \mbox{p\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_6,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_2,\mbox{axiom})$

$( \mbox{p\_2}_3\mbox{ and }\mbox{q\_1}_2 ) \ \Rightarrow \ \mbox{q\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_3,\mbox{axiom})$

$( \mbox{q\_2}_3\mbox{ and }\mbox{p\_1}_2 ) \ \Rightarrow \ \mbox{q\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_4,\mbox{axiom})$

$( \mbox{p\_3}_3\mbox{ and }\mbox{q\_2}_2 ) \ \Rightarrow \ \mbox{q\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_5,\mbox{axiom})$

$( \mbox{q\_3}_3\mbox{ and }\mbox{p\_2}_2 ) \ \Rightarrow \ \mbox{q\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_6,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_1,\mbox{axiom})$

$\mbox{p\_2}_3 \ \Rightarrow \ \mbox{p\_3}_3\qquad\mbox{cnf}(\mbox{s2\_type22}_2,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_3,\mbox{axiom})$

$\mbox{q\_2}_3 \ \Rightarrow \ \mbox{q\_3}_3\qquad\mbox{cnf}(\mbox{s2\_type22}_4,\mbox{axiom})$

$\mbox{sym\_p\_1}_3\qquad\mbox{cnf}(\mbox{sym\_s2\_goal}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_2 \ \Rightarrow \ ( \mbox{sym\_p\_2}_2\mbox{ or }\mbox{sym\_p\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_2 \ \Rightarrow \ ( \mbox{sym\_q\_2}_2\mbox{ or }\mbox{sym\_q\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_2,\mbox{axiom})$

$\mbox{sym\_p\_1}_3 \ \Rightarrow \ ( \mbox{sym\_p\_2}_3\mbox{ or }\mbox{sym\_p\_1}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_3,\mbox{axiom})$

$\mbox{sym\_p\_1}_3 \ \Rightarrow \ ( \mbox{sym\_q\_2}_3\mbox{ or }\mbox{sym\_q\_1}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_4,\mbox{axiom})$

$\mbox{sym\_p\_2}_3 \ \Rightarrow \ ( \mbox{sym\_p\_3}_3\mbox{ or }\mbox{sym\_p\_2}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_5,\mbox{axiom})$

$\mbox{sym\_p\_2}_3 \ \Rightarrow \ ( \mbox{sym\_q\_3}_3\mbox{ or }\mbox{sym\_q\_2}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type11}_6,\mbox{axiom})$

$\mbox{sym\_q\_1}_2 \ \Rightarrow \ ( \mbox{sym\_p\_2}_2\mbox{ or }\mbox{sym\_q\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_1,\mbox{axiom})$

$\mbox{sym\_q\_1}_2 \ \Rightarrow \ ( \mbox{sym\_q\_2}_2\mbox{ or }\mbox{sym\_p\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_2,\mbox{axiom})$

$\mbox{sym\_q\_1}_3 \ \Rightarrow \ ( \mbox{sym\_p\_2}_3\mbox{ or }\mbox{sym\_q\_1}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_3,\mbox{axiom})$

$\mbox{sym\_q\_1}_3 \ \Rightarrow \ ( \mbox{sym\_q\_2}_3\mbox{ or }\mbox{sym\_p\_1}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_4,\mbox{axiom})$

$\mbox{sym\_q\_2}_3 \ \Rightarrow \ ( \mbox{sym\_p\_3}_3\mbox{ or }\mbox{sym\_q\_2}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_5,\mbox{axiom})$

$\mbox{sym\_q\_2}_3 \ \Rightarrow \ ( \mbox{sym\_q\_3}_3\mbox{ or }\mbox{sym\_p\_2}_2 )\qquad\mbox{cnf}(\mbox{sym\_s2\_type12}_6,\mbox{axiom})$

$\mbox{sym\_p\_1}_1 \ \Rightarrow \ \mbox{sym\_p\_1}_2\qquad\mbox{cnf}(\mbox{sym\_s2\_type21}_1,\mbox{axiom})$

$\mbox{sym\_q\_1}_1 \ \Rightarrow \ \mbox{sym\_q\_1}_2\qquad\mbox{cnf}(\mbox{sym\_s2\_type21}_2,\mbox{axiom})$

$\mbox{sym\_p\_2}_2 \ \Rightarrow \ \mbox{sym\_p\_1}_2\qquad\mbox{cnf}(\mbox{sym\_s2\_type22}_1,\mbox{axiom})$

$\mbox{sym\_p\_3}_3 \ \Rightarrow \ \mbox{sym\_p\_2}_3\qquad\mbox{cnf}(\mbox{sym\_s2\_type22}_2,\mbox{axiom})$

$\mbox{sym\_q\_2}_2 \ \Rightarrow \ \mbox{sym\_q\_1}_2\qquad\mbox{cnf}(\mbox{sym\_s2\_type22}_3,\mbox{axiom})$

$\mbox{sym\_q\_3}_3 \ \Rightarrow \ \mbox{sym\_q\_2}_3\qquad\mbox{cnf}(\mbox{sym\_s2\_type22}_4,\mbox{axiom})$

\medskip\textbf{SYN173-1.p} RPT63 synthetic problem 71 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(e,a,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN312-1.p} Problem for testing satisfiability






$( p(x,x_1,x_2)\mbox{ and }p(x_1,x_3,x_2) ) \ \Rightarrow \ p(x,x_3,x_2)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(x,x_1,x_2) \ \Rightarrow \ p(x_2,x_1,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(x,x_1,x_2) \ \Rightarrow \ p(x_1,x,x_2)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(x,x_1,x_2) \ \Rightarrow \ p(x,x_1,f(x_2))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$p(x,x_1,x_2) \ \Rightarrow \ p(g(x),x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$p(a,f(b),c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$p(f(b),d,c)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$\neg\,p(f(g(a)),f(g(d)),f(g(c)))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN388$\wedge$4.p} Expanded Law of Excluded Middle






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_7,\mbox{conjecture})$

\medskip\textbf{SYN358+1.p} Peter Andrews Problem X2109






$\exists x{:}\;( p\mbox{ and }\mbox{big\_q}(x) )\iff ( p\mbox{ and }\exists x{:}\;\mbox{big\_q}(x) )\qquad\mbox{fof}(x_{2109},\mbox{conjecture})$

\medskip\textbf{SYN063-1.p} Pelletier Problem 33






$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{big\_p}(a) \ \Rightarrow \ ( \mbox{big\_p}(x)\mbox{ or }\mbox{big\_p}(c)\mbox{ or }\mbox{big\_p}(y) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(a)\mbox{ and }\mbox{big\_p}(b) ) \ \Rightarrow \ \mbox{big\_p}(c)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_p}(c)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(e)\mbox{ and }\mbox{big\_p}(d) ) \ \Rightarrow \ \mbox{big\_p}(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(a)\mbox{ and }\mbox{big\_p}(d) ) \ \Rightarrow \ ( \mbox{big\_p}(x)\mbox{ or }\mbox{big\_p}(c)\mbox{ or }\mbox{big\_p}(b) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(e)\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(b)\mbox{ or }\mbox{big\_p}(x)\mbox{ or }\mbox{big\_p}(c) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN120-1.p} RPT63 synthetic problem 18 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_1(e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN066-1.p} Pelletier Problem 37






$\mbox{big\_p}(y,x) \ \Rightarrow \ \mbox{big\_p}(f(x,y),g(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_p}(f(x,y),x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_p}(f(x,y),g(x)) \ \Rightarrow \ \mbox{big\_q}(h(x,y),g(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_p}(x,y)\mbox{ or }\mbox{big\_q}(i(x,y),x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_q}(x,y) \ \Rightarrow \ \mbox{big\_r}(z,z)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\neg\,\mbox{big\_r}(a,z)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN260-1.p} RPT63 synthetic problem 45 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(x,d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN367$\wedge$7.p} Peter Andrews Problem X2118






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{big\_r}{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{big\_r\_type},\mbox{type})$

$\mbox{big\_q}{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{big\_q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_q}@ x ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ p ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_r}@ x ) ) ) ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_q}@ x ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_r}@ x ) ) ) ) ) ) )\qquad\mbox{thf}(x_{2118},\mbox{conjecture})$

\medskip\textbf{SYN341+1.p} Church problem 46.15 (6)






$\exists x_1{:}\;\forall x_2{:}\;\exists x_3{:}\;\forall x_4{:}\;( \mbox{big\_f}(x_1,x_2,x_3) \ \Rightarrow \ \mbox{big\_f}(x_2,x_3,x_4) )\qquad\mbox{fof}(\mbox{church\_46\_15}_6,\mbox{conjecture})$

\medskip\textbf{SYN211-1.p} RPT63 synthetic problem 109 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_4(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN052+1.p} Pelletier Problem 22






$\forall x{:}\;( p\iff \mbox{big\_f}(x) ) \ \Rightarrow \ ( p\iff \forall x_1{:}\;\mbox{big\_f}(x_1) )\qquad\mbox{fof}(\mbox{pel}_{22},\mbox{conjecture})$

\medskip\textbf{SYN578-1.p} Harrison problem 3328






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_3(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_9(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p9}_5,\mbox{negated\_conjecture})$

$p_{11}(c_{14},c_{15})\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_2(c_{16},f_5(c_{14}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_3(f_4(f_5(x_{17})),f_4(x_{17}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_3(f_6(f_5(x_{20})),f_6(x_{20}))\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$p_9(x_{32},f_4(c_{16})) \ \Rightarrow \ p_9(x_{32},c_{13})\qquad\mbox{cnf}(\mbox{p9}_{10},\mbox{negated\_conjecture})$

$p_9(x_{32},c_{13}) \ \Rightarrow \ p_9(x_{32},f_4(c_{16}))\qquad\mbox{cnf}(\mbox{p9}_{11},\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_5(x_{10}),f_5(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_3(f_6(x_{18}),f_6(x_{19}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_3(f_4(x_{15}),f_4(x_{16}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$( p_7(x_{21},x_{22})\mbox{ and }p_7(x_{21},x_{23}) ) \ \Rightarrow \ p_7(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{13})\mbox{ and }p_3(x_{12},x_{14}) ) \ \Rightarrow \ p_3(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{31},x_{28})\mbox{ and }p_9(x_{31},x_{30})\mbox{ and }p_3(x_{30},x_{29}) ) \ \Rightarrow \ p_9(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p9}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_7(f_8(x_{24},x_{25}),f_8(x_{26},x_{27}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_{34},c_{15})\mbox{ and }p_9(c_{12},x_{33}) ) \ \Rightarrow \ ( p_9(f_8(x_{33},x_{34}),x_{33})\mbox{ or }p_9(f_8(x_{33},x_{34}),f_4(x_{34})) )\qquad\mbox{cnf}(\mbox{p9}_{22},\mbox{negated\_conjecture})$

$( p_{11}(x_{34},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_9(f_8(x_{33},x_{34}),x_{33}) ) \ \Rightarrow \ \neg\,p_9(f_8(x_{33},x_{34}),f_4(x_{34}))\qquad\mbox{cnf}(\mbox{not\_p11}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{SYN389$\wedge$4.p} Pierce's Law






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{iatom}@ p ) )@ ( \mbox{iatom}@ p ) )\qquad\mbox{thf}(\mbox{pel}_8,\mbox{conjecture})$

\medskip\textbf{SYN294-1.p} RPT63 synthetic problem 79 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(d,d,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN212-1.p} RPT63 synthetic problem 110 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_4(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN384+1.p} Peter Andrews Problem X2136






$\forall z{:}\;\exists x,y{:}\;\forall u{:}\;( \mbox{big\_p}(x,y,z) \ \Rightarrow \ \mbox{big\_p}(u,x,x) )\qquad\mbox{fof}(x_{2136},\mbox{conjecture})$

\medskip\textbf{SYN262-1.p} RPT63 synthetic problem 47 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(b,c,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN175-1.p} RPT63 synthetic problem 73 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_2(x,d,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN406+1.p} Kalish and Montague Problem 240

Laws of alphabetic variance.




$( \forall x{:}\;( f(x) \ \Rightarrow \ g(x) )\mbox{ and }\exists y{:}\;( f(y)\mbox{ and }h(y) ) ) \ \Rightarrow \ \exists z{:}\;( g(z)\mbox{ and }h(z) )\qquad\mbox{fof}(\mbox{kalish}_{240},\mbox{conjecture})$

\medskip\textbf{SYN281-1.p} RPT63 synthetic problem 66 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN934+1.p} From Smullyan






$\exists x{:}\;( c \ \Rightarrow \ p(x) )\iff ( c \ \Rightarrow \ \exists x{:}\;p(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN052-1.p} Pelletier Problem 22






$\mbox{big\_f}(x) \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p \ \Rightarrow \ \mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{big\_f}(y)\mbox{ or }p\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p \ \Rightarrow \ \neg\,\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(a) \ \Rightarrow \ \mbox{big\_f}(y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN009-3.p} A problem to demonstrate the usefulness of relevancy testing






$p(x,y,z) \ \Rightarrow \ \neg\,t(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q(x,y,z) \ \Rightarrow \ \neg\,t(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(x,y,z) \ \Rightarrow \ \neg\,t(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$s(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$s(c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,v(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$u(c,c,c)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( s(x)\mbox{ and }s(y)\mbox{ and }s(z) ) \ \Rightarrow \ ( p(x,y,z)\mbox{ or }q(x,y,z)\mbox{ or }r(x,y,z) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$u(x,y,z) \ \Rightarrow \ ( t(x,y,z)\mbox{ or }v(x,y,z) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{SYN953+1.p} Syntactic from Shults






$\forall x{:}\;( p(x) \ \Rightarrow \ q(x) ) \ \Rightarrow \ ( \forall x{:}\;p(x) \ \Rightarrow \ \forall x{:}\;q(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN595-1.p} Harrison problem 3634






$p_2(x_{10},x_{10})\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_7(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_2(c_{21},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{16},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{not\_p2}_4,\mbox{negated\_conjecture})$

$p_{14}(c_{18},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_5,\mbox{negated\_conjecture})$

$p_{14}(x_4,x_5) \ \Rightarrow \ p_{15}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p15}_6,\mbox{negated\_conjecture})$

$p_{14}(x_4,x_5) \ \Rightarrow \ \neg\,p_2(x_4,x_5)\qquad\mbox{cnf}(\mbox{not\_p14}_7,\mbox{negated\_conjecture})$

$p_{14}(c_{19},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_8,\mbox{negated\_conjecture})$

$p_7(x_{36},x_{37}) \ \Rightarrow \ p_7(f_{12}(x_{36}),f_{12}(x_{37}))\qquad\mbox{cnf}(\mbox{p7}_9,\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_2(f_3(x_{17}),f_3(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{19},x_{20}) \ \Rightarrow \ p_2(f_4(x_{19}),f_4(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_5(x_{21}),f_5(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{34},x_{35}) \ \Rightarrow \ p_7(f_{11}(x_{34}),f_{11}(x_{35}))\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$p_{15}(x_4,x_5) \ \Rightarrow \ ( p_{14}(x_4,x_5)\mbox{ or }p_2(x_4,x_5) )\qquad\mbox{cnf}(\mbox{p14}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{27},x_{28})\mbox{ and }p_7(x_{27},x_{29}) ) \ \Rightarrow \ p_7(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{11})\mbox{ and }p_2(x_{10},x_{12}) ) \ \Rightarrow \ p_2(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$p_2(c_{19},f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{14}(x_2,x_3) ) \ \Rightarrow \ p_{14}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p14}_{18},\mbox{negated\_conjecture})$

$( p_2(x_8,x_6)\mbox{ and }p_2(x_9,x_7)\mbox{ and }p_{15}(x_8,x_9) ) \ \Rightarrow \ p_{15}(x_6,x_7)\qquad\mbox{cnf}(\mbox{p15}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{15})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_2(f_{13}(x_{13},x_{14}),f_{13}(x_{15},x_{16}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_7(x_{42},x_{44})\mbox{ and }p_7(x_{43},x_{45}) ) \ \Rightarrow \ p_7(f_9(x_{42},x_{43}),f_9(x_{44},x_{45}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_7(x_{38},x_{40})\mbox{ and }p_7(x_{39},x_{41}) ) \ \Rightarrow \ p_7(f_8(x_{38},x_{39}),f_8(x_{40},x_{41}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{31},x_{33})\mbox{ and }p_7(x_{30},x_{32}) ) \ \Rightarrow \ p_7(f_{10}(x_{30},x_{31}),f_{10}(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{23},x_{25})\mbox{ and }p_2(x_{24},x_{26}) ) \ \Rightarrow \ p_2(f_6(x_{23},x_{24}),f_6(x_{25},x_{26}))\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{48},f_3(c_{17}))\mbox{ and }p_{14}(x_{46},f_3(f_4(f_5(c_{17}))))\mbox{ and }p_{15}(x_{47},f_6(f_3(f_4(f_5(c_{17}))),c_{16})) ) \ \Rightarrow \ \neg\,p_7(f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),f_3(c_{17})),f_9(f_{11}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17}))))),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))),f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),x_{46}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),x_{48}),f_9(f_{11}(x_{47}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p2}_{26},\mbox{negated\_conjecture})$

$( p_{14}(x_{46},f_3(f_4(f_5(c_{17}))))\mbox{ and }p_{15}(x_{47},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\mbox{ and }p_{14}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))),x_{47}) ) \ \Rightarrow \ \neg\,p_7(f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),f_3(c_{17})),f_9(f_{11}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17}))))),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))),f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),x_{46}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),x_{48}),f_9(f_{11}(x_{47}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p14}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN104-1.p} RPT63 synthetic problem 2 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_0(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN594-1.p} Harrison problem 3633






$p_2(x_{10},x_{10})\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_7(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{16},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{not\_p2}_3,\mbox{negated\_conjecture})$

$p_{14}(c_{18},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_4,\mbox{negated\_conjecture})$

$p_{14}(x_4,x_5) \ \Rightarrow \ p_{15}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p15}_5,\mbox{negated\_conjecture})$

$p_{14}(x_4,x_5) \ \Rightarrow \ \neg\,p_2(x_4,x_5)\qquad\mbox{cnf}(\mbox{not\_p14}_6,\mbox{negated\_conjecture})$

$p_{14}(c_{19},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_7(x_{36},x_{37}) \ \Rightarrow \ p_7(f_{12}(x_{36}),f_{12}(x_{37}))\qquad\mbox{cnf}(\mbox{p7}_8,\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_2(f_3(x_{17}),f_3(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{19},x_{20}) \ \Rightarrow \ p_2(f_4(x_{19}),f_4(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_5(x_{21}),f_5(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{34},x_{35}) \ \Rightarrow \ p_7(f_{11}(x_{34}),f_{11}(x_{35}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_{15}(x_4,x_5) \ \Rightarrow \ ( p_{14}(x_4,x_5)\mbox{ or }p_2(x_4,x_5) )\qquad\mbox{cnf}(\mbox{p14}_{13},\mbox{negated\_conjecture})$

$( p_7(x_{27},x_{28})\mbox{ and }p_7(x_{27},x_{29}) ) \ \Rightarrow \ p_7(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p7}_{14},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{11})\mbox{ and }p_2(x_{10},x_{12}) ) \ \Rightarrow \ p_2(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{15}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))),c_{19})\qquad\mbox{cnf}(\mbox{p15}_{16},\mbox{negated\_conjecture})$

$\neg\,p_2(c_{19},f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))))\qquad\mbox{cnf}(\mbox{not\_p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{14}(x_2,x_3) ) \ \Rightarrow \ p_{14}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p14}_{18},\mbox{negated\_conjecture})$

$( p_2(x_8,x_6)\mbox{ and }p_2(x_9,x_7)\mbox{ and }p_{15}(x_8,x_9) ) \ \Rightarrow \ p_{15}(x_6,x_7)\qquad\mbox{cnf}(\mbox{p15}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{15})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_2(f_{13}(x_{13},x_{14}),f_{13}(x_{15},x_{16}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_7(x_{42},x_{44})\mbox{ and }p_7(x_{43},x_{45}) ) \ \Rightarrow \ p_7(f_9(x_{42},x_{43}),f_9(x_{44},x_{45}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_7(x_{38},x_{40})\mbox{ and }p_7(x_{39},x_{41}) ) \ \Rightarrow \ p_7(f_8(x_{38},x_{39}),f_8(x_{40},x_{41}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{31},x_{33})\mbox{ and }p_7(x_{30},x_{32}) ) \ \Rightarrow \ p_7(f_{10}(x_{30},x_{31}),f_{10}(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{23},x_{25})\mbox{ and }p_2(x_{24},x_{26}) ) \ \Rightarrow \ p_2(f_6(x_{23},x_{24}),f_6(x_{25},x_{26}))\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{48},f_3(c_{17}))\mbox{ and }p_{14}(x_{46},f_3(f_4(f_5(c_{17}))))\mbox{ and }p_{15}(x_{47},f_6(f_3(f_4(f_5(c_{17}))),c_{16})) ) \ \Rightarrow \ \neg\,p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),x_{46}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),x_{48}),f_9(f_{11}(x_{47}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p2}_{26},\mbox{negated\_conjecture})$

$( p_{14}(x_{46},f_3(f_4(f_5(c_{17}))))\mbox{ and }p_{15}(x_{47},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\mbox{ and }p_{14}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))),x_{47}) ) \ \Rightarrow \ \neg\,p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),x_{46}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),x_{48}),f_9(f_{11}(x_{47}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p14}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN037-2.p} Andrews Challenge Problem Variant






$( m_1\mbox{ and }m_3\mbox{ and }p(x_1) ) \ \Rightarrow \ \neg\,p(\mbox{fy}(x_1))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( m_1\mbox{ and }m_3 ) \ \Rightarrow \ ( p(x_1)\mbox{ or }p(\mbox{fy}(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( m_1\mbox{ and }p(\mbox{cx}) ) \ \Rightarrow \ ( m_3\mbox{ or }p(y_4) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( m_1\mbox{ and }p(y_5) ) \ \Rightarrow \ ( m_3\mbox{ or }p(\mbox{cx}) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( m_2\mbox{ and }m_3\mbox{ and }q(\mbox{cw}) ) \ \Rightarrow \ q(z_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( m_2\mbox{ and }m_3\mbox{ and }q(z) ) \ \Rightarrow \ q(\mbox{cw})\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( m_2\mbox{ and }q(w_2)\mbox{ and }q(\mbox{fz}_5(w_2)) ) \ \Rightarrow \ m_3\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$m_2 \ \Rightarrow \ ( m_3\mbox{ or }q(w_2)\mbox{ or }q(\mbox{fz}_5(w_2)) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( m_3\mbox{ and }p(\mbox{cx}) ) \ \Rightarrow \ ( m_1\mbox{ or }p(y_1) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( m_3\mbox{ and }p(y_2) ) \ \Rightarrow \ ( m_1\mbox{ or }p(\mbox{cx}) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( m_3\mbox{ and }q(w_1)\mbox{ and }q(\mbox{fz}_2(w_1)) ) \ \Rightarrow \ m_2\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$m_3 \ \Rightarrow \ ( m_2\mbox{ or }q(w_1)\mbox{ or }q(\mbox{fz}_2(w_1)) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( p(x_2)\mbox{ and }p(\mbox{fy}_3(x_2)) ) \ \Rightarrow \ ( m_1\mbox{ or }m_3 )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$q(\mbox{cw}) \ \Rightarrow \ ( m_2\mbox{ or }m_3\mbox{ or }q(z_4) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$q(z_3) \ \Rightarrow \ ( m_2\mbox{ or }m_3\mbox{ or }q(\mbox{cw}) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$m_1\mbox{ or }m_3\mbox{ or }p(x_2)\mbox{ or }p(\mbox{fy}_3(x_2))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( m_1\mbox{ and }q(u_1) ) \ \Rightarrow \ q(\mbox{uu}_1)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( m_2\mbox{ and }p(v_1) ) \ \Rightarrow \ p(\mbox{vv}_1)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$p(\mbox{cvv}) \ \Rightarrow \ m_2\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$q(\mbox{cuu}) \ \Rightarrow \ m_1\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$m_2\mbox{ or }p(\mbox{cv})\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$m_1\mbox{ or }q(\mbox{cu})\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN045+1.p} Pelletier Problem 13






$( p\mbox{ or }( q\mbox{ and }r ) )\iff ( ( p\mbox{ or }q )\mbox{ and }( p\mbox{ or }r ) )\qquad\mbox{fof}(\mbox{pel}_{13},\mbox{conjecture})$

\medskip\textbf{SYN296-1.p} RPT63 synthetic problem 81 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(e,b,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN172-1.p} RPT63 synthetic problem 70 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_0(c,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN573-1.p} Harrison problem 3208






$p_{12}(c_{15})\qquad\mbox{cnf}(\mbox{c15\_is\_p12}_1,\mbox{negated\_conjecture})$

$p_4(x_{30},x_{30})\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$p_2(x_{17},x_{17})\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_{12}(x_{16}) \ \Rightarrow \ p_2(x_{16},f_9(f_{10}(x_{16})))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$( p_{12}(x_{11})\mbox{ and }p_2(x_{11},x_{10}) ) \ \Rightarrow \ p_{12}(x_{10})\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{negated\_conjecture})$

$p_4(x_{37},x_{38}) \ \Rightarrow \ p_4(f_7(x_{37}),f_7(x_{38}))\qquad\mbox{cnf}(\mbox{p4}_6,\mbox{negated\_conjecture})$

$p_4(x_{24},x_{25}) \ \Rightarrow \ p_2(f_5(x_{24}),f_5(x_{25}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_2(x_{26},x_{27}) \ \Rightarrow \ p_2(f_8(x_{26}),f_8(x_{27}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_4(x_{28},x_{29}) \ \Rightarrow \ p_2(f_9(x_{28}),f_9(x_{29}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{33},x_{34}) \ \Rightarrow \ p_4(f_{10}(x_{33}),f_{10}(x_{34}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_4(x_{35},x_{36}) \ \Rightarrow \ p_4(f_6(x_{35}),f_6(x_{36}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$( p_4(x_{30},x_{31})\mbox{ and }p_4(x_{30},x_{32}) ) \ \Rightarrow \ p_4(x_{31},x_{32})\qquad\mbox{cnf}(\mbox{p4}_{12},\mbox{negated\_conjecture})$

$( p_2(x_{17},x_{18})\mbox{ and }p_2(x_{17},x_{19}) ) \ \Rightarrow \ p_2(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_{12}(x_9) \ \Rightarrow \ ( p_{11}(x_9,c_{16})\mbox{ or }p_2(x_9,f_9(c_{17})) )\qquad\mbox{cnf}(\mbox{p11}_{14},\mbox{negated\_conjecture})$

$p_{12}(x_{16}) \ \Rightarrow \ ( p_{13}(f_{10}(x_{16}),c_{17})\mbox{ or }p_4(f_{10}(x_{16}),c_{17}) )\qquad\mbox{cnf}(\mbox{p13}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_7,f_3(f_8(x_8),f_3(f_8(x_7),f_5(f_6(f_7(c_{14}))))))\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{11}(x_2,x_3) ) \ \Rightarrow \ p_{11}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p11}_{17},\mbox{negated\_conjecture})$

$( p_4(x_{14},x_{12})\mbox{ and }p_4(x_{15},x_{13})\mbox{ and }p_{13}(x_{14},x_{15}) ) \ \Rightarrow \ p_{13}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p13}_{18},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{15},f_3(f_8(c_{16}),f_3(f_8(f_9(c_{17})),f_5(f_6(f_7(c_{14}))))))\qquad\mbox{cnf}(\mbox{not\_p11}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{22})\mbox{ and }p_2(x_{21},x_{23}) ) \ \Rightarrow \ p_2(f_3(x_{20},x_{21}),f_3(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$p_{11}(x_4,x_5) \ \Rightarrow \ p_{11}(x_4,f_3(f_8(x_5),f_3(f_8(x_6),f_5(f_6(f_7(c_{14}))))))\qquad\mbox{cnf}(\mbox{p11}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{SYN943+1.p} Syntactic from Shults






$\forall a{:}\;\exists x,x_2,x_3,x_4,y{:}\;( ( p(a)\mbox{ and }e(a)\mbox{ and }( e(x) \ \Rightarrow \ ( g(x)\mbox{ or }s(x,f(x)) ) )\mbox{ and }( e(x_2) \ \Rightarrow \ ( g(x_2)\mbox{ or }c(f(x_2)) ) )\mbox{ and }( s(a,y) \ \Rightarrow \ p(y) ) ) \ \Rightarrow \ ( ( p(x_3)\mbox{ and }g(x_3) )\mbox{ or }( p(x_4)\mbox{ and }c(x_4) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN073-1.p} Pelletier Problem 50






$\mbox{big\_f}(a,x)\mbox{ or }\mbox{big\_f}(x,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_f}(x,f(x))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN070-1.p} Pelletier Problem 46






$\mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_f}(f(x))\mbox{ or }\mbox{big\_g}(x) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_h}(f(x),x)\mbox{ or }\mbox{big\_g}(x) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(f(x)) ) \ \Rightarrow \ \mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_g}(x)\mbox{ or }\mbox{big\_f}(a) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(a) ) \ \Rightarrow \ \mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_f}(y) ) \ \Rightarrow \ ( \mbox{big\_g}(x)\mbox{ or }\mbox{big\_g}(y)\mbox{ or }\mbox{big\_j}(a,y) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ \neg\,\mbox{big\_j}(y,x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$\mbox{big\_f}(b)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_g}(b)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN156-1.p} RPT63 synthetic problem 54 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_4(a,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN375+1.p} Peter Andrews Problem X2126






$\forall x{:}\;( \mbox{big\_p}(x)\iff \exists y{:}\;\mbox{big\_p}(y) )\iff ( \forall x{:}\;\mbox{big\_p}(x)\iff \exists y{:}\;\mbox{big\_p}(y) )\qquad\mbox{fof}(x_{2126},\mbox{conjecture})$

\medskip\textbf{SYN139-1.p} RPT63 synthetic problem 37 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_4(a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN076+1.p} Pelletier Problem 53






$\exists x,y{:}\;( x\ne y\mbox{ and }\forall z{:}\;( z = x\mbox{ or }z = y ) )\qquad\mbox{fof}(\mbox{pel53}_1,\mbox{axiom})$

$\exists z{:}\;\forall x{:}\;( \exists w{:}\;\forall y{:}\;( \mbox{big\_f}(x,y)\iff y = w )\iff x = z )\iff \exists w_1{:}\;\forall y_1{:}\;( \exists z_1{:}\;\forall x_1{:}\;( \mbox{big\_f}(x_1,y_1)\iff x_1 = z_1 )\iff y_1 = w_1 )\qquad\mbox{fof}(\mbox{pel}_{53},\mbox{conjecture})$

\medskip\textbf{SYN162-1.p} RPT63 synthetic problem 60 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.001.p} Orevkov formula - size 1

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_1,z_0{:}\;( r(0,0,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN734-1.p} PSAT - K=4 C=20 V=4 D=1.2






$\mbox{ssRr}(u,\mbox{skf}_1(u))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v) ) \ \Rightarrow \ ( \mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v) ) \ \Rightarrow \ ( \mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(v)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_4(v)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_2(u)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_3(u)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_1(w)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(w)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_1(u) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_1(u) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(w) ) \ \Rightarrow \ ( \mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_2(w)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_4(u)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_1(x) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_3(x) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_3(x)\mbox{ or }\mbox{ssPv}_1(u) )\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_3(x)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{SYN331-1.p} Church problem 46.14 (3)






$f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(z(x,y),z(x,y)) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(y,x)\mbox{ or }f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\neg\,f(z(x,y),x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,f(z(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN193-1.p} RPT63 synthetic problem 91 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_4(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN999$\wedge$1.p} Leibniz equality is symmetric






$\mbox{leibeq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{leibeq}@ x@ y ) \ \Rightarrow \ ( \mbox{leibeq}@ y@ x ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN945+1.p} Syntactic from Shults






$\forall x{:}\;p(x) \ \Rightarrow \ \forall a,b{:}\;( p(a)\mbox{ and }p(b) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN586-1.p} Harrison problem 3571






$p_{12}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{negated\_conjecture})$

$p_5(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_3(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{13},x_{13})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(c_{21},c_{19})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$\neg\,p_{12}(f_{13}(c_{20}),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{not\_p12}_6,\mbox{negated\_conjecture})$

$p_2(x_3,x_4) \ \Rightarrow \ p_{12}(f_{13}(x_3),f_{13}(x_4))\qquad\mbox{cnf}(\mbox{p12}_7,\mbox{negated\_conjecture})$

$p_3(x_{42},x_{43}) \ \Rightarrow \ p_5(f_7(x_{42}),f_7(x_{43}))\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$p_5(x_{40},x_{41}) \ \Rightarrow \ p_5(f_{10}(x_{40}),f_{10}(x_{41}))\qquad\mbox{cnf}(\mbox{p5}_9,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{36}) \ \Rightarrow \ p_3(f_9(x_{35}),f_9(x_{36}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_3(x_{33},x_{34}) \ \Rightarrow \ p_3(f_8(x_{33}),f_8(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_3(x_{27},x_{28}) \ \Rightarrow \ p_3(f_{11}(x_{27}),f_{11}(x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$( p_{12}(x_0,x_1)\mbox{ and }p_{12}(x_0,x_2) ) \ \Rightarrow \ p_{12}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_{13},\mbox{negated\_conjecture})$

$( p_5(x_{37},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_5(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_3(x_{24},x_{25})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_3(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_9)\mbox{ and }p_2(x_{12},x_{10})\mbox{ and }p_{15}(x_{11},x_{12}) ) \ \Rightarrow \ p_{15}(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p15}_{17},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),c_{19}),c_{20})\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{14}(x_{29},x_{30}),f_{14}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{18})\mbox{ and }p_3(x_{17},x_{19}) ) \ \Rightarrow \ p_2(f_4(x_{16},x_{17}),f_4(x_{18},x_{19}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{23})\mbox{ and }p_5(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_6(x_{20},x_{21}),f_6(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(c_{21})),c_{20})\qquad\mbox{cnf}(\mbox{p15}_{22},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(c_{19})),c_{20})\qquad\mbox{cnf}(\mbox{not\_p15}_{23},\mbox{negated\_conjecture})$

$p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{14}(x_5,x_6)),x_5)\qquad\mbox{cnf}(\mbox{p12}_{24},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(f_{14}(x_5,x_6))),x_5) \ \Rightarrow \ p_{12}(f_{13}(x_5),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{p12}_{25},\mbox{negated\_conjecture})$

$( p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_7),x_5)\mbox{ and }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_8),x_5) ) \ \Rightarrow \ ( p_3(x_7,x_8)\mbox{ or }p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_8)),x_5)\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_7)),x_5) )\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{SYN353-1.p} Church problem 46.18 (5)






$( f(y_1,y_2,y_3)\mbox{ and }f(a,a,z(y_1,y_2,y_3)) ) \ \Rightarrow \ ( f(y_2,y_3,y_1)\mbox{ or }f(y_3,y_1,y_2) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(y_3,y_1,y_2)\mbox{ and }f(y_2,y_1,z(y_1,y_2,y_3)) ) \ \Rightarrow \ f(y_1,y_2,y_3)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(y_3,y_1,y_2)\mbox{ and }f(y_2,y_1,z(y_1,y_2,y_3)) ) \ \Rightarrow \ f(y_2,y_3,y_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(y_3,y_1,y_2)\mbox{ or }f(y_2,y_1,z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( f(y_1,y_2,y_3)\mbox{ and }f(y_2,y_3,y_1) ) \ \Rightarrow \ f(y_2,y_1,z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(y_2,y_3,y_1)\mbox{ and }f(y_1,z(y_1,y_2,y_3),y_2) ) \ \Rightarrow \ f(y_1,y_2,y_3)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( f(y_2,y_3,y_1)\mbox{ and }f(y_1,z(y_1,y_2,y_3),y_2) ) \ \Rightarrow \ f(y_3,y_1,y_2)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$f(y_2,y_3,y_1)\mbox{ or }f(y_1,z(y_1,y_2,y_3),y_2)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( f(y_1,y_2,y_3)\mbox{ and }f(y_3,y_1,y_2) ) \ \Rightarrow \ f(y_1,z(y_1,y_2,y_3),y_2)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$f(z(y_1,y_2,y_3),y_2,y_1) \ \Rightarrow \ ( f(y_3,y_1,y_2)\mbox{ or }f(y_1,y_2,y_3) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$f(z(y_1,y_2,y_3),y_2,y_1) \ \Rightarrow \ ( f(y_2,y_3,y_1)\mbox{ or }f(y_1,y_2,y_3) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( f(y_3,y_1,y_2)\mbox{ and }f(y_2,y_3,y_1) ) \ \Rightarrow \ f(z(y_1,y_2,y_3),y_2,y_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$f(y_1,y_2,y_3) \ \Rightarrow \ f(z(y_1,y_2,y_3),y_2,y_1)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$f(y_1,y_2,y_3)\mbox{ or }f(z(y_1,y_2,y_3),z(y_1,y_2,y_3),z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$f(y_2,y_3,y_1)\mbox{ or }f(z(y_1,y_2,y_3),z(y_1,y_2,y_3),z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$f(y_3,y_1,y_2)\mbox{ or }f(z(y_1,y_2,y_3),z(y_1,y_2,y_3),z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( f(y_1,y_2,y_3)\mbox{ and }f(y_2,y_3,y_1)\mbox{ and }f(y_3,y_1,y_2) ) \ \Rightarrow \ \neg\,f(z(y_1,y_2,y_3),z(y_1,y_2,y_3),z(y_1,y_2,y_3))\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

\medskip\textbf{SYN346+1.p} Church problem 46.17 (2)






$\forall x_1,x_2{:}\;\exists y_1,y_2{:}\;\forall z_1,z_2{:}\;( \mbox{big\_f}(x_2,z_1) \ \Rightarrow \ ( \mbox{big\_f}(y_1,z_2) \ \Rightarrow \ ( ( \mbox{big\_f}(y_1,z_1)\mbox{ and }\mbox{big\_f}(y_2,z_1) )\mbox{ or }( \mbox{big\_f}(x_2,z_2)\mbox{ and }\mbox{big\_f}(y_2,z_2) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_17}_2,\mbox{conjecture})$

\medskip\textbf{SYN209-1.p} RPT63 synthetic problem 107 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_4(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN101-1.002.002.p} Plaisted problem n(t(2,2),2)






$\neg\,\mbox{p\_1}_2(a,a)\qquad\mbox{cnf}(\mbox{n\_s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2(x_1,x_2)\mbox{ and }\mbox{p\_1}_1(x_1,x_2) ) \ \Rightarrow \ \mbox{p\_1}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x_1,x_2)\mbox{ and }\mbox{q\_1}_1(x_1,x_2) ) \ \Rightarrow \ \mbox{p\_1}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_2(x_1,x_2)\mbox{ and }\mbox{q\_1}_1(x_1,x_2) ) \ \Rightarrow \ \mbox{q\_1}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x_1,x_2)\mbox{ and }\mbox{p\_1}_1(x_1,x_2) ) \ \Rightarrow \ \mbox{q\_1}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type12}_2,\mbox{axiom})$

$\mbox{p\_1}_2(x_1,x_2) \ \Rightarrow \ \mbox{p\_1}_1(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2(x_1,x_2) \ \Rightarrow \ \mbox{q\_1}_1(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2(x_1,x_2) \ \Rightarrow \ \mbox{p\_2}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type22}_1,\mbox{axiom})$

$\mbox{q\_1}_2(x_1,x_2) \ \Rightarrow \ \mbox{q\_2}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_s2\_type22}_2,\mbox{axiom})$

$( \mbox{nq}_1(x_1)\mbox{ and }\mbox{nq}_2(x_2) ) \ \Rightarrow \ \mbox{p\_1}_1(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_t2}_1,\mbox{axiom})$

$( \mbox{nq}_1(x_1)\mbox{ and }\mbox{nq}_2(x_2) ) \ \Rightarrow \ \mbox{q\_1}_1(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_t2}_2,\mbox{axiom})$

$( \mbox{nq}_1(x_1)\mbox{ and }\mbox{nq}_2(x_2) ) \ \Rightarrow \ \mbox{p\_2}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_t2}_3,\mbox{axiom})$

$( \mbox{nq}_1(x_1)\mbox{ and }\mbox{nq}_2(x_2) ) \ \Rightarrow \ \mbox{q\_2}_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{n\_t2}_4,\mbox{axiom})$

$\mbox{nq}_1(a)\qquad\mbox{cnf}(n_1,\mbox{axiom})$

$\mbox{nq}_1(b)\qquad\mbox{cnf}(n_2,\mbox{axiom})$

$\mbox{nq}_2(a)\qquad\mbox{cnf}(n_3,\mbox{axiom})$

$\mbox{nq}_2(b)\qquad\mbox{cnf}(n_4,\mbox{axiom})$

\medskip\textbf{SYN011-1.p} A problem to demonstrate C-reduction






$n \ \Rightarrow \ \neg\,t\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$m\mbox{ or }q\mbox{ or }n\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$m \ \Rightarrow \ l\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ l\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$l \ \Rightarrow \ \neg\,p\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$r\mbox{ or }p\mbox{ or }n\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ \neg\,l\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$t\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN975+1.p} Syntactic from Shults






$\forall x,y{:}\;p(x,y) \ \Rightarrow \ \forall x{:}\;p(x,x)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN000-2.p} Advanced TPTP CNF syntax

Advanced TPTP CNF syntax that you will encounter some time.




$\mbox{"An Apple"}\ne \mbox{"A \"Microsoft \\ escape\""}\qquad\mbox{cnf}(\mbox{distinct\_object},\mbox{axiom})$

$p(12)\mbox{ or }p(-12)\qquad\mbox{cnf}(\mbox{integers},\mbox{axiom})$

$p(123/456)\mbox{ or }p(-123/456)\mbox{ or }p(+123/456)\qquad\mbox{cnf}(\mbox{rationals},\mbox{axiom})$

$p(\mbox{123.}_{456})\mbox{ or }p(-\mbox{123.}_{456})\mbox{ or }p(\mbox{123.456E}_{78})\mbox{ or }p(\mbox{123.456e}_{78})\mbox{ or }p(-\mbox{123.456E}_{78})\mbox{ or }p(\mbox{123.456E-}_{78})\mbox{ or }p(-\mbox{123.456E-}_{78})\qquad\mbox{cnf}(\mbox{reals},\mbox{axiom})$

$f(d) = f(x)\qquad\mbox{cnf}(\mbox{role\_definition},\mbox{definition})$

$p(a)\qquad\mbox{cnf}(\mbox{role\_assumption},\mbox{assumption})$

$p(l)\qquad\mbox{cnf}(\mbox{role\_lemma},\mbox{lemma})$

$p(t)\qquad\mbox{cnf}(\mbox{role\_theorem},\mbox{theorem})$

$p(u)\qquad\mbox{cnf}(\mbox{role\_unknown},\mbox{unknown})$

$\mbox{include}(\mbox{'Axioms/SYN000-0.ax'},[\mbox{ia}_1,\mbox{ia}_3])$

$p(x)\qquad\mbox{cnf}(\mbox{source\_unknown},\mbox{axiom})$

$p(x)\qquad\mbox{cnf}(\mbox{source},\mbox{axiom})$

$p(x)\qquad\mbox{cnf}(\mbox{source\_name},\mbox{axiom})$

$p(x)\qquad\mbox{cnf}(\mbox{source\_copy},\mbox{axiom})$

$p(x)\qquad\mbox{cnf}(\mbox{source\_introduced\_assumption},\mbox{axiom})$

$p(a)\qquad\mbox{cnf}(\mbox{source\_inference},\mbox{axiom})$

$p(a)\qquad\mbox{cnf}(\mbox{source\_inference\_with\_bind},\mbox{axiom})$

$p(x)\qquad\mbox{cnf}(\mbox{useful\_info},\mbox{axiom})$

\medskip\textbf{SYN566-1.p} Harrison problem 2967






$p_9(x_{25},x_{25})\qquad\mbox{cnf}(\mbox{p9}_1,\mbox{negated\_conjecture})$

$p_2(x_4,x_4)\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_3(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_7(x_{16},x_{16})\qquad\mbox{cnf}(\mbox{p7}_4,\mbox{negated\_conjecture})$

$p_{10}(c_{15},c_{11})\qquad\mbox{cnf}(\mbox{p10}_5,\mbox{negated\_conjecture})$

$p_8(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p8}_6,\mbox{negated\_conjecture})$

$p_2(c_{14},f_6(c_{15}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_3(c_{13},f_4(c_{14}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_2(x_{14},x_{15}) \ \Rightarrow \ p_3(f_4(x_{14}),f_4(x_{15}))\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_5(x_7),f_5(x_8))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_9,x_{10}) \ \Rightarrow \ p_2(f_6(x_9),f_6(x_{10}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_9(x_{25},x_{26})\mbox{ and }p_9(x_{25},x_{27}) ) \ \Rightarrow \ p_9(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p9}_{12},\mbox{negated\_conjecture})$

$( p_2(x_4,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(x_5,x_6)\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_3(x_{11},x_{12})\mbox{ and }p_3(x_{11},x_{13}) ) \ \Rightarrow \ p_3(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{16},x_{17})\mbox{ and }p_7(x_{16},x_{18}) ) \ \Rightarrow \ p_7(x_{17},x_{18})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$p_8(x_{19},f_4(f_6(x_{20}))) \ \Rightarrow \ p_8(x_{19},f_4(f_5(x_{20})))\qquad\mbox{cnf}(\mbox{p8}_{16},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_9(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{17},\mbox{negated\_conjecture})$

$( p_7(x_{24},x_{21})\mbox{ and }p_8(x_{24},x_{23})\mbox{ and }p_3(x_{23},x_{22}) ) \ \Rightarrow \ p_8(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p8}_{18},\mbox{negated\_conjecture})$

$( p_{10}(x_{28},c_{11})\mbox{ and }p_8(c_{12},x_{30})\mbox{ and }p_3(x_{30},f_4(x_{29})) ) \ \Rightarrow \ \neg\,p_2(x_{29},f_5(x_{28}))\qquad\mbox{cnf}(\mbox{not\_p10}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{SYN604-1.p} Harrison problem 3820






$p_{14}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p14}_1,\mbox{negated\_conjecture})$

$p_9(x_{49},x_{49})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_5(x_{42},x_{42})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_{20},x_{20})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{15}(c_{17},c_{18})\qquad\mbox{cnf}(\mbox{p15}_6,\mbox{negated\_conjecture})$

$p_3(f_4(f_8(x_{41})),f_{12}(x_{41}))\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{10}(x_{38})),f_{10}(x_{38}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_{16}(f_{10}(f_{11}(c_{21})),f_{12}(x_{16}))\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_{16}(x_{14},x_{15})\mbox{ or }p_{16}(x_{15},f_{12}(x_{14}))\qquad\mbox{cnf}(\mbox{p16}_{10},\mbox{negated\_conjecture})$

$p_9(x_{52},x_{53}) \ \Rightarrow \ p_9(f_{11}(x_{52}),f_{11}(x_{53}))\qquad\mbox{cnf}(\mbox{p9}_{11},\mbox{negated\_conjecture})$

$p_2(x_{23},x_{24}) \ \Rightarrow \ p_2(f_{13}(x_{23}),f_{13}(x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_3(x_{29},x_{30}) \ \Rightarrow \ p_2(f_8(x_{29}),f_8(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_9(x_{34},x_{35}) \ \Rightarrow \ p_3(f_{10}(x_{34}),f_{10}(x_{35}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{36},x_{37}) \ \Rightarrow \ p_3(f_{12}(x_{36}),f_{12}(x_{37}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_2(x_{39},x_{40}) \ \Rightarrow \ p_3(f_4(x_{39}),f_4(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_{16}(f_4(f_6(c_{18},c_{22})),f_4(f_6(c_{18},x_{17})))\qquad\mbox{cnf}(\mbox{p16}_{17},\mbox{negated\_conjecture})$

$( p_{14}(x_0,x_1)\mbox{ and }p_{14}(x_0,x_2) ) \ \Rightarrow \ p_{14}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p14}_{18},\mbox{negated\_conjecture})$

$( p_9(x_{49},x_{50})\mbox{ and }p_9(x_{49},x_{51}) ) \ \Rightarrow \ p_9(x_{50},x_{51})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_5(x_{42},x_{43})\mbox{ and }p_5(x_{42},x_{44}) ) \ \Rightarrow \ p_5(x_{43},x_{44})\qquad\mbox{cnf}(\mbox{p5}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{31},x_{32})\mbox{ and }p_3(x_{31},x_{33}) ) \ \Rightarrow \ p_3(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{21})\mbox{ and }p_2(x_{20},x_{22}) ) \ \Rightarrow \ p_2(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_{16}(x_{11},x_{13})\mbox{ and }p_{16}(x_{13},x_{12}) ) \ \Rightarrow \ p_{16}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p16}_{23},\mbox{negated\_conjecture})$

$( p_{15}(x_5,x_6)\mbox{ and }p_5(x_6,x_4)\mbox{ and }p_{14}(x_5,x_3) ) \ \Rightarrow \ p_{15}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p15}_{24},\mbox{negated\_conjecture})$

$( p_3(x_9,x_7)\mbox{ and }p_3(x_{10},x_8)\mbox{ and }p_{16}(x_9,x_{10}) ) \ \Rightarrow \ p_{16}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p16}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{45},x_{47})\mbox{ and }p_5(x_{46},x_{48}) ) \ \Rightarrow \ p_5(f_7(x_{45},x_{46}),f_7(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p5}_{26},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_5(x_{25},x_{27}) ) \ \Rightarrow \ p_2(f_6(x_{25},x_{26}),f_6(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{16}(f_4(f_6(f_7(c_{20},c_{18}),x_{54})),f_4(f_6(f_7(c_{20},c_{18}),f_{13}(x_{54}))))\qquad\mbox{cnf}(\mbox{not\_p16}_{28},\mbox{negated\_conjecture})$

$p_{16}(f_4(x_{18}),f_{12}(c_{19})) \ \Rightarrow \ p_{16}(f_4(f_6(f_7(c_{20},c_{18}),c_{23})),f_4(f_6(f_7(c_{20},c_{18}),x_{18})))\qquad\mbox{cnf}(\mbox{p16}_{29},\mbox{negated\_conjecture})$

$p_{16}(c_{19},f_4(x_{19})) \ \Rightarrow \ p_{16}(f_4(f_6(f_7(c_{20},c_{18}),f_8(f_{10}(f_{11}(c_{21}))))),f_4(f_6(f_7(c_{20},c_{18}),x_{19})))\qquad\mbox{cnf}(\mbox{p16}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN127-1.p} RPT63 synthetic problem 25 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_4(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN186-1.p} RPT63 synthetic problem 84 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_1(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN916$\wedge$4.p} FALSE






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$\mbox{ivalid}@ ( \mbox{iatom}@ \mbox{ifalse} )\qquad\mbox{thf}(\mbox{falsity},\mbox{conjecture})$

\medskip\textbf{SYN095-1.002.p} Plaisted problem m(t(2,2))






$\neg\,\mbox{p\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2(x)\mbox{ and }\mbox{p\_1}_1(x) ) \ \Rightarrow \ \mbox{p\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x)\mbox{ and }\mbox{q\_1}_1(x) ) \ \Rightarrow \ \mbox{p\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_2(x)\mbox{ and }\mbox{q\_1}_1(x) ) \ \Rightarrow \ \mbox{q\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2(x)\mbox{ and }\mbox{p\_1}_1(x) ) \ \Rightarrow \ \mbox{q\_1}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type12}_2,\mbox{axiom})$

$\mbox{p\_1}_2(x) \ \Rightarrow \ \mbox{p\_1}_1(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2(x) \ \Rightarrow \ \mbox{q\_1}_1(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2(x) \ \Rightarrow \ \mbox{p\_2}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_1,\mbox{axiom})$

$\mbox{q\_1}_2(x) \ \Rightarrow \ \mbox{q\_2}_2(x)\qquad\mbox{cnf}(\mbox{m\_s2\_type22}_2,\mbox{axiom})$

$\mbox{p\_1}_1(a)\qquad\mbox{cnf}(\mbox{m\_t2}_1,\mbox{axiom})$

$\mbox{q\_1}_1(a)\qquad\mbox{cnf}(\mbox{m\_t2}_2,\mbox{axiom})$

$\mbox{p\_2}_2(a)\qquad\mbox{cnf}(\mbox{m\_t2}_3,\mbox{axiom})$

$\mbox{q\_2}_2(a)\qquad\mbox{cnf}(\mbox{m\_t2}_4,\mbox{axiom})$

\medskip\textbf{SYN168-1.p} RPT63 synthetic problem 66 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_3(x,y,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN325-1.p} Church problem 46.9 (2)






$f(x,x)\mbox{ or }f(x,y(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,x)\mbox{ or }g(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y(x),y(x)) \ \Rightarrow \ f(x,y(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(y(x),y(x)) \ \Rightarrow \ g(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,g(y(x))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.000.p} Orevkov formula - size 0

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_0{:}\;r(0,0,z_0)\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN111-1.p} RPT63 synthetic problem 9 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(c,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN593-1.p} Harrison problem 3632






$p_{12}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{negated\_conjecture})$

$p_5(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_3(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{13},x_{13})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{21},c_{19})\qquad\mbox{cnf}(\mbox{not\_p3}_5,\mbox{negated\_conjecture})$

$\neg\,p_{12}(f_{13}(c_{20}),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{not\_p12}_6,\mbox{negated\_conjecture})$

$p_2(x_3,x_4) \ \Rightarrow \ p_{12}(f_{13}(x_3),f_{13}(x_4))\qquad\mbox{cnf}(\mbox{p12}_7,\mbox{negated\_conjecture})$

$p_3(x_{42},x_{43}) \ \Rightarrow \ p_5(f_7(x_{42}),f_7(x_{43}))\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$p_5(x_{40},x_{41}) \ \Rightarrow \ p_5(f_{10}(x_{40}),f_{10}(x_{41}))\qquad\mbox{cnf}(\mbox{p5}_9,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{36}) \ \Rightarrow \ p_3(f_9(x_{35}),f_9(x_{36}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_3(x_{33},x_{34}) \ \Rightarrow \ p_3(f_8(x_{33}),f_8(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_3(x_{27},x_{28}) \ \Rightarrow \ p_3(f_{11}(x_{27}),f_{11}(x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$( p_{12}(x_0,x_1)\mbox{ and }p_{12}(x_0,x_2) ) \ \Rightarrow \ p_{12}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_{13},\mbox{negated\_conjecture})$

$( p_5(x_{37},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_5(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_3(x_{24},x_{25})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_3(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_9)\mbox{ and }p_2(x_{12},x_{10})\mbox{ and }p_{15}(x_{11},x_{12}) ) \ \Rightarrow \ p_{15}(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p15}_{17},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),c_{19}),c_{20})\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),c_{21}),c_{20})\qquad\mbox{cnf}(\mbox{p15}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{14}(x_{29},x_{30}),f_{14}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{18})\mbox{ and }p_3(x_{17},x_{19}) ) \ \Rightarrow \ p_2(f_4(x_{16},x_{17}),f_4(x_{18},x_{19}))\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{23})\mbox{ and }p_5(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_6(x_{20},x_{21}),f_6(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(c_{21})),c_{20})\qquad\mbox{cnf}(\mbox{not\_p15}_{23},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(c_{19})),c_{20})\qquad\mbox{cnf}(\mbox{not\_p15}_{24},\mbox{negated\_conjecture})$

$p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{14}(x_5,x_6)),x_5)\qquad\mbox{cnf}(\mbox{p12}_{25},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(f_{14}(x_5,x_6))),x_5) \ \Rightarrow \ p_{12}(f_{13}(x_5),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{p12}_{26},\mbox{negated\_conjecture})$

$( p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_7),x_5)\mbox{ and }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_8),x_5) ) \ \Rightarrow \ ( p_3(x_7,x_8)\mbox{ or }p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_8)),x_5)\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_7)),x_5) )\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN249-1.p} RPT63 synthetic problem 34 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(d,e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN944+1.p} Syntactic from Shults






$\forall a,b,c{:}\;( ( s(a)\mbox{ and }s(b)\mbox{ and }r(b,c)\mbox{ and }\forall x{:}\;( s(x) \ \Rightarrow \ p(x) )\mbox{ and }\forall x,y{:}\;( r(x,y) \ \Rightarrow \ q(x,y) ) ) \ \Rightarrow \ \exists x,y{:}\;( p(x)\mbox{ and }q(x,y) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN255-1.p} RPT63 synthetic problem 40 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_2(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN184-1.p} RPT63 synthetic problem 82 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_0(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN726-1.p} Peter Andrews Problem THM400






$( p(a,b)\mbox{ and }p(b,c) ) \ \Rightarrow \ p(a,c)\qquad\mbox{cnf}(\mbox{thm400}_1,\mbox{negated\_conjecture})$

$( q(a,b)\mbox{ and }q(b,c) ) \ \Rightarrow \ q(a,c)\qquad\mbox{cnf}(\mbox{thm400}_2,\mbox{negated\_conjecture})$

$q(a,b) \ \Rightarrow \ q(b,a)\qquad\mbox{cnf}(\mbox{thm400}_3,\mbox{negated\_conjecture})$

$p(a,b)\mbox{ or }q(a,b)\qquad\mbox{cnf}(\mbox{thm400}_4,\mbox{negated\_conjecture})$

$\neg\,p(\mbox{sk}_1,\mbox{sk}_2)\qquad\mbox{cnf}(\mbox{thm400}_5,\mbox{negated\_conjecture})$

$\neg\,q(\mbox{sk}_3,\mbox{sk}_4)\qquad\mbox{cnf}(\mbox{thm400}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN323+1.p} Church problem 46.4 (2)






$\exists y{:}\;( \forall x{:}\;( ( \mbox{big\_f}(x,y) \ \Rightarrow \ \mbox{big\_f}(y,x) ) \ \Rightarrow \ \mbox{big\_g}(x,y) ) \ \Rightarrow \ \neg\,\forall x{:}\;( ( \mbox{big\_f}(x,y) \ \Rightarrow \ \mbox{big\_f}(y,x) ) \ \Rightarrow \ \neg\,\mbox{big\_g}(x,y) ) )\qquad\mbox{fof}(\mbox{church\_46\_4}_2,\mbox{conjecture})$

\medskip\textbf{SYN605-1.p} Harrison problem 3825






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_7,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),c_{22})),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{28},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN053+1.p} Pelletier Problem 23






$\forall x{:}\;( p\mbox{ or }\mbox{big\_f}(x) )\iff ( p\mbox{ or }\forall x_1{:}\;\mbox{big\_f}(x_1) )\qquad\mbox{fof}(\mbox{pel}_{23},\mbox{conjecture})$

\medskip\textbf{SYN987$\wedge$2.p} All things are true






$\forall x{:}\;\mbox{\$o}{:}\;x\qquad\mbox{thf}(\mbox{thm},\mbox{axiom})$

\medskip\textbf{SYN721-1.p} Peter Andrews Problem LX1






$r(a,b)\qquad\mbox{cnf}(\mbox{lx1}_1,\mbox{negated\_conjecture})$

$r(a,b) \ \Rightarrow \ q(a,a)\qquad\mbox{cnf}(\mbox{lx1}_2,\mbox{negated\_conjecture})$

$q(a,b) \ \Rightarrow \ r(c,b)\qquad\mbox{cnf}(\mbox{lx1}_3,\mbox{negated\_conjecture})$

$q(a,a) \ \Rightarrow \ \neg\,r(b,a)\qquad\mbox{cnf}(\mbox{lx1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN557-1.p} Harrison problem 2313






$p_2(x_3,x_3)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_4(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$\neg\,p_2(c_6,f_3(c_8,f_5(c_9)))\qquad\mbox{cnf}(\mbox{not\_p2}_3,\mbox{negated\_conjecture})$

$p_2(f_3(x_{10},x_{11}),f_3(x_{11},x_{10}))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_4(x_{15},x_{16}) \ \Rightarrow \ p_2(f_5(x_{15}),f_5(x_{16}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$( p_2(x_3,x_4)\mbox{ and }p_2(x_3,x_5) ) \ \Rightarrow \ p_2(x_4,x_5)\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$( p_4(x_{19},x_{17})\mbox{ and }p_4(x_{19},x_{18}) ) \ \Rightarrow \ p_4(x_{17},x_{18})\qquad\mbox{cnf}(\mbox{p4}_7,\mbox{negated\_conjecture})$

$p_2(f_3(x_0,x_2),f_3(x_1,x_2)) \ \Rightarrow \ p_2(x_0,x_1)\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_2(x_0,x_1) \ \Rightarrow \ p_2(f_3(x_0,x_2),f_3(x_1,x_2))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(f_3(x_{12},f_3(x_{13},x_{14})),f_3(x_{13},f_3(x_{12},x_{14})))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(f_3(f_3(x_{12},x_{13}),x_{14}),f_3(x_{12},f_3(x_{13},x_{14})))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(f_3(c_6,f_5(c_7)),f_3(c_8,f_3(f_5(c_7),f_5(c_9))))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_2(x_6,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(f_3(x_6,x_7),f_3(x_8,x_9))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

\medskip\textbf{SYN923+1.p} From Smullyan






$( \forall x{:}\;p(x)\mbox{ or }\forall x{:}\;q(x) ) \ \Rightarrow \ \forall x{:}\;( p(x)\mbox{ or }q(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN204-1.p} RPT63 synthetic problem 102 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(a,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN933+1.p} From Smullyan






$\forall x{:}\;c\iff c\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN045$\wedge$7.p} Pelletier Problem 13






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ q )@ ( \mbox{mbox\_s}_4@ r ) ) )@ ( \mbox{mand}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ r ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ r ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ q )@ ( \mbox{mbox\_s}_4@ r ) ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{13},\mbox{conjecture})$

\medskip\textbf{SYN723+1.p} Peter Andrews Problem THM138






$\exists x{:}\;\forall y{:}\;( p(x)\iff p(y) )\iff ( ( \exists x{:}\;q(x)\iff \forall y{:}\;r(y) )\iff ( ( \exists x{:}\;\forall y{:}\;( q(x)\iff q(y) )\iff ( \exists x{:}\;r(x)\iff \forall y{:}\;s(y) ) )\iff ( \exists x{:}\;\forall y{:}\;( r(x)\iff r(y) )\iff ( ( \exists x{:}\;s(x)\iff \forall y{:}\;p(y) )\iff ( \exists x{:}\;\forall y{:}\;( s(x)\iff s(y) )\iff ( \exists x{:}\;p(x)\iff \forall y{:}\;q(y) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{thm}_{138},\mbox{conjecture})$

\medskip\textbf{SYN978$\wedge$4.p} Syntactic from Shults






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) ) )\qquad\mbox{thf}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN055+1.p} Pelletier Problem 25






$\exists x{:}\;\mbox{big\_p}(x)\qquad\mbox{fof}(\mbox{pel25}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ ( \neg\,\mbox{big\_g}(x)\mbox{ and }\mbox{big\_r}(x) ) )\qquad\mbox{fof}(\mbox{pel25}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_f}(x) ) )\qquad\mbox{fof}(\mbox{pel25}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_q}(x) )\mbox{ or }\exists z{:}\;( \mbox{big\_p}(z)\mbox{ and }\mbox{big\_r}(z) )\qquad\mbox{fof}(\mbox{pel25}_4,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_q}(x)\mbox{ and }\mbox{big\_p}(x) )\qquad\mbox{fof}(\mbox{pel}_{25},\mbox{conjecture})$

\medskip\textbf{SYN178-1.p} RPT63 synthetic problem 76 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_3(b,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN970+1.p} Syntactic from Shults






$\forall a,b{:}\;\exists x,y{:}\;( ( p(x) \ \Rightarrow \ r(y) ) \ \Rightarrow \ ( p(a) \ \Rightarrow \ r(b) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN574-1.p} Harrison problem 3263






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_3(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_9(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p9}_5,\mbox{negated\_conjecture})$

$p_{11}(c_{14},c_{15})\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_3(f_4(f_5(x_{17})),f_4(x_{17}))\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_3(f_6(f_5(x_{20})),f_6(x_{20}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_9(x_{32},f_4(c_{14})) \ \Rightarrow \ p_9(x_{32},c_{13})\qquad\mbox{cnf}(\mbox{p9}_9,\mbox{negated\_conjecture})$

$p_9(x_{32},c_{13}) \ \Rightarrow \ p_9(x_{32},f_4(c_{14}))\qquad\mbox{cnf}(\mbox{p9}_{10},\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_5(x_{10}),f_5(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_3(f_6(x_{18}),f_6(x_{19}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_3(f_4(x_{15}),f_4(x_{16}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{21},x_{22})\mbox{ and }p_7(x_{21},x_{23}) ) \ \Rightarrow \ p_7(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{13})\mbox{ and }p_3(x_{12},x_{14}) ) \ \Rightarrow \ p_3(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{31},x_{28})\mbox{ and }p_9(x_{31},x_{30})\mbox{ and }p_3(x_{30},x_{29}) ) \ \Rightarrow \ p_9(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_7(f_8(x_{24},x_{25}),f_8(x_{26},x_{27}))\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_{11}(x_{35},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_2(x_{34},f_5(x_{35})) ) \ \Rightarrow \ ( p_9(f_8(x_{33},x_{34}),x_{33})\mbox{ or }p_9(f_8(x_{33},x_{34}),f_4(x_{34})) )\qquad\mbox{cnf}(\mbox{p9}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_{35},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_2(x_{34},f_5(x_{35}))\mbox{ and }p_9(f_8(x_{33},x_{34}),x_{33}) ) \ \Rightarrow \ \neg\,p_9(f_8(x_{33},x_{34}),f_4(x_{34}))\qquad\mbox{cnf}(\mbox{not\_p11}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN261-1.p} RPT63 synthetic problem 46 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(b,x,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN387$\wedge$4.p} The Law of Excluded Middle






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) ) )\qquad\mbox{thf}(\mbox{pel}_6,\mbox{conjecture})$

\medskip\textbf{SYN199-1.p} RPT63 synthetic problem 97 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_1(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN415+1.p} Kalish and Montague Problem 317






$( \exists x{:}\;f(x)\mbox{ and }\forall y,z{:}\;( ( f(y)\mbox{ and }f(z) ) \ \Rightarrow \ y = z ) )\iff \exists u{:}\;( f(u)\mbox{ and }\forall v{:}\;( f(v) \ \Rightarrow \ u = v ) )\qquad\mbox{fof}(\mbox{kalish}_{317},\mbox{conjecture})$

\medskip\textbf{SYN032-1.p} Ances






$j \ \Rightarrow \ ( h\mbox{ or }a )\qquad\mbox{cnf}(\mbox{three},\mbox{hypothesis})$

$h\mbox{ or }k\mbox{ or }j\qquad\mbox{cnf}(\mbox{four},\mbox{hypothesis})$

$k \ \Rightarrow \ ( h\mbox{ or }j )\qquad\mbox{cnf}(\mbox{five},\mbox{hypothesis})$

$a \ \Rightarrow \ b\qquad\mbox{cnf}(\mbox{six},\mbox{hypothesis})$

$h \ \Rightarrow \ c\qquad\mbox{cnf}(\mbox{seven},\mbox{hypothesis})$

$h \ \Rightarrow \ \neg\,c\qquad\mbox{cnf}(\mbox{one},\mbox{hypothesis})$

$a \ \Rightarrow \ \neg\,b\qquad\mbox{cnf}(\mbox{prove\_something},\mbox{negated\_conjecture})$

\medskip\textbf{SYN046-1.p} Pelletier Problem 15






$p \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN959+1.p} Syntactic from Shults






$\exists x{:}\;b(x) \ \Rightarrow \ \exists x{:}\;( a(x)\mbox{ or }b(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN399+1.p} Kalish and Montague Problem 223

Laws of confinement.




$\forall x{:}\;( f(x)\iff p ) \ \Rightarrow \ ( \forall y{:}\;f(y)\iff p )\qquad\mbox{fof}(\mbox{kalish}_{223},\mbox{conjecture})$

\medskip\textbf{SYN123-1.p} RPT63 synthetic problem 21 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN106-1.p} RPT63 synthetic problem 4 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_1(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN152-1.p} RPT63 synthetic problem 50 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_2(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN360$\wedge$5.p} TPS problem X2111A

Variant of X2111 for testing infix mechanism.




$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{c\_less\_}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_less\_},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{c\_less\_}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{c\_less\_}@ \mbox{xz}@ \mbox{xy} ) ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cX2111A\_pme},\mbox{conjecture})$

\medskip\textbf{SYN297-1.p} RPT63 synthetic problem 82 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(e,e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN966+1.p} Syntactic from Shults






$\forall x,y{:}\;( \mbox{eq}(x,y)\iff \forall z{:}\;( \mbox{a\_member\_of}(z,x)\iff \mbox{a\_member\_of}(z,y) ) ) \ \Rightarrow \ \forall a,b{:}\;( \mbox{eq}(a,b) \ \Rightarrow \ \mbox{eq}(b,a) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN416$\wedge$7.p} Pelletier Problem 16






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ q )@ ( \mbox{mbox\_s}_4@ p ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{16},\mbox{conjecture})$

\medskip\textbf{SYN313-1.001.002.p} Problem for testing satisfiability






$p(x_0,f_0(x_0),x_1) \ \Rightarrow \ \neg\,p(x_0,f_1(x_1),x_2)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(h_0,y_0,j_0(y_2))\mbox{ or }p(h_1(y_0),y_1,j_1(y_2))\mbox{ or }p(h_2(y_1),y_2,j_2(y_2))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN625-1.p} Harrison problem 4030






$p_{15}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p15}_1,\mbox{negated\_conjecture})$

$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_2,\mbox{negated\_conjecture})$

$p_8(x_{71},x_{71})\qquad\mbox{cnf}(\mbox{p8}_3,\mbox{negated\_conjecture})$

$p_6(x_{68},x_{68})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_5(x_{47},x_{47})\qquad\mbox{cnf}(\mbox{p5}_5,\mbox{negated\_conjecture})$

$p_3(x_{40},x_{40})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_2(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_7(c_{19},c_{20}),c_{21}))\qquad\mbox{cnf}(\mbox{f4\_is\_not\_p15}_8,\mbox{negated\_conjecture})$

$p_5(x_{15},x_{16}) \ \Rightarrow \ p_{16}(x_{14},x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_{15}(x_7) \ \Rightarrow \ \neg\,p_{18}(f_9(x_7),x_{76},x_{76})\qquad\mbox{cnf}(\mbox{not\_p15}_{10},\mbox{negated\_conjecture})$

$( p_{15}(x_6)\mbox{ and }p_3(x_6,x_5) ) \ \Rightarrow \ p_{15}(x_5)\qquad\mbox{cnf}(\mbox{p15}_{11},\mbox{negated\_conjecture})$

$p_6(x_3,x_4) \ \Rightarrow \ p_{10}(f_{11}(x_3),f_{11}(x_4))\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$p_3(x_{74},x_{75}) \ \Rightarrow \ p_8(f_9(x_{74}),f_9(x_{75}))\qquad\mbox{cnf}(\mbox{p8}_{13},\mbox{negated\_conjecture})$

$p_3(x_{66},x_{67}) \ \Rightarrow \ p_5(f_{14}(x_{66}),f_{14}(x_{67}))\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$p_{15}(x_7)\mbox{ or }p_{18}(f_9(x_7),f_{14}(x_7),f_{14}(x_7))\qquad\mbox{cnf}(\mbox{p15}_{15},\mbox{negated\_conjecture})$

$p_{17}(x_{77},f_{11}(c_{20})) \ \Rightarrow \ \neg\,p_{16}(f_9(c_{21}),x_{77},c_{19})\qquad\mbox{cnf}(\mbox{not\_p17}_{16},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{17},\mbox{negated\_conjecture})$

$( p_8(x_{71},x_{72})\mbox{ and }p_8(x_{71},x_{73}) ) \ \Rightarrow \ p_8(x_{72},x_{73})\qquad\mbox{cnf}(\mbox{p8}_{18},\mbox{negated\_conjecture})$

$( p_6(x_{68},x_{69})\mbox{ and }p_6(x_{68},x_{70}) ) \ \Rightarrow \ p_6(x_{69},x_{70})\qquad\mbox{cnf}(\mbox{p6}_{19},\mbox{negated\_conjecture})$

$( p_5(x_{47},x_{48})\mbox{ and }p_5(x_{47},x_{49}) ) \ \Rightarrow \ p_5(x_{48},x_{49})\qquad\mbox{cnf}(\mbox{p5}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{40},x_{41})\mbox{ and }p_3(x_{40},x_{42}) ) \ \Rightarrow \ p_3(x_{41},x_{42})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{34})\mbox{ and }p_2(x_{33},x_{35}) ) \ \Rightarrow \ p_2(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_{16}(x_{14},x_{15},x_{17})\mbox{ and }p_{16}(x_{14},x_{17},x_{16}) ) \ \Rightarrow \ p_{16}(x_{14},x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p16}_{23},\mbox{negated\_conjecture})$

$( p_{17}(x_{26},x_{25})\mbox{ and }p_5(x_{26},x_{23})\mbox{ and }p_{10}(x_{25},x_{24}) ) \ \Rightarrow \ p_{17}(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p17}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{43},x_{45})\mbox{ and }p_3(x_{44},x_{46}) ) \ \Rightarrow \ p_3(f_4(x_{43},x_{44}),f_4(x_{45},x_{46}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_5(x_{36},x_{38})\mbox{ and }p_6(x_{37},x_{39}) ) \ \Rightarrow \ p_2(f_7(x_{36},x_{37}),f_7(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$p_{16}(x_{14},x_{15},x_{16}) \ \Rightarrow \ ( p_5(x_{15},x_{16})\mbox{ or }p_{16}(x_{14},x_{15},f_{13}(x_{14},x_{15},x_{16})) )\qquad\mbox{cnf}(\mbox{p5}_{27},\mbox{negated\_conjecture})$

$p_{16}(x_{14},x_{15},x_{16}) \ \Rightarrow \ ( p_5(x_{15},x_{16})\mbox{ or }p_{16}(x_{14},f_{13}(x_{14},x_{15},x_{16}),x_{16}) )\qquad\mbox{cnf}(\mbox{p5}_{28},\mbox{negated\_conjecture})$

$( p_8(x_{11},x_8)\mbox{ and }p_5(x_{13},x_{10})\mbox{ and }p_5(x_{12},x_9)\mbox{ and }p_{16}(x_{11},x_{12},x_{13}) ) \ \Rightarrow \ p_{16}(x_8,x_9,x_{10})\qquad\mbox{cnf}(\mbox{p16}_{29},\mbox{negated\_conjecture})$

$( p_8(x_{30},x_{27})\mbox{ and }p_5(x_{32},x_{29})\mbox{ and }p_5(x_{31},x_{28})\mbox{ and }p_{18}(x_{30},x_{31},x_{32}) ) \ \Rightarrow \ p_{18}(x_{27},x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p18}_{30},\mbox{negated\_conjecture})$

$( p_5(x_{62},x_{65})\mbox{ and }p_8(x_{60},x_{63})\mbox{ and }p_5(x_{61},x_{64}) ) \ \Rightarrow \ p_5(f_{13}(x_{60},x_{61},x_{62}),f_{13}(x_{63},x_{64},x_{65}))\qquad\mbox{cnf}(\mbox{p5}_{31},\mbox{negated\_conjecture})$

$p_{18}(f_9(f_4(f_7(x_{20},x_{22}),x_{18})),x_{19},x_{21}) \ \Rightarrow \ ( p_{16}(f_9(x_{18}),x_{19},x_{20})\mbox{ or }p_{18}(f_9(x_{18}),x_{19},x_{21}) )\qquad\mbox{cnf}(\mbox{p16}_{32},\mbox{negated\_conjecture})$

$p_{18}(f_9(f_4(f_7(x_{20},x_{22}),x_{18})),x_{19},x_{21}) \ \Rightarrow \ ( p_{18}(f_9(x_{18}),x_{19},x_{21})\mbox{ or }p_{17}(f_{12}(x_{18},x_{20},x_{22},x_{19},x_{21}),f_{11}(x_{22})) )\qquad\mbox{cnf}(\mbox{p18}_{33},\mbox{negated\_conjecture})$

$p_{18}(f_9(f_4(f_7(x_{20},x_{22}),x_{18})),x_{19},x_{21}) \ \Rightarrow \ ( p_{18}(f_9(x_{18}),x_{19},x_{21})\mbox{ or }p_{16}(f_9(x_{18}),f_{12}(x_{18},x_{20},x_{22},x_{19},x_{21}),x_{21}) )\qquad\mbox{cnf}(\mbox{p18}_{34},\mbox{negated\_conjecture})$

$( p_5(x_{54},x_{59})\mbox{ and }p_6(x_{52},x_{57})\mbox{ and }p_3(x_{50},x_{55})\mbox{ and }p_5(x_{51},x_{56})\mbox{ and }p_5(x_{53},x_{58}) ) \ \Rightarrow \ p_5(f_{12}(x_{50},x_{51},x_{52},x_{53},x_{54}),f_{12}(x_{55},x_{56},x_{57},x_{58},x_{59}))\qquad\mbox{cnf}(\mbox{p5}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{SYN631-1.p} Harrison problem 4139






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_9(x_{74},x_{74})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_7(x_{65},x_{65})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_6(x_{62},x_{62})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_5(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p5}_5,\mbox{negated\_conjecture})$

$p_3(x_{32},x_{32})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_2(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_{12}(x_5,x_5)\qquad\mbox{cnf}(\mbox{p12}_8,\mbox{negated\_conjecture})$

$p_{22}(c_{25},c_{28})\qquad\mbox{cnf}(\mbox{p22}_9,\mbox{negated\_conjecture})$

$p_5(f_{17}(f_{18}(x_{48},x_{49})),x_{48})\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(f_{16}(f_{18}(x_{44},x_{45})),x_{45})\qquad\mbox{cnf}(\mbox{p5}_{11},\mbox{negated\_conjecture})$

$p_{23}(f_{13}(c_{27},f_{14}(c_{28})),f_{11}(f_{15}(c_{24})))\qquad\mbox{cnf}(\mbox{p23}_{12},\mbox{negated\_conjecture})$

$\neg\,p_{23}(f_{13}(f_{16}(c_{27}),f_{14}(c_{28})),f_{11}(c_{24}))\qquad\mbox{cnf}(\mbox{not\_p23}_{13},\mbox{negated\_conjecture})$

$p_9(x_{77},x_{78}) \ \Rightarrow \ p_9(f_{15}(x_{77}),f_{15}(x_{78}))\qquad\mbox{cnf}(\mbox{p9}_{14},\mbox{negated\_conjecture})$

$p_9(x_3,x_4) \ \Rightarrow \ p_{10}(f_{11}(x_3),f_{11}(x_4))\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$p_2(x_{35},x_{36}) \ \Rightarrow \ p_3(f_4(x_{35}),f_4(x_{36}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_6(x_{40},x_{41}) \ \Rightarrow \ p_5(f_{14}(x_{40}),f_{14}(x_{41}))\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$p_5(x_{42},x_{43}) \ \Rightarrow \ p_5(f_{16}(x_{42}),f_{16}(x_{43}))\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_{17}(x_{46}),f_{17}(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$p_5(x_{60},x_{61}) \ \Rightarrow \ p_5(f_{20}(x_{60}),f_{20}(x_{61}))\qquad\mbox{cnf}(\mbox{p5}_{20},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{21},\mbox{negated\_conjecture})$

$( p_9(x_{74},x_{75})\mbox{ and }p_9(x_{74},x_{76}) ) \ \Rightarrow \ p_9(x_{75},x_{76})\qquad\mbox{cnf}(\mbox{p9}_{22},\mbox{negated\_conjecture})$

$( p_7(x_{65},x_{66})\mbox{ and }p_7(x_{65},x_{67}) ) \ \Rightarrow \ p_7(x_{66},x_{67})\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_6(x_{62},x_{63})\mbox{ and }p_6(x_{62},x_{64}) ) \ \Rightarrow \ p_6(x_{63},x_{64})\qquad\mbox{cnf}(\mbox{p6}_{24},\mbox{negated\_conjecture})$

$( p_5(x_{37},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_5(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p5}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{32},x_{33})\mbox{ and }p_3(x_{32},x_{34}) ) \ \Rightarrow \ p_3(x_{33},x_{34})\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{13})\mbox{ and }p_2(x_{12},x_{14}) ) \ \Rightarrow \ p_2(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_{12}(x_5,x_6)\mbox{ and }p_{12}(x_5,x_7) ) \ \Rightarrow \ p_{12}(x_6,x_7)\qquad\mbox{cnf}(\mbox{p12}_{28},\mbox{negated\_conjecture})$

$( p_{12}(x_{28},x_{25})\mbox{ and }p_{23}(x_{28},x_{27})\mbox{ and }p_{10}(x_{27},x_{26}) ) \ \Rightarrow \ p_{23}(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p23}_{29},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{16})\mbox{ and }p_7(x_{17},x_{15})\mbox{ and }p_{21}(x_{17},x_{18}) ) \ \Rightarrow \ p_{21}(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p21}_{30},\mbox{negated\_conjecture})$

$( p_3(x_{23},x_{21})\mbox{ and }p_6(x_{24},x_{22})\mbox{ and }p_{22}(x_{23},x_{24}) ) \ \Rightarrow \ p_{22}(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p22}_{31},\mbox{negated\_conjecture})$

$( p_5(x_8,x_{10})\mbox{ and }p_5(x_9,x_{11}) ) \ \Rightarrow \ p_{12}(f_{13}(x_8,x_9),f_{13}(x_{10},x_{11}))\qquad\mbox{cnf}(\mbox{p12}_{32},\mbox{negated\_conjecture})$

$( p_5(x_{50},x_{52})\mbox{ and }p_5(x_{51},x_{53}) ) \ \Rightarrow \ p_5(f_{18}(x_{50},x_{51}),f_{18}(x_{52},x_{53}))\qquad\mbox{cnf}(\mbox{p5}_{33},\mbox{negated\_conjecture})$

$p_{23}(f_{13}(x_{30},x_{31}),f_{11}(f_{15}(x_{29}))) \ \Rightarrow \ p_5(x_{30},f_{18}(f_{20}(c_{29}),f_{19}(x_{29},x_{30},x_{31})))\qquad\mbox{cnf}(\mbox{p5}_{34},\mbox{negated\_conjecture})$

$p_{23}(f_{13}(x_{30},x_{31}),f_{11}(f_{15}(x_{29}))) \ \Rightarrow \ p_{23}(f_{13}(f_{19}(x_{29},x_{30},x_{31}),x_{31}),f_{11}(x_{29}))\qquad\mbox{cnf}(\mbox{p23}_{35},\mbox{negated\_conjecture})$

$( p_{22}(c_{25},x_{20})\mbox{ and }p_{23}(f_{13}(x_{19},f_{14}(x_{20})),f_{11}(c_{24})) ) \ \Rightarrow \ p_{21}(f_8(c_{26},x_{19},x_{20}),f_4(c_{26}))\qquad\mbox{cnf}(\mbox{p21}_{36},\mbox{negated\_conjecture})$

$( p_5(x_{69},x_{72})\mbox{ and }p_6(x_{70},x_{73})\mbox{ and }p_2(x_{68},x_{71}) ) \ \Rightarrow \ p_7(f_8(x_{68},x_{69},x_{70}),f_8(x_{71},x_{72},x_{73}))\qquad\mbox{cnf}(\mbox{p7}_{37},\mbox{negated\_conjecture})$

$( p_5(x_{56},x_{59})\mbox{ and }p_9(x_{54},x_{57})\mbox{ and }p_5(x_{55},x_{58}) ) \ \Rightarrow \ p_5(f_{19}(x_{54},x_{55},x_{56}),f_{19}(x_{57},x_{58},x_{59}))\qquad\mbox{cnf}(\mbox{p5}_{38},\mbox{negated\_conjecture})$

\medskip\textbf{SYN050+1.p} Pelletier Problem 20






$\forall x,y{:}\;\exists z{:}\;\forall w{:}\;( ( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(y) ) \ \Rightarrow \ ( \mbox{big\_r}(z)\mbox{ and }\mbox{big\_s}(w) ) ) \ \Rightarrow \ \exists x_1,y_1{:}\;( ( \mbox{big\_p}(x_1)\mbox{ and }\mbox{big\_q}(y_1) ) \ \Rightarrow \ \exists z_1{:}\;\mbox{big\_r}(z_1) )\qquad\mbox{fof}(\mbox{pel}_{20},\mbox{conjecture})$

\medskip\textbf{SYN257-1.p} RPT63 synthetic problem 42 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_0(c,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN226-1.p} RPT63 synthetic problem 11 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN333+1.p} Church problem 46.14 (5)






$\exists x,y{:}\;\forall z{:}\;( \mbox{big\_f}(x,y) \ \Rightarrow \ ( \mbox{big\_f}(y,z)\mbox{ and }\mbox{big\_f}(z,z)\mbox{ and }( ( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_g}(x,y) ) \ \Rightarrow \ ( \mbox{big\_g}(x,z)\mbox{ and }\mbox{big\_g}(z,z) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_5,\mbox{conjecture})$

\medskip\textbf{SYN366+1.p} Peter Andrews Problem X2117






$( \forall u,v{:}\;( \mbox{big\_r}(u,u)\iff \mbox{big\_r}(u,v) )\mbox{ and }\forall w,z{:}\;( \mbox{big\_r}(w,w)\iff \mbox{big\_r}(z,w) ) ) \ \Rightarrow \ ( \exists x{:}\;\mbox{big\_r}(x,x) \ \Rightarrow \ \forall y{:}\;\mbox{big\_r}(y,y) )\qquad\mbox{fof}(x_{2117},\mbox{conjecture})$

\medskip\textbf{SYN611-1.p} Harrison problem 3831






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_7,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{28},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_4(f_6(f_7(f_8(c_{20})))))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN939+1.p} Syntactic from Shults






$\forall c,b{:}\;( \forall z{:}\;q(f(z)) \ \Rightarrow \ \exists x,y{:}\;( ( p(f(y)) \ \Rightarrow \ p(x) )\mbox{ and }( r(y) \ \Rightarrow \ ( r(b)\mbox{ and }r(c) ) )\mbox{ and }q(x) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN411+1.p} Kalish and Montague Problem 250






$\forall x,y,z{:}\;f(x,y,z)\iff \neg\,\exists u,v,w{:}\;\neg\,f(u,v,w)\qquad\mbox{fof}(\mbox{kalish}_{250},\mbox{conjecture})$

\medskip\textbf{SYN279-1.p} RPT63 synthetic problem 64 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,a,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN367+1.p} Peter Andrews Problem X2118






$\forall x{:}\;( ( p\mbox{ and }\mbox{big\_q}(x) )\mbox{ or }( \neg\,p\mbox{ and }\mbox{big\_r}(x) ) ) \ \Rightarrow \ ( \forall x{:}\;\mbox{big\_q}(x)\mbox{ or }\forall x{:}\;\mbox{big\_r}(x) )\qquad\mbox{fof}(x_{2118},\mbox{conjecture})$

\medskip\textbf{SYN252-1.p} RPT63 synthetic problem 37 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_4(x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN288-1.p} RPT63 synthetic problem 73 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(b,b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN324+1.p} Church problem 46.9 (1)






$\exists x{:}\;\forall y{:}\;( ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(x,x) ) \ \Rightarrow \ ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(y,y) ) )\qquad\mbox{fof}(\mbox{church\_46\_9}_1,\mbox{conjecture})$

\medskip\textbf{SYN357$\wedge$7.p} Peter Andrews Problem X2108






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$\mbox{big\_p}{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{big\_p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(x_{2108},\mbox{conjecture})$

\medskip\textbf{SYN272-1.p} RPT63 synthetic problem 57 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(d,e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN072-1.p} Pelletier Problem 49






$x = c\mbox{ or }x = d\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_p}(b)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$a\ne b\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\neg\,\mbox{big\_p}(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN289-1.p} RPT63 synthetic problem 74 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(b,d,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN338+1.p} Church problem 46.15 (3)






$\exists x{:}\;\forall y{:}\;\exists z{:}\;( \mbox{big\_f}(x,y) \ \Rightarrow \ ( \mbox{big\_f}(z,x) \ \Rightarrow \ \mbox{big\_f}(y,y) ) )\qquad\mbox{fof}(\mbox{church\_46\_15}_3,\mbox{conjecture})$

\medskip\textbf{SYN286-1.p} RPT63 synthetic problem 71 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(a,e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN158-1.p} RPT63 synthetic problem 56 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN361$\wedge$5.p} TPS problem X2112






$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \exists \mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xv} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{cQ}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cS}@ \mbox{xu}\qquad\mbox{thf}(\mbox{cX}_{2112},\mbox{conjecture})$

\medskip\textbf{SYN049$\wedge$5.p} TPS problem PELL19






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( ( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{xz} ) ) \ \Rightarrow \ ( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{19},\mbox{conjecture})$

\medskip\textbf{SYN386$\wedge$5.p} TPS problem X2138






$\mbox{cD}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cD},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xn}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xw}@ \mbox{xx}@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xd}{:}\;\mbox{\$i}{:}\;\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;( ( \mbox{cD}@ \mbox{xa}@ \mbox{xb}@ \mbox{xd} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xa}@ \mbox{xy}\mbox{ and }\mbox{cF}@ \mbox{xb}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xy}@ \mbox{xz}@ \mbox{xe} ) ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xm}@ \mbox{xw} ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xw}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xz}@ \mbox{xy}@ \mbox{xe} ) ) )\qquad\mbox{thf}(\mbox{cX}_{2138},\mbox{conjecture})$

\medskip\textbf{SYN205-1.p} RPT63 synthetic problem 103 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(a,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN401+1.p} Kalish and Montague Problem 229

Laws of vacuous quantification.




$\forall x{:}\;( \forall y{:}\;f(y) \ \Rightarrow \ f(x) )\qquad\mbox{fof}(\mbox{kalish}_{229},\mbox{conjecture})$

\medskip\textbf{SYN341-1.p} Church problem 46.15 (6)






$f(y_1,y_2(y_1),y_3)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,f(y_2(y_1),y_3,y_4(y_1,y_3))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN368+1.p} Peter Andrews Problem X2119






$\exists y{:}\;\forall x{:}\;( \mbox{big\_p}(y) \ \Rightarrow \ \mbox{big\_p}(x) )\qquad\mbox{fof}(x_{2119},\mbox{conjecture})$

\medskip\textbf{SYN054-1.p} Pelletier Problem 24






$\mbox{big\_s}(x) \ \Rightarrow \ \neg\,\mbox{big\_q}(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_r}(x) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_p}(a)\mbox{ or }\mbox{big\_q}(b)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_q}(x) \ \Rightarrow \ \mbox{big\_s}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_r}(x) \ \Rightarrow \ \mbox{big\_s}(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\mbox{big\_p}(x) \ \Rightarrow \ \neg\,\mbox{big\_r}(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN038-1.p} Syntactic formula

Exists X, Exists Y, All Z (((p(X,Y) $\to$ (p(X,Z) $<$$\to$ q(Y,Z)))
 \& (p(X,Y) $<$$\to$ (p(Z,Z) $\to$ q(Z,Z)))) $\to$ (q(Z,Y) $<$$\to$ q(Z,Z)))




$( p(x,f(x,y))\mbox{ and }p(x,y) ) \ \Rightarrow \ q(y,f(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( p(x,y)\mbox{ and }q(y,f(x,y)) ) \ \Rightarrow \ p(x,f(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( p(x,y)\mbox{ and }p(f(x,y),f(x,y)) ) \ \Rightarrow \ q(f(x,y),f(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(x,y)\mbox{ or }p(f(x,y),f(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$q(f(x,y),f(x,y)) \ \Rightarrow \ p(x,y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$q(x,y)\mbox{ or }q(f(x,y),f(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$q(x,y) \ \Rightarrow \ \neg\,q(f(x,y),f(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN727-1.p} Peter Andrews Problem THM68






$\mbox{likes}(a,\mbox{bruce})\qquad\mbox{cnf}(\mbox{thm68}_1,\mbox{negated\_conjecture})$

$\mbox{likes}(a,b) \ \Rightarrow \ \mbox{likes}(\mbox{lyle},a)\qquad\mbox{cnf}(\mbox{thm68}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{likes}(a,\mbox{sk}_1(a))\qquad\mbox{cnf}(\mbox{thm68}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN584-1.p} Harrison problem 3511






$p_{12}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{negated\_conjecture})$

$p_6(x_{42},x_{42})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{26},x_{26})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_5(x_{36},x_{37}) \ \Rightarrow \ p_5(f_{11}(x_{36}),f_{11}(x_{37}))\qquad\mbox{cnf}(\mbox{p5}_6,\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_3(f_4(x_{29}),f_4(x_{30}))\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_5(x_{34},x_{35}) \ \Rightarrow \ p_5(f_{10}(x_{34}),f_{10}(x_{35}))\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$( p_{12}(x_0,x_1)\mbox{ and }p_{12}(x_0,x_2) ) \ \Rightarrow \ p_{12}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_9,\mbox{negated\_conjecture})$

$( p_6(x_{42},x_{43})\mbox{ and }p_6(x_{42},x_{44}) ) \ \Rightarrow \ p_6(x_{43},x_{44})\qquad\mbox{cnf}(\mbox{p6}_{10},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{11},\mbox{negated\_conjecture})$

$( p_3(x_{26},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(x_{27},x_{28})\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{20})\mbox{ and }p_2(x_{19},x_{21}) ) \ \Rightarrow \ p_2(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_{12}(x_{18},x_{16})\mbox{ and }p_{16}(x_{17},x_{18})\mbox{ and }p_{12}(x_{17},x_{15}) ) \ \Rightarrow \ p_{16}(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p16}_{14},\mbox{negated\_conjecture})$

$( p_{15}(x_{13},x_{14})\mbox{ and }p_3(x_{14},x_{12})\mbox{ and }p_{12}(x_{13},x_{11}) ) \ \Rightarrow \ p_{15}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p15}_{15},\mbox{negated\_conjecture})$

$( p_{12}(x_3,x_5)\mbox{ and }p_{12}(x_4,x_6) ) \ \Rightarrow \ p_{12}(f_{13}(x_3,x_4),f_{13}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$( p_5(x_{45},x_{47})\mbox{ and }p_5(x_{46},x_{48}) ) \ \Rightarrow \ p_6(f_8(x_{45},x_{46}),f_8(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p6}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{38},x_{40})\mbox{ and }p_5(x_{39},x_{41}) ) \ \Rightarrow \ p_5(f_9(x_{38},x_{39}),f_9(x_{40},x_{41}))\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{22},x_{24})\mbox{ and }p_6(x_{23},x_{25}) ) \ \Rightarrow \ p_2(f_7(x_{22},x_{23}),f_7(x_{24},x_{25}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_{12}(x_8,x_{10})\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_{12}(f_{14}(x_7,x_8),f_{14}(x_9,x_{10}))\qquad\mbox{cnf}(\mbox{p12}_{20},\mbox{negated\_conjecture})$

$p_{15}(c_{22},f_4(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20})))))))\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$p_{15}(c_{17},f_4(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20})))))))\qquad\mbox{cnf}(\mbox{p15}_{22},\mbox{negated\_conjecture})$

$p_{12}(f_{13}(c_{17},f_{14}(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20}))))),c_{17})),c_{22})\qquad\mbox{cnf}(\mbox{p12}_{23},\mbox{negated\_conjecture})$

$( p_{16}(x_{49},c_{17})\mbox{ and }p_{16}(c_{22},x_{49}) ) \ \Rightarrow \ \neg\,p_{15}(x_{49},f_4(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20})))))))\qquad\mbox{cnf}(\mbox{not\_p16}_{24},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_{13}(c_{17},f_{14}(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20}))))),c_{17})),f_4(f_7(c_{18},f_8(f_9(c_{19},f_{10}(f_{11}(c_{20}))),f_9(c_{21},f_{10}(f_{11}(c_{20})))))))\qquad\mbox{cnf}(\mbox{not\_p15}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{SYN728+1.p} Peter Andrews Problem THM69






$( \forall x{:}\;( \exists y{:}\;p(x,y) \ \Rightarrow \ \forall z{:}\;p(z,z) )\mbox{ and }\forall r{:}\;\exists s{:}\;( p(r,s)\mbox{ or }( m(r)\mbox{ and }q(f(r,s)) ) )\mbox{ and }\forall w{:}\;( q(w) \ \Rightarrow \ \neg\,m(g(w)) ) ) \ \Rightarrow \ \forall u{:}\;\exists v{:}\;( p(g(u),v)\mbox{ and }p(u,u) )\qquad\mbox{fof}(\mbox{thm}_{69},\mbox{conjecture})$

\medskip\textbf{SYN182-1.p} RPT63 synthetic problem 80 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_5(x,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN394+1.p} Kalish and Montague Problem 201

Laws of distribution.




$\forall x{:}\;( f(x) \ \Rightarrow \ g(x) ) \ \Rightarrow \ ( \forall y{:}\;f(y) \ \Rightarrow \ \forall z{:}\;g(z) )\qquad\mbox{fof}(\mbox{kalish}_{201},\mbox{conjecture})$

\medskip\textbf{SYN334+1.p} Church problem 46.14 (6)






$\exists x,y{:}\;\forall z{:}\;( ( \mbox{big\_f}(x,y) \ \Rightarrow \ ( \mbox{big\_f}(x,z)\iff \mbox{big\_g}(y,z) ) ) \ \Rightarrow \ ( ( \mbox{big\_f}(x,y)\iff ( \mbox{big\_f}(z,z) \ \Rightarrow \ \mbox{big\_g}(z,z) ) ) \ \Rightarrow \ ( \mbox{big\_g}(x,y)\iff \mbox{big\_g}(z,z) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_6,\mbox{conjecture})$

\medskip\textbf{SYN064$\wedge$5.p} TPS problem PELL35






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i},\mbox{xy}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0@ \mbox{xy}_0 ) )\qquad\mbox{thf}(\mbox{cPELL}_{35},\mbox{conjecture})$

\medskip\textbf{SYN040-1.p} Pelletier Problem 1

A biconditional version of the 'most difficult' theorem
 proved by the original Logic Theorist.




$p \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN990$\wedge$1.p} Simple test for satisfiability

There are two individuals, a and b. Everything is either a or b.
 There are three functions from individuals to individuals: f,g,h.
 They are all different.




$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;( x = a\mbox{ or }x = b )\qquad\mbox{thf}(\mbox{ab},\mbox{axiom})$

$f\ne g\qquad\mbox{thf}(\mbox{fg},\mbox{axiom})$

$g\ne h\qquad\mbox{thf}(\mbox{gh},\mbox{axiom})$

$f\ne h\qquad\mbox{thf}(\mbox{fh},\mbox{axiom})$

\medskip\textbf{SYN925+1.p} From Smullyan






$\exists y{:}\;( p(y) \ \Rightarrow \ \forall x{:}\;p(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN735-1.p} PSAT - K=4 C=20 V=4 D=1.8






$\mbox{ssRr}(u,\mbox{skf}_1(u))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{ssRr}(u,v) \ \Rightarrow \ ( \mbox{ssPv}_1(v)\mbox{ or }\mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_3(u)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_3(u)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w) ) \ \Rightarrow \ ( \mbox{ssPv}_3(v)\mbox{ or }\mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w) ) \ \Rightarrow \ ( \mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_1(u) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_3(w) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_2(w) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(w)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w) ) \ \Rightarrow \ ( \mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_3(w)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_3(x)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_1(x) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_2(x) )\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_2(x)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_3(x)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_2(x) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_2(x)\mbox{ or }\mbox{ssPv}_1(u) )\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_1(u) ) \ \Rightarrow \ \mbox{ssPv}_2(x)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{SYN962+1.p} Syntactic from Shults






$\forall x{:}\;p(x) \ \Rightarrow \ ( \forall x{:}\;p(x)\mbox{ and }\forall y{:}\;p(y) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN242-1.p} RPT63 synthetic problem 27 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(a,a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN585-1.p} Harrison problem 3538






$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_{11}(x_8,x_8)\qquad\mbox{cnf}(\mbox{p11}_2,\mbox{negated\_conjecture})$

$p_{10}(c_{14},c_{15})\qquad\mbox{cnf}(\mbox{p10}_3,\mbox{negated\_conjecture})$

$p_{10}(c_{17},c_{14})\qquad\mbox{cnf}(\mbox{p10}_4,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{14},f_3(c_{13}))\qquad\mbox{cnf}(\mbox{not\_p2}_5,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{12},f_3(c_{13}))\qquad\mbox{cnf}(\mbox{not\_p2}_6,\mbox{negated\_conjecture})$

$p_{11}(c_{18},f_9(c_{12},c_{14}))\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_2(f_6(c_{15},c_{14}),c_{17})\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_2(f_5(c_{15},c_{14}),c_{16})\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{12},f_3(f_4(c_{13})))\qquad\mbox{cnf}(\mbox{not\_p2}_{10},\mbox{negated\_conjecture})$

$p_2(f_7(f_8(c_{16},c_{14}),c_{17}),c_{15})\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{18},f_9(c_{12},f_8(c_{16},c_{14})))\qquad\mbox{cnf}(\mbox{not\_p11}_{12},\mbox{negated\_conjecture})$

$p_{11}(x_9,x_{11}) \ \Rightarrow \ p_{11}(x_9,f_8(x_{10},x_{11}))\qquad\mbox{cnf}(\mbox{p11}_{13},\mbox{negated\_conjecture})$

$p_2(x_{23},x_{24}) \ \Rightarrow \ p_2(f_4(x_{23}),f_4(x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_3(x_{21}),f_3(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$( p_{11}(x_{12},x_{14})\mbox{ and }p_{11}(x_{14},x_{13}) ) \ \Rightarrow \ p_{11}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{18},\mbox{negated\_conjecture})$

$( p_2(x_6,x_4)\mbox{ and }p_2(x_7,x_5)\mbox{ and }p_{11}(x_6,x_7) ) \ \Rightarrow \ p_{11}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{41},x_{43})\mbox{ and }p_2(x_{42},x_{44}) ) \ \Rightarrow \ p_2(f_9(x_{41},x_{42}),f_9(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_2(x_{26},x_{28}) ) \ \Rightarrow \ p_2(f_5(x_{25},x_{26}),f_5(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{32})\mbox{ and }p_2(x_{29},x_{31}) ) \ \Rightarrow \ p_2(f_6(x_{29},x_{30}),f_6(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{35})\mbox{ and }p_2(x_{34},x_{36}) ) \ \Rightarrow \ p_2(f_7(x_{33},x_{34}),f_7(x_{35},x_{36}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{37},x_{39})\mbox{ and }p_2(x_{38},x_{40}) ) \ \Rightarrow \ p_2(f_8(x_{37},x_{38}),f_8(x_{39},x_{40}))\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$p_{11}(f_9(x_{17},x_{15}),f_9(x_{17},x_{16})) \ \Rightarrow \ ( p_{11}(x_{15},x_{16})\mbox{ or }p_2(x_{17},f_3(c_{13})) )\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

$p_{11}(x_{15},x_{16}) \ \Rightarrow \ ( p_2(x_{17},f_3(c_{13}))\mbox{ or }p_{11}(f_9(x_{17},x_{15}),f_9(x_{17},x_{16})) )\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{SYN915+1.p} TRUE






$\mbox{\$true}\qquad\mbox{fof}(\mbox{truth},\mbox{conjecture})$

\medskip\textbf{SYN354-1.p} Church problem 46.20 (1)






$f(a,b)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$g(a,b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y_1,y_2) \ \Rightarrow \ ( g(b,z(y_1,y_2))\mbox{ or }g(y_2,z(y_1,y_2))\mbox{ or }f(b,y_2) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( g(b,z(y_1,y_2))\mbox{ and }g(y_2,z(y_1,y_2))\mbox{ and }f(y_1,y_2) ) \ \Rightarrow \ f(b,y_2)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$g(y_1,z(y_1,y_2)) \ \Rightarrow \ g(b,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$g(b,z(y_1,y_2)) \ \Rightarrow \ g(y_1,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( f(a,y_1)\mbox{ and }f(b,y_1) ) \ \Rightarrow \ \neg\,f(y_1,y_2)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN362+1.p} Peter Andrews Problem X2113






$( \forall y{:}\;\exists w{:}\;\mbox{big\_r}(y,w)\mbox{ and }\exists z{:}\;\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \neg\,\mbox{big\_r}(z,x) ) ) \ \Rightarrow \ \exists x{:}\;\neg\,\mbox{big\_p}(x)\qquad\mbox{fof}(x_{2113},\mbox{conjecture})$

\medskip\textbf{SYN306-1.p} Problem for testing satisfiability






$p(x)\mbox{ or }q(g(x,x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q(y) \ \Rightarrow \ r(x,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(a,a) \ \Rightarrow \ \neg\,r(f(b),a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$r(f(x),y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$p(f(x)) \ \Rightarrow \ p(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\neg\,p(a)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN583-1.p} Harrison problem 3468






$p_7(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p7}_1,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_2(c_{13},c_{18})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(c_{18},c_{14})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(c_{18}),f_8(f_9(c_{17})))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{14})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_8,\mbox{negated\_conjecture})$

$\neg\,p_{11}(f_4(f_6(c_{15},c_{18})),c_{16})\qquad\mbox{cnf}(\mbox{not\_p11}_9,\mbox{negated\_conjecture})$

$p_{12}(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_2(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_7(x_{37},x_{38}) \ \Rightarrow \ p_7(f_9(x_{37}),f_9(x_{38}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_3(x_{21}),f_3(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_2(x_{23},x_{24}) \ \Rightarrow \ p_2(f_4(x_{23}),f_4(x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_7(x_{29},x_{30}) \ \Rightarrow \ p_2(f_8(x_{29}),f_8(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_{11},x_{12}) \ \Rightarrow \ ( p_{12}(x_{11},x_{12})\mbox{ or }p_2(x_{11},x_{12}) )\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$( p_7(x_{34},x_{35})\mbox{ and }p_7(x_{34},x_{36}) ) \ \Rightarrow \ p_7(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_6,c_{14})\mbox{ and }p_{11}(c_{13},x_6) ) \ \Rightarrow \ p_{10}(c_{15},f_3(x_6),x_6)\qquad\mbox{cnf}(\mbox{p10}_{20},\mbox{negated\_conjecture})$

$( p_2(x_9,x_7)\mbox{ and }p_2(x_{10},x_8)\mbox{ and }p_{11}(x_9,x_{10}) ) \ \Rightarrow \ p_{11}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p11}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{14})\mbox{ and }p_2(x_{17},x_{15})\mbox{ and }p_{12}(x_{16},x_{17}) ) \ \Rightarrow \ p_{12}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p12}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_5(x_{25},x_{27}) ) \ \Rightarrow \ p_2(f_6(x_{25},x_{26}),f_6(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_5(x_3,x_0)\mbox{ and }p_2(x_5,x_2)\mbox{ and }p_2(x_4,x_1)\mbox{ and }p_{10}(x_3,x_4,x_5) ) \ \Rightarrow \ p_{10}(x_0,x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{24},\mbox{negated\_conjecture})$

$( p_{12}(x_{13},c_{14})\mbox{ and }p_{12}(c_{13},x_{13})\mbox{ and }p_2(f_3(x_{13}),f_8(f_9(c_{17}))) ) \ \Rightarrow \ p_{11}(f_4(f_6(c_{15},x_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{SYN063+1.p} Pelletier Problem 33






$\forall x{:}\;( ( \mbox{big\_p}(a)\mbox{ and }( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_p}(b) ) ) \ \Rightarrow \ \mbox{big\_p}(c) )\iff \forall x_1{:}\;( ( \neg\,\mbox{big\_p}(a)\mbox{ or }\mbox{big\_p}(x_1)\mbox{ or }\mbox{big\_p}(c) )\mbox{ and }( \neg\,\mbox{big\_p}(a)\mbox{ or }\neg\,\mbox{big\_p}(b)\mbox{ or }\mbox{big\_p}(c) ) )\qquad\mbox{fof}(\mbox{pel}_{33},\mbox{conjecture})$

\medskip\textbf{SYN382+1.p} Peter Andrews Problem X2134






$\forall z{:}\;\exists x{:}\;( \forall y{:}\;\mbox{big\_p}(x,y)\mbox{ or }\mbox{big\_q}(x,z) ) \ \Rightarrow \ \forall y{:}\;\exists x{:}\;( \mbox{big\_p}(x,y)\mbox{ or }\mbox{big\_q}(x,y) )\qquad\mbox{fof}(x_{2134},\mbox{conjecture})$

\medskip\textbf{SYN081-1.p} Pelletier Problem 59






$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,\mbox{big\_f}(f(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(f(x))\mbox{ or }\mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_f}(f(x))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN407+1.p} Kalish and Montague Problem 241

Laws of alphabetic variance.




$\forall x{:}\;( f(x) \ \Rightarrow \ ( g(x)\mbox{ or }h(x) ) ) \ \Rightarrow \ ( \forall y{:}\;( f(y) \ \Rightarrow \ g(y) )\mbox{ or }\exists z{:}\;( f(z)\mbox{ and }h(z) ) )\qquad\mbox{fof}(\mbox{kalish}_{241},\mbox{conjecture})$

\medskip\textbf{SYN607-1.p} Harrison problem 3827






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_7,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{15}(c_{22}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{27},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{28},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN976+1.p} Syntactic from Shults






$\forall a{:}\;( ( ( f\mbox{ or }g )\mbox{ and }\forall x{:}\;( p(x)\mbox{ and }q(x) ) ) \ \Rightarrow \ q(a) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN927+1.p} From Smullyan






$\forall y{:}\;( \forall x{:}\;p(x) \ \Rightarrow \ p(y) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN315+1.p} Church problem 46.2 (1)






$\exists x{:}\;\forall y{:}\;( ( \mbox{big\_f}(x)\iff p ) \ \Rightarrow \ ( \mbox{big\_f}(y)\iff p ) )\qquad\mbox{fof}(\mbox{church\_46\_2}_1,\mbox{conjecture})$

\medskip\textbf{SYN336+1.p} Church problem 46.15 (1)






$\forall x{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( \mbox{big\_f}(x,z) \ \Rightarrow \ ( \mbox{big\_f}(y_1,z) \ \Rightarrow \ ( \mbox{big\_f}(y_2,z) \ \Rightarrow \ ( \mbox{big\_f}(y_1,x) \ \Rightarrow \ \mbox{big\_f}(z,y_2) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_15}_1,\mbox{conjecture})$

\medskip\textbf{SYN917+1.p} Combined problems from Smullyan






$( ( \forall x{:}\;( ( ( f(x)\mbox{ and }g(x) ) \ \Rightarrow \ h(x) ) \ \Rightarrow \ \exists y{:}\;( f(y)\mbox{ and }\neg\,g(y) ) )\mbox{ and }( \forall w{:}\;( f(w) \ \Rightarrow \ g(w) )\mbox{ or }\forall z{:}\;( f(z) \ \Rightarrow \ h(z) ) ) ) \ \Rightarrow \ ( \forall r{:}\;( ( f(r)\mbox{ and }h(r) ) \ \Rightarrow \ g(r) ) \ \Rightarrow \ \exists v{:}\;( f(v)\mbox{ and }g(v)\mbox{ and }\neg\,h(v) ) ) )\mbox{ and }( ( \forall x,y{:}\;( r(x,y) \ \Rightarrow \ r(y,x) )\mbox{ and }\forall x,y,z{:}\;( ( r(x,y)\mbox{ and }r(y,z) ) \ \Rightarrow \ r(x,z) ) ) \ \Rightarrow \ \forall x,y{:}\;( r(x,y) \ \Rightarrow \ r(x,x) ) )\mbox{ and }( ( ( \forall x{:}\;( ( f(x)\mbox{ and }g(x) ) \ \Rightarrow \ h(x) ) \ \Rightarrow \ \exists x{:}\;( f(x)\mbox{ and }\neg\,g(x) ) )\mbox{ and }( \forall w{:}\;( f(w) \ \Rightarrow \ g(w) )\mbox{ or }\forall z{:}\;( f(z) \ \Rightarrow \ h(z) ) ) ) \ \Rightarrow \ ( \forall r{:}\;( ( f(r)\mbox{ and }h(r) ) \ \Rightarrow \ g(r) ) \ \Rightarrow \ \exists v{:}\;( f(v)\mbox{ and }g(v)\mbox{ and }\neg\,h(v) ) ) )\mbox{ and }\exists x{:}\;\forall y{:}\;( ( p(y) \ \Rightarrow \ q(x) ) \ \Rightarrow \ ( p(x) \ \Rightarrow \ q(x) ) )\mbox{ and }( \forall x{:}\;( p(x)\mbox{ and }q(x) )\iff ( \forall x{:}\;p(x)\mbox{ and }\forall x{:}\;q(x) ) )\mbox{ and }( ( \forall x{:}\;p(x)\mbox{ or }\forall x{:}\;q(x) ) \ \Rightarrow \ \forall x{:}\;( p(x)\mbox{ or }q(x) ) )\mbox{ and }( \exists x{:}\;( p(x)\mbox{ or }q(x) )\iff ( \exists x{:}\;p(x)\mbox{ or }\exists x{:}\;q(x) ) )\mbox{ and }\exists y{:}\;( p(y) \ \Rightarrow \ \forall x{:}\;p(x) )\mbox{ and }( \exists x{:}\;( p(x)\mbox{ and }q(x) ) \ \Rightarrow \ ( \exists x{:}\;p(x)\mbox{ and }\exists x{:}\;q(x) ) )\mbox{ and }\forall y{:}\;( \forall x{:}\;p(x) \ \Rightarrow \ p(y) )\mbox{ and }( \forall x{:}\;p(x) \ \Rightarrow \ \exists x{:}\;p(x) )\mbox{ and }( \neg\,\exists y{:}\;p(y) \ \Rightarrow \ \forall y{:}\;( \exists x{:}\;p(x) \ \Rightarrow \ p(y) ) )\mbox{ and }( \forall x{:}\;( p(x)\mbox{ or }c )\iff ( \forall x{:}\;p(x)\mbox{ or }c ) )\mbox{ and }( \exists x{:}\;( p(x)\mbox{ and }c )\iff ( \exists x{:}\;p(x)\mbox{ and }c ) )\mbox{ and }( \exists x{:}\;c\iff c )\mbox{ and }( \forall x{:}\;c\iff c )\mbox{ and }( \exists x{:}\;( c \ \Rightarrow \ p(x) )\iff ( c \ \Rightarrow \ \exists x{:}\;p(x) ) )\mbox{ and }( \exists x{:}\;( p(x) \ \Rightarrow \ c )\iff ( \forall x{:}\;p(x) \ \Rightarrow \ c ) )\mbox{ and }( \forall x{:}\;( c \ \Rightarrow \ p(x) )\iff ( c \ \Rightarrow \ \forall x{:}\;p(x) ) )\mbox{ and }( \forall x{:}\;( p(x) \ \Rightarrow \ c )\iff ( \exists x{:}\;p(x) \ \Rightarrow \ c ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN731$\wedge$5.p} TPS problem X2150






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists w{:}\;\mbox{\$i}{:}\;( \forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ w@ x@ y ) \ \Rightarrow \ \exists z{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ z@ z@ w ) )\qquad\mbox{thf}(\mbox{cX}_{2150},\mbox{conjecture})$

\medskip\textbf{SYN291-1.p} RPT63 synthetic problem 76 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(d,x,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN051$\wedge$5.p} TPS problem PELL21






$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( p \ \Rightarrow \ ( \mbox{cF}@ \mbox{xx} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ p ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( p \ \Rightarrow \ ( \mbox{cF}@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ p ) )\qquad\mbox{thf}(\mbox{cPELL}_{21},\mbox{conjecture})$

\medskip\textbf{SYN275-1.p} RPT63 synthetic problem 60 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_0(d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN983$\wedge$1.p} Factoring application over conjunction






$a{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{a\_const},\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{b\_const},\mbox{type})$

$p{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_const},\mbox{type})$

$( p@ a\mbox{ and }p@ b ) \ \Rightarrow \ ( p@ ( a\mbox{ and }b ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN230-1.p} RPT63 synthetic problem 15 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN949+1.p} Syntactic from Shults






$\forall a,b,c{:}\;( ( s(a)\mbox{ and }s(b)\mbox{ and }r(b,c)\mbox{ and }\forall x{:}\;( s(x) \ \Rightarrow \ p(x) )\mbox{ and }\forall x,y{:}\;( r(x,y) \ \Rightarrow \ q(x,y) ) ) \ \Rightarrow \ \exists x,y{:}\;( p(x)\mbox{ and }q(x,y) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN059$\wedge$5.p} TPS problem PELL29






$\mbox{cJ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cH}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cH},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx} )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ \mbox{xx} ) ) \ \Rightarrow \ ( ( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cH}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cG}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cJ}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx}\mbox{ and }\mbox{cG}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cH}@ \mbox{xx}\mbox{ and }\mbox{cJ}@ \mbox{xy} ) ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx}\mbox{ and }\mbox{cG}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cH}@ \mbox{xx}\mbox{ and }\mbox{cJ}@ \mbox{xy} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cG}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cJ}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{29},\mbox{conjecture})$

\medskip\textbf{SYN036-3.p} Andrews Challenge Problem






$( n_2\mbox{ and }n_9\mbox{ and }n_6 ) \ \Rightarrow \ \neg\,n_{10}\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_9 ) \ \Rightarrow \ ( n_6\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( n_6\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_2\mbox{ or }n_9 )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$n_2\mbox{ or }n_9\mbox{ or }n_6\mbox{ or }n_{10}\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_6 ) \ \Rightarrow \ ( n_9\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_9\mbox{ or }n_6 )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( n_9\mbox{ and }n_6 ) \ \Rightarrow \ ( n_2\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( n_9\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_2\mbox{ or }n_6 )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$p(s_8) \ \Rightarrow \ n_8\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$n_8 \ \Rightarrow \ p(x_8)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$p(x_7) \ \Rightarrow \ n_7\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$n_7 \ \Rightarrow \ p(s_7)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$n_5(x_6) \ \Rightarrow \ n_6\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$n_6 \ \Rightarrow \ n_5(s_6)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( q(x)\mbox{ and }q(s_5(x)) ) \ \Rightarrow \ n_5(x)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$q(x)\mbox{ or }q(s_5(x))\mbox{ or }n_5(x)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( n_5(x)\mbox{ and }q(x) ) \ \Rightarrow \ q(x_5)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( n_5(x)\mbox{ and }q(x_5) ) \ \Rightarrow \ q(x)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$q(s_4) \ \Rightarrow \ n_4\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$n_4 \ \Rightarrow \ q(x_4)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$q(x_3) \ \Rightarrow \ n_3\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$n_3 \ \Rightarrow \ q(s_3)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

$n_1(x_2) \ \Rightarrow \ n_2\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{negated\_conjecture})$

$n_2 \ \Rightarrow \ n_1(s_2)\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{negated\_conjecture})$

$( p(x)\mbox{ and }p(s_1(x)) ) \ \Rightarrow \ n_1(x)\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{negated\_conjecture})$

$p(x)\mbox{ or }p(s_1(x))\mbox{ or }n_1(x)\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{negated\_conjecture})$

$( n_1(x)\mbox{ and }p(x) ) \ \Rightarrow \ p(x_1)\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{negated\_conjecture})$

$( n_1(x)\mbox{ and }p(x_1) ) \ \Rightarrow \ p(x)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{negated\_conjecture})$

$( n_3\mbox{ and }n_4 ) \ \Rightarrow \ n_9\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{negated\_conjecture})$

$n_3\mbox{ or }n_4\mbox{ or }n_9\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{negated\_conjecture})$

$( n_3\mbox{ and }n_9 ) \ \Rightarrow \ n_4\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{negated\_conjecture})$

$( n_4\mbox{ and }n_9 ) \ \Rightarrow \ n_3\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{negated\_conjecture})$

$( n_7\mbox{ and }n_8 ) \ \Rightarrow \ n_{10}\qquad\mbox{cnf}(\mbox{clause}_{33},\mbox{negated\_conjecture})$

$n_7\mbox{ or }n_8\mbox{ or }n_{10}\qquad\mbox{cnf}(\mbox{clause}_{34},\mbox{negated\_conjecture})$

$( n_7\mbox{ and }n_{10} ) \ \Rightarrow \ n_8\qquad\mbox{cnf}(\mbox{clause}_{35},\mbox{negated\_conjecture})$

$( n_8\mbox{ and }n_{10} ) \ \Rightarrow \ n_7\qquad\mbox{cnf}(\mbox{clause}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{SYN153-1.p} RPT63 synthetic problem 51 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_3(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN074+1.p} Pelletier Problem 51






$\exists z,w{:}\;\forall x,y{:}\;( \mbox{big\_f}(x,y)\iff ( x = z\mbox{ and }y = w ) )\qquad\mbox{fof}(\mbox{pel51}_1,\mbox{axiom})$

$\exists z{:}\;\forall x{:}\;( \exists w{:}\;\forall y{:}\;( \mbox{big\_f}(x,y)\iff y = w )\iff x = z )\qquad\mbox{fof}(\mbox{pel}_{51},\mbox{conjecture})$

\medskip\textbf{SYN568-1.p} Harrison problem 3076






$p_2(x_{17},x_{17})\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_3(x_{28},x_{28})\qquad\mbox{cnf}(\mbox{p3}_2,\mbox{negated\_conjecture})$

$\neg\,p_{11}(f_9(c_{13}),f_9(c_{14}))\qquad\mbox{cnf}(\mbox{not\_p11}_3,\mbox{negated\_conjecture})$

$p_{11}(f_5(f_6(c_{12})),f_9(x_{16}))\qquad\mbox{cnf}(\mbox{p11}_4,\mbox{negated\_conjecture})$

$p_{10}(x_4,x_5)\mbox{ or }p_{11}(x_5,x_4)\qquad\mbox{cnf}(\mbox{p10}_5,\mbox{negated\_conjecture})$

$p_{10}(x_4,x_5) \ \Rightarrow \ \neg\,p_{11}(x_5,x_4)\qquad\mbox{cnf}(\mbox{not\_p10}_6,\mbox{negated\_conjecture})$

$\neg\,p_3(f_6(f_7(f_8(c_{12}))),f_6(c_{12}))\qquad\mbox{cnf}(\mbox{not\_p3}_7,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{36}) \ \Rightarrow \ p_3(f_8(x_{35}),f_8(x_{36}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_3(x_{24},x_{25}) \ \Rightarrow \ p_2(f_5(x_{24}),f_5(x_{25}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{26},x_{27}) \ \Rightarrow \ p_2(f_9(x_{26}),f_9(x_{27}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_3(x_{31},x_{32}) \ \Rightarrow \ p_3(f_6(x_{31}),f_6(x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_3(x_{33},x_{34}) \ \Rightarrow \ p_3(f_7(x_{33}),f_7(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$( p_2(x_{17},x_{18})\mbox{ and }p_2(x_{17},x_{19}) ) \ \Rightarrow \ p_2(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_3(x_{28},x_{29})\mbox{ and }p_3(x_{28},x_{30}) ) \ \Rightarrow \ p_3(x_{29},x_{30})\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_9)\mbox{ and }p_2(x_{12},x_{10})\mbox{ and }p_{11}(x_{11},x_{12}) ) \ \Rightarrow \ p_{11}(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{22})\mbox{ and }p_3(x_{21},x_{23}) ) \ \Rightarrow \ p_2(f_4(x_{20},x_{21}),f_4(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$p_{11}(f_4(f_9(c_{13}),f_6(f_7(f_8(c_{12})))),f_4(f_9(c_{14}),f_6(f_7(f_8(c_{12})))))\qquad\mbox{cnf}(\mbox{p11}_{18},\mbox{negated\_conjecture})$

$( p_{11}(x_{13},x_{15})\mbox{ and }p_{11}(f_5(f_6(c_{12})),x_{13}) ) \ \Rightarrow \ p_{11}(f_4(x_{13},x_{14}),f_4(x_{15},x_{14}))\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$( p_{10}(x_6,x_8)\mbox{ and }p_{11}(f_5(f_6(c_{12})),x_6) ) \ \Rightarrow \ ( p_3(x_7,f_6(c_{12}))\mbox{ or }p_{10}(f_4(x_6,x_7),f_4(x_8,x_7)) )\qquad\mbox{cnf}(\mbox{p3}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{SYN241-1.p} RPT63 synthetic problem 26 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(a,a,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN037-1.p} Andrews Challenge Problem Variant






$( n_2\mbox{ and }n_9\mbox{ and }n_6 ) \ \Rightarrow \ \neg\,n_{10}\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_9 ) \ \Rightarrow \ ( n_6\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( n_6\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_2\mbox{ or }n_9 )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$n_2\mbox{ or }n_9\mbox{ or }n_6\mbox{ or }n_{10}\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_6 ) \ \Rightarrow \ ( n_9\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( n_2\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_9\mbox{ or }n_6 )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( n_9\mbox{ and }n_6 ) \ \Rightarrow \ ( n_2\mbox{ or }n_{10} )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( n_9\mbox{ and }n_{10} ) \ \Rightarrow \ ( n_2\mbox{ or }n_6 )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$q(s_8) \ \Rightarrow \ n_8\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$n_8 \ \Rightarrow \ q(x_{10})\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$p(x_9) \ \Rightarrow \ n_7\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$n_7 \ \Rightarrow \ p(s_7)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$n_5(x_8) \ \Rightarrow \ n_6\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$n_6 \ \Rightarrow \ n_5(s_6)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( q(s_5(w))\mbox{ and }q(w) ) \ \Rightarrow \ n_5(w)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$q(s_5(w))\mbox{ or }q(w)\mbox{ or }n_5(w)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( n_5(w)\mbox{ and }q(x_7) ) \ \Rightarrow \ q(w)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( n_5(w)\mbox{ and }q(w) ) \ \Rightarrow \ q(x_7)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$p(s_4) \ \Rightarrow \ n_4\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$n_4 \ \Rightarrow \ p(x_6)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$q(x_5) \ \Rightarrow \ n_3\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$n_3 \ \Rightarrow \ q(s_3)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

$n_1(x_4) \ \Rightarrow \ n_2\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{negated\_conjecture})$

$n_2 \ \Rightarrow \ n_1(s_2)\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{negated\_conjecture})$

$( p(x)\mbox{ and }p(s_1(x)) ) \ \Rightarrow \ n_1(x)\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{negated\_conjecture})$

$p(x)\mbox{ or }p(s_1(x))\mbox{ or }n_1(x)\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{negated\_conjecture})$

$( n_1(x)\mbox{ and }p(x) ) \ \Rightarrow \ p(x_3)\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{negated\_conjecture})$

$( n_1(x)\mbox{ and }p(x_3) ) \ \Rightarrow \ p(x)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{negated\_conjecture})$

$( n_3\mbox{ and }n_4 ) \ \Rightarrow \ n_9\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{negated\_conjecture})$

$n_3\mbox{ or }n_4\mbox{ or }n_9\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{negated\_conjecture})$

$( n_3\mbox{ and }n_9 ) \ \Rightarrow \ n_4\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{negated\_conjecture})$

$( n_4\mbox{ and }n_9 ) \ \Rightarrow \ n_3\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{negated\_conjecture})$

$( n_7\mbox{ and }n_8 ) \ \Rightarrow \ n_{10}\qquad\mbox{cnf}(\mbox{clause}_{33},\mbox{negated\_conjecture})$

$n_7\mbox{ or }n_8\mbox{ or }n_{10}\qquad\mbox{cnf}(\mbox{clause}_{34},\mbox{negated\_conjecture})$

$( n_7\mbox{ and }n_{10} ) \ \Rightarrow \ n_8\qquad\mbox{cnf}(\mbox{clause}_{35},\mbox{negated\_conjecture})$

$( n_8\mbox{ and }n_{10} ) \ \Rightarrow \ n_7\qquad\mbox{cnf}(\mbox{clause}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{SYN924+1.p} From Smullyan






$\exists x{:}\;( p(x)\mbox{ or }q(x) )\iff ( \exists x{:}\;p(x)\mbox{ or }\exists x{:}\;q(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN599-1.p} Harrison problem 3757






$p_{17}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p17}_1,\mbox{negated\_conjecture})$

$p_{18}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_p18}_2,\mbox{negated\_conjecture})$

$p_7(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(x_9,x_9)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_6,\mbox{negated\_conjecture})$

$p_3(f_4(f_6(c_{21})),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$( p_{17}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{17}(x_0)\qquad\mbox{cnf}(\mbox{p17}_8,\mbox{negated\_conjecture})$

$( p_{18}(x_3)\mbox{ and }p_5(x_3,x_2) ) \ \Rightarrow \ p_{18}(x_2)\qquad\mbox{cnf}(\mbox{p18}_9,\mbox{negated\_conjecture})$

$p_5(x_{41},x_{42}) \ \Rightarrow \ p_7(f_{16}(x_{41}),f_{16}(x_{42}))\qquad\mbox{cnf}(\mbox{p7}_{10},\mbox{negated\_conjecture})$

$p_5(x_{12},x_{13}) \ \Rightarrow \ p_2(f_6(x_{12}),f_6(x_{13}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{21},x_{22}) \ \Rightarrow \ p_3(f_{10}(x_{21}),f_{10}(x_{22}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_3(x_{23},x_{24}) \ \Rightarrow \ p_3(f_{11}(x_{23}),f_{11}(x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{25},x_{26}) \ \Rightarrow \ p_3(f_{13}(x_{25}),f_{13}(x_{26}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{27},x_{28}) \ \Rightarrow \ p_3(f_{14}(x_{27}),f_{14}(x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_3(f_4(x_{29}),f_4(x_{30}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_3(x_{37},x_{38}) \ \Rightarrow \ p_7(f_{12}(x_{37}),f_{12}(x_{38}))\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$p_7(x_{39},x_{40}) \ \Rightarrow \ p_7(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p7}_{18},\mbox{negated\_conjecture})$

$( p_2(x_9,x_{10})\mbox{ and }p_2(x_9,x_{11}) ) \ \Rightarrow \ p_2(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{34},x_{35})\mbox{ and }p_7(x_{34},x_{36}) ) \ \Rightarrow \ p_7(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{21},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_7(x_6,x_4)\mbox{ and }p_7(x_7,x_5)\mbox{ and }p_{19}(x_6,x_7) ) \ \Rightarrow \ p_{19}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p19}_{23},\mbox{negated\_conjecture})$

$( p_7(x_{16},x_{14})\mbox{ and }p_7(x_{17},x_{15})\mbox{ and }p_{20}(x_{16},x_{17}) ) \ \Rightarrow \ p_{20}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p20}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{48},x_{50})\mbox{ and }p_7(x_{47},x_{49}) ) \ \Rightarrow \ p_7(f_9(x_{47},x_{48}),f_9(x_{49},x_{50}))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$( p_7(x_{43},x_{45})\mbox{ and }p_7(x_{44},x_{46}) ) \ \Rightarrow \ p_7(f_8(x_{43},x_{44}),f_8(x_{45},x_{46}))\qquad\mbox{cnf}(\mbox{p7}_{26},\mbox{negated\_conjecture})$

$\neg\,p_{19}(f_{15}(f_{16}(c_{21})),f_8(f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_{14}(c_{22})),f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_{10}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(c_{23})))))))))))\qquad\mbox{cnf}(\mbox{not\_p19}_{27},\mbox{negated\_conjecture})$

$p_{18}(x_8) \ \Rightarrow \ p_{20}(f_8(f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_4(f_6(x_8))),f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_{10}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(c_{23})))))))))),f_{15}(f_{16}(x_8)))\qquad\mbox{cnf}(\mbox{p20}_{28},\mbox{negated\_conjecture})$

$p_{18}(x_8) \ \Rightarrow \ p_{19}(f_{15}(f_{16}(x_8)),f_8(f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_{14}(f_4(f_6(x_8)))),f_9(f_{12}(f_{10}(f_{13}(f_{11}(c_{23})))),f_{10}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(f_{11}(c_{23})))))))))))\qquad\mbox{cnf}(\mbox{p19}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{SYN988$\wedge$1.p} All things are false






$\forall x{:}\;\mbox{\$o}{:}\;\neg\,x\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN133-1.p} RPT63 synthetic problem 31 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_0(d,x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN345+1.p} Church problem 46.16 (4)






$\forall x_1,x_2{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( ( ( \mbox{big\_f}(x_1,x_2,y_1) \ \Rightarrow \ \mbox{big\_f}(y_2,y_1,z) ) \ \Rightarrow \ ( \mbox{big\_f}(x_1,x_1,x_2) \ \Rightarrow \ \mbox{big\_f}(x_1,x_2,x_2) ) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(x_2,y_1,y_2) \ \Rightarrow \ \mbox{big\_f}(y_1,z,z) ) \ \Rightarrow \ ( \mbox{big\_f}(x_1,x_2,x_2) \ \Rightarrow \ \mbox{big\_f}(x_1,x_1,x_2) ) ) \ \Rightarrow \ ( \mbox{big\_f}(y_1,y_2,z) \ \Rightarrow \ ( \mbox{big\_f}(x_2,x_2,y_1)\mbox{ and }( \mbox{big\_f}(x_1,x_1,x_2)\iff \mbox{big\_f}(x_1,x_2,x_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_16}_4,\mbox{conjecture})$

\medskip\textbf{SYN047$\wedge$4.p} Pelletier Problem 17






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{s\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iatom}@ p )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ r ) ) )@ ( \mbox{iatom}@ s ) )@ ( \mbox{iand}@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ s ) ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ r ) )@ ( \mbox{iatom}@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{17},\mbox{conjecture})$

\medskip\textbf{SYN134-1.p} RPT63 synthetic problem 32 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_2(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN570-1.p} Harrison problem 3089






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_6(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_3(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{11}(c_{12},c_{14})\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_9(c_{15},f_8(c_{13}))\qquad\mbox{cnf}(\mbox{p9}_7,\mbox{negated\_conjecture})$

$p_2(c_{12},f_5(c_{15}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_3(f_4(f_5(x_{17})),x_{17})\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$\neg\,p_9(f_4(c_{12}),f_8(c_{13}))\qquad\mbox{cnf}(\mbox{not\_p9}_{10},\mbox{negated\_conjecture})$

$p_6(x_{24},x_{25}) \ \Rightarrow \ p_7(f_8(x_{24}),f_8(x_{25}))\qquad\mbox{cnf}(\mbox{p7}_{11},\mbox{negated\_conjecture})$

$p_3(x_{10},x_{11}) \ \Rightarrow \ p_2(f_5(x_{10}),f_5(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_3(f_4(x_{15}),f_4(x_{16}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{21},x_{22})\mbox{ and }p_7(x_{21},x_{23}) ) \ \Rightarrow \ p_7(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_6(x_{18},x_{19})\mbox{ and }p_6(x_{18},x_{20}) ) \ \Rightarrow \ p_6(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p6}_{16},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{13})\mbox{ and }p_3(x_{12},x_{14}) ) \ \Rightarrow \ p_3(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{29},x_{27})\mbox{ and }p_9(x_{28},x_{29})\mbox{ and }p_3(x_{28},x_{26}) ) \ \Rightarrow \ p_9(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{SYN009-4.p} A problem to demonstrate the usefulness of relevancy testing






$p(x,y,z) \ \Rightarrow \ \neg\,m(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q(x,y,z) \ \Rightarrow \ \neg\,m(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(x,y,z) \ \Rightarrow \ \neg\,m(x,y,z)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$s(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$s(c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,n(c,c,c)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( s(x)\mbox{ and }s(y)\mbox{ and }s(z) ) \ \Rightarrow \ ( p(x,y,z)\mbox{ or }q(x,y,z)\mbox{ or }r(x,y,z) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( s(x)\mbox{ and }s(y)\mbox{ and }s(z) ) \ \Rightarrow \ ( m(x,y,z)\mbox{ or }n(x,y,z) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN114-1.p} RPT63 synthetic problem 12 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_3(x,x,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN008-1.p} A problem to demonstrate the usefulness of relevancy testing






$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,q\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ ( a\mbox{ or }b )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ ( p\mbox{ or }c\mbox{ or }d )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ ( p\mbox{ or }q )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN564-1.p} Harrison problem 2888






$p_2(x_9,x_9)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_4(x_{23},x_{23})\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$p_2(f_5(f_7(x_{18})),f_5(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_4(f_8(x_{38},x_{39}),f_6(x_{38},f_7(x_{39})))\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_4(f_7(f_8(x_{32},x_{33})),f_8(x_{33},x_{32}))\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_4(x_{16},x_{17}) \ \Rightarrow \ p_2(f_5(x_{16}),f_5(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_4(x_{30},x_{31}) \ \Rightarrow \ p_4(f_7(x_{30}),f_7(x_{31}))\qquad\mbox{cnf}(\mbox{p4}_7,\mbox{negated\_conjecture})$

$( p_2(x_9,x_{10})\mbox{ and }p_2(x_9,x_{11}) ) \ \Rightarrow \ p_2(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_4(x_{23},x_{24})\mbox{ and }p_4(x_{23},x_{25}) ) \ \Rightarrow \ p_4(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p4}_9,\mbox{negated\_conjecture})$

$p_{10}(f_5(x_7),f_3(f_5(f_6(x_7,x_8)),f_5(x_8)))\qquad\mbox{cnf}(\mbox{p10}_{10},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_9(f_5(c_{11}),f_5(c_{12})),f_5(f_8(c_{11},c_{12})))\qquad\mbox{cnf}(\mbox{not\_p10}_{11},\mbox{negated\_conjecture})$

$p_{10}(f_9(x_4,x_6),x_5) \ \Rightarrow \ p_{10}(x_4,f_3(x_5,x_6))\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$p_{10}(x_4,f_3(x_5,x_6)) \ \Rightarrow \ p_{10}(f_9(x_4,x_6),x_5)\qquad\mbox{cnf}(\mbox{p10}_{13},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_4(x_{34},x_{36})\mbox{ and }p_4(x_{35},x_{37}) ) \ \Rightarrow \ p_4(f_8(x_{34},x_{35}),f_8(x_{36},x_{37}))\qquad\mbox{cnf}(\mbox{p4}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(f_3(x_{12},x_{13}),f_3(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{21})\mbox{ and }p_2(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_9(x_{19},x_{20}),f_9(x_{21},x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_4(x_{26},x_{28})\mbox{ and }p_4(x_{27},x_{29}) ) \ \Rightarrow \ p_4(f_6(x_{26},x_{27}),f_6(x_{28},x_{29}))\qquad\mbox{cnf}(\mbox{p4}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{SYN582-1.p} Harrison problem 3467






$p_7(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p7}_1,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_{12}(c_{18},c_{14})\qquad\mbox{cnf}(\mbox{p12}_4,\mbox{negated\_conjecture})$

$p_2(c_{13},c_{18})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(c_{18}),f_8(f_9(c_{17})))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{14})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_8,\mbox{negated\_conjecture})$

$\neg\,p_{11}(f_4(f_6(c_{15},c_{18})),c_{16})\qquad\mbox{cnf}(\mbox{not\_p11}_9,\mbox{negated\_conjecture})$

$p_{12}(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_2(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_7(x_{37},x_{38}) \ \Rightarrow \ p_7(f_9(x_{37}),f_9(x_{38}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_3(x_{21}),f_3(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_2(x_{23},x_{24}) \ \Rightarrow \ p_2(f_4(x_{23}),f_4(x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_7(x_{29},x_{30}) \ \Rightarrow \ p_2(f_8(x_{29}),f_8(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_{11},x_{12}) \ \Rightarrow \ ( p_{12}(x_{11},x_{12})\mbox{ or }p_2(x_{11},x_{12}) )\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$( p_7(x_{34},x_{35})\mbox{ and }p_7(x_{34},x_{36}) ) \ \Rightarrow \ p_7(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_6,c_{14})\mbox{ and }p_{11}(c_{13},x_6) ) \ \Rightarrow \ p_{10}(c_{15},f_3(x_6),x_6)\qquad\mbox{cnf}(\mbox{p10}_{20},\mbox{negated\_conjecture})$

$( p_2(x_9,x_7)\mbox{ and }p_2(x_{10},x_8)\mbox{ and }p_{11}(x_9,x_{10}) ) \ \Rightarrow \ p_{11}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p11}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{14})\mbox{ and }p_2(x_{17},x_{15})\mbox{ and }p_{12}(x_{16},x_{17}) ) \ \Rightarrow \ p_{12}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p12}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_5(x_{25},x_{27}) ) \ \Rightarrow \ p_2(f_6(x_{25},x_{26}),f_6(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_5(x_3,x_0)\mbox{ and }p_2(x_5,x_2)\mbox{ and }p_2(x_4,x_1)\mbox{ and }p_{10}(x_3,x_4,x_5) ) \ \Rightarrow \ p_{10}(x_0,x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{24},\mbox{negated\_conjecture})$

$( p_{12}(x_{13},c_{14})\mbox{ and }p_{12}(c_{13},x_{13})\mbox{ and }p_2(f_3(x_{13}),f_8(f_9(c_{17}))) ) \ \Rightarrow \ p_{11}(f_4(f_6(c_{15},x_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{SYN014-1.p} A problem in quantification theory






$m\ne n\qquad\mbox{cnf}(c_1,\mbox{negated\_conjecture})$

$n = k\mbox{ or }m = k\qquad\mbox{cnf}(c_2,\mbox{negated\_conjecture})$

$y = k \ \Rightarrow \ ( y = j\mbox{ or }\mbox{element}(y,j) )\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$\mbox{element}(y,j) \ \Rightarrow \ ( y = j\mbox{ or }y = k )\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = m ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = y ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,f(y)) )\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(f(y),y) )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$( \mbox{element}(y,v_1)\mbox{ and }\mbox{element}(v_1,y) ) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,m)\mbox{ or }v_1 = m\mbox{ or }v_1 = y )\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$g(y) = n \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$g(y) = y \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(y,g(y))\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(g(y),y)\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$( \mbox{element}(y,n)\mbox{ and }\mbox{element}(y,v)\mbox{ and }\mbox{element}(v,y) ) \ \Rightarrow \ ( y = n\mbox{ or }v = n\mbox{ or }v = y )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$y = m \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$y = n \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

$\mbox{element}(y,k) \ \Rightarrow \ ( y = k\mbox{ or }y = m\mbox{ or }y = n )\qquad\mbox{cnf}(c_{17},\mbox{negated\_conjecture})$

\medskip\textbf{SYN318+1.p} Church problem 46.2 (4)






$\exists y{:}\;( \forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_g}(x) ) ) \ \Rightarrow \ ( p \ \Rightarrow \ \forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(y) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_2}_4,\mbox{conjecture})$

\medskip\textbf{SYN347-1.p} Church problem 46.17 (3)






$f(x_2,y(x_1,x_2)) \ \Rightarrow \ ( f(x_1,y(x_1,x_2))\mbox{ or }f(a,b) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x_1,y(x_1,x_2)) \ \Rightarrow \ ( f(x_2,y(x_1,x_2))\mbox{ or }f(a,b) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(x_1,y(x_1,x_2))\mbox{ and }f(x_2,y(x_1,x_2)) ) \ \Rightarrow \ \neg\,f(a,b)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(a,b) \ \Rightarrow \ ( f(x_1,y(x_1,x_2))\mbox{ or }f(x_2,y(x_1,x_2)) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(a,y(x_1,x_2))\mbox{ or }f(b,y(x_1,x_2))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$f(a,y(x_1,x_2)) \ \Rightarrow \ \neg\,f(b,y(x_1,x_2))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN929+1.p} From Smullyan






$\neg\,\exists y{:}\;p(y) \ \Rightarrow \ \forall y{:}\;( \exists x{:}\;p(x) \ \Rightarrow \ p(y) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN131-1.p} RPT63 synthetic problem 29 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_0(x,y,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN036$\wedge$5.p} TPS problem X2129






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} )\iff ( \mbox{cP}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} )\iff ( \mbox{cQ}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cX}_{2129},\mbox{conjecture})$

\medskip\textbf{SYN320-1.p} Church problem 46.3 (1)






$f(x,a)\mbox{ or }f(a,y_1(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y_2(x),a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,f(a,y_2(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN215-1.p} RPT63 synthetic problem 113 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_5(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN351+1.p} Church problem 46.18 (3)






$\forall x_1,x_2{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( \mbox{big\_f}(x_1,y_2,x_1,z) \ \Rightarrow \ ( ( \mbox{big\_f}(x_1,y_1,x_1,y_2)\iff \mbox{big\_f}(y_1,x_2,y_1,y_2) ) \ \Rightarrow \ ( \mbox{big\_f}(x_1,y_1,x_1,y_2) \ \Rightarrow \ ( ( \mbox{big\_f}(x_1,y_2,y_1,y_2) \ \Rightarrow \ \mbox{big\_f}(x_1,z,y_1,z) )\mbox{ and }( \mbox{big\_f}(x_1,z,y_1,z) \ \Rightarrow \ ( \mbox{big\_f}(x_1,y_1,x_1,y_2)\iff \mbox{big\_f}(x_1,y_2,y_1,y_2) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_18}_3,\mbox{conjecture})$

\medskip\textbf{SYN192-1.p} RPT63 synthetic problem 90 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_4(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN045$\wedge$4.p} Pelletier Problem 13






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iand}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ r ) ) )@ ( \mbox{iand}@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ r ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{13},\mbox{conjecture})$

\medskip\textbf{SYN994$\wedge$1.p} Skolemization test 2






$q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_decl},\mbox{type})$

$\exists y{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( \forall z{:}\;\mbox{\$i}{:}\;( q@ x@ z )\mbox{ or }\neg\,q@ x@ y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN328+1.p} Church problem 46.12 (3)






$\exists x{:}\;\forall y,z{:}\;( ( ( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_g}(y) )\iff \mbox{big\_f}(x) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_h}(y) )\iff \mbox{big\_g}(x) ) \ \Rightarrow \ ( ( ( ( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_g}(y) ) \ \Rightarrow \ \mbox{big\_h}(y) )\iff \mbox{big\_h}(x) ) \ \Rightarrow \ ( \mbox{big\_f}(z)\mbox{ and }\mbox{big\_g}(z)\mbox{ and }\mbox{big\_h}(z) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_12}_3,\mbox{conjecture})$

\medskip\textbf{SYN085-1.010.p} Plaisted problem s(1,10)






$\neg\,p_0\qquad\mbox{cnf}(\mbox{s1\_goal}_1,\mbox{negated\_conjecture})$

$( p_1\mbox{ and }p_2\mbox{ and }p_3\mbox{ and }p_4\mbox{ and }p_5\mbox{ and }p_6\mbox{ and }p_7\mbox{ and }p_8\mbox{ and }p_9\mbox{ and }p_{10} ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{s1}_1,\mbox{axiom})$

$p_1\qquad\mbox{cnf}(\mbox{s1}_2,\mbox{axiom})$

$p_2\qquad\mbox{cnf}(\mbox{s1}_3,\mbox{axiom})$

$p_3\qquad\mbox{cnf}(\mbox{s1}_4,\mbox{axiom})$

$p_4\qquad\mbox{cnf}(\mbox{s1}_5,\mbox{axiom})$

$p_5\qquad\mbox{cnf}(\mbox{s1}_6,\mbox{axiom})$

$p_6\qquad\mbox{cnf}(\mbox{s1}_7,\mbox{axiom})$

$p_7\qquad\mbox{cnf}(\mbox{s1}_8,\mbox{axiom})$

$p_8\qquad\mbox{cnf}(\mbox{s1}_9,\mbox{axiom})$

$p_9\qquad\mbox{cnf}(\mbox{s1}_{10},\mbox{axiom})$

$p_{10}\qquad\mbox{cnf}(\mbox{s1}_{11},\mbox{axiom})$

\medskip\textbf{SYN113-1.p} RPT63 synthetic problem 11 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(e,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN322+1.p} Church problem 46.4 (1)






$\exists y{:}\;( \forall x{:}\;( \mbox{big\_f}(x,y) \ \Rightarrow \ \mbox{big\_f}(y,x) ) \ \Rightarrow \ \neg\,\forall x{:}\;( \mbox{big\_f}(x,y) \ \Rightarrow \ \neg\,\mbox{big\_f}(y,x) ) )\qquad\mbox{fof}(\mbox{church\_46\_4}_1,\mbox{conjecture})$

\medskip\textbf{SYN057-1.p} Pelletier Problem 27






$\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\neg\,\mbox{big\_g}(a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( \mbox{big\_j}(x)\mbox{ and }\mbox{big\_i}(x) ) \ \Rightarrow \ \mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( \mbox{big\_h}(x)\mbox{ and }\mbox{big\_i}(y)\mbox{ and }\mbox{big\_h}(y) ) \ \Rightarrow \ \mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_j}(b)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\mbox{big\_i}(b)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN051-1.p} Pelletier Problem 21

A moderately tricky problem, especially for 'natural' systems
 with 'strong' restrictions on variables generated from
 existential quantifiers.




$p \ \Rightarrow \ \mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(b) \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$p\mbox{ or }\mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,p\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN577-1.p} Harrison problem 3300






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_8(c_{14},c_{13})\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_7(c_{11},c_{12})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_7(c_{13},c_{12})\qquad\mbox{cnf}(\mbox{p7}_4,\mbox{negated\_conjecture})$

$p_7(c_9,c_{10})\qquad\mbox{cnf}(\mbox{p7}_5,\mbox{negated\_conjecture})$

$p_8(c_{14},c_{11})\qquad\mbox{cnf}(\mbox{p8}_6,\mbox{negated\_conjecture})$

$p_2(f_3(c_{12},c_{11}),c_{15})\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_2(f_4(c_{16},c_{14}),c_{18})\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_2(f_4(c_{15},c_{14}),c_{17})\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(f_3(c_{12},c_{13}),c_{16})\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(f_5(f_6(c_9),c_{19}),c_{17})\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(f_5(f_6(c_9),c_{20}),c_{18})\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{22},x_{23}) \ \Rightarrow \ p_2(f_6(x_{22}),f_6(x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$( p_2(x_{31},x_{29})\mbox{ and }p_8(x_{30},x_{31})\mbox{ and }p_2(x_{30},x_{28}) ) \ \Rightarrow \ p_8(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p8}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{25})\mbox{ and }p_7(x_{26},x_{27})\mbox{ and }p_2(x_{26},x_{24}) ) \ \Rightarrow \ p_7(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(f_3(x_{10},x_{11}),f_3(x_{12},x_{13}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{21})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(f_5(x_{18},x_{19}),f_5(x_{20},x_{21}))\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{14},x_{16})\mbox{ and }p_2(x_{15},x_{17}) ) \ \Rightarrow \ p_2(f_4(x_{14},x_{15}),f_4(x_{16},x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{34},c_{10})\mbox{ and }p_2(c_{15},x_{32})\mbox{ and }p_2(c_{16},x_{33})\mbox{ and }p_2(f_4(x_{32},c_{14}),f_5(f_6(x_{34}),c_{19})) ) \ \Rightarrow \ \neg\,p_2(f_4(x_{33},c_{14}),f_5(f_6(x_{34}),c_{20}))\qquad\mbox{cnf}(\mbox{not\_p7}_{20},\mbox{negated\_conjecture})$

$( p_7(x_7,c_{10})\mbox{ and }p_7(x_3,c_{12})\mbox{ and }p_7(x_4,c_{12})\mbox{ and }p_8(c_{14},x_4)\mbox{ and }p_8(c_{14},x_3)\mbox{ and }p_2(f_3(c_{12},x_4),x_6)\mbox{ and }p_2(f_3(c_{12},x_3),x_5)\mbox{ and }p_2(f_4(x_5,c_{14}),f_5(f_6(x_7),x_8))\mbox{ and }p_2(f_4(x_6,c_{14}),f_5(f_6(x_7),x_9)) ) \ \Rightarrow \ p_2(x_3,x_4)\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_7(x_7,c_{10})\mbox{ and }p_7(x_3,c_{12})\mbox{ and }p_7(x_4,c_{12})\mbox{ and }p_8(c_{14},x_4)\mbox{ and }p_8(c_{14},x_3)\mbox{ and }p_2(f_3(c_{12},x_4),x_6)\mbox{ and }p_2(f_3(c_{12},x_3),x_5)\mbox{ and }p_2(f_4(x_5,c_{14}),f_5(f_6(x_7),x_8))\mbox{ and }p_2(f_4(x_6,c_{14}),f_5(f_6(x_7),x_9)) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN219-1.p} RPT63 synthetic problem 4 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(e,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN337+1.p} Church problem 46.15 (2)






$\forall x_1,x_2{:}\;\exists y{:}\;\forall z{:}\;( \mbox{big\_f}(x_1,y) \ \Rightarrow \ ( \mbox{big\_f}(z,x_1) \ \Rightarrow \ ( \mbox{big\_f}(z,y) \ \Rightarrow \ ( \mbox{big\_f}(x_2,y)\mbox{ or }\mbox{big\_f}(x_2,z) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_15}_2,\mbox{conjecture})$

\medskip\textbf{SYN277-1.p} RPT63 synthetic problem 62 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN965+1.p} Syntactic from Shults






$\exists z{:}\;\forall x{:}\;\exists y{:}\;( ( p(y,x) \ \Rightarrow \ \exists w{:}\;p(w,y) )\mbox{ and }( ( p(z,y)\mbox{ and }p(y,z) ) \ \Rightarrow \ p(y,x) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN332-1.p} Church problem 46.14 (4)






$f(x,y)\mbox{ or }f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y,x)\mbox{ or }f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(x,y)\mbox{ and }f(y,x) ) \ \Rightarrow \ \neg\,f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(z(x,y),x) \ \Rightarrow \ f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(x,z(x,y)) \ \Rightarrow \ f(z(x,y),x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$f(y,z(x,y)) \ \Rightarrow \ f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$f(x,z(x,y)) \ \Rightarrow \ f(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( f(y,x)\mbox{ and }f(z(x,y),z(x,y)) ) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$f(y,x)\mbox{ or }f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ f(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$f(y,x) \ \Rightarrow \ ( f(x,y)\mbox{ or }f(z(x,y),y) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$f(x,y) \ \Rightarrow \ ( f(y,x)\mbox{ or }f(z(x,y),y) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$f(z(x,y),y) \ \Rightarrow \ ( f(x,y)\mbox{ or }f(y,x) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( f(x,y)\mbox{ and }f(y,x) ) \ \Rightarrow \ \neg\,f(z(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

\medskip\textbf{SYN044-1.p} Pelletier Problem 10

A reasonably simple probloem designed to see whether 'natural'
 systems correctly manipulate premises.




$q \ \Rightarrow \ r\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$r \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$r \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$p \ \Rightarrow \ ( q\mbox{ or }r )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$p \ \Rightarrow \ \neg\,q\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$p\mbox{ or }q\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN969+1.p} Syntactic from Shults






$\forall b{:}\;( ( \forall x{:}\;( p(x) \ \Rightarrow \ q(x) )\mbox{ and }r(b) ) \ \Rightarrow \ ( \forall y{:}\;( r(y) \ \Rightarrow \ p(y) ) \ \Rightarrow \ q(b) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN189-1.p} RPT63 synthetic problem 87 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_2(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN622-1.p} Harrison problem 4012






$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{negated\_conjecture})$

$p_9(x_{60},x_{60})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_8(x_{36},x_{36})\qquad\mbox{cnf}(\mbox{p8}_3,\mbox{negated\_conjecture})$

$p_3(x_{25},x_{25})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_9,x_9)\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_3(f_4(c_{23}),c_{24})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_{19}(f_5(f_6(f_7(c_{25}))),c_{24})\qquad\mbox{cnf}(\mbox{p19}_7,\mbox{negated\_conjecture})$

$p_8(f_{16}(x_{50},f_{17}(x_{51},x_{50})),x_{51})\qquad\mbox{cnf}(\mbox{p8}_8,\mbox{negated\_conjecture})$

$( p_{21}(x_{18})\mbox{ and }p_{11}(x_{18},x_{17}) ) \ \Rightarrow \ p_{21}(x_{17})\qquad\mbox{cnf}(\mbox{p21}_9,\mbox{negated\_conjecture})$

$p_2(x_3,x_4) \ \Rightarrow \ p_{11}(f_{13}(x_3),f_{13}(x_4))\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_3(x_{65},x_{66}) \ \Rightarrow \ p_9(f_{15}(x_{65}),f_{15}(x_{66}))\qquad\mbox{cnf}(\mbox{p9}_{11},\mbox{negated\_conjecture})$

$p_8(x_{63},x_{64}) \ \Rightarrow \ p_9(f_{10}(x_{63}),f_{10}(x_{64}))\qquad\mbox{cnf}(\mbox{p9}_{12},\mbox{negated\_conjecture})$

$p_9(x_{44},x_{45}) \ \Rightarrow \ p_8(f_{14}(x_{44}),f_{14}(x_{45}))\qquad\mbox{cnf}(\mbox{p8}_{13},\mbox{negated\_conjecture})$

$p_3(x_{34},x_{35}) \ \Rightarrow \ p_3(f_7(x_{34}),f_7(x_{35}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{32},x_{33}) \ \Rightarrow \ p_3(f_6(x_{32}),f_6(x_{33}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_3(x_{30},x_{31}) \ \Rightarrow \ p_3(f_5(x_{30}),f_5(x_{31}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_2(x_{28},x_{29}) \ \Rightarrow \ p_3(f_4(x_{28}),f_4(x_{29}))\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$\neg\,p_8(f_{12}(f_{13}(c_{26}),c_{27}),f_{14}(f_{15}(f_5(c_{25}))))\qquad\mbox{cnf}(\mbox{not\_p8}_{18},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$( p_9(x_{60},x_{61})\mbox{ and }p_9(x_{60},x_{62}) ) \ \Rightarrow \ p_9(x_{61},x_{62})\qquad\mbox{cnf}(\mbox{p9}_{20},\mbox{negated\_conjecture})$

$( p_8(x_{36},x_{37})\mbox{ and }p_8(x_{36},x_{38}) ) \ \Rightarrow \ p_8(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p8}_{21},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{26})\mbox{ and }p_3(x_{25},x_{27}) ) \ \Rightarrow \ p_3(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_2(x_9,x_{10})\mbox{ and }p_2(x_9,x_{11}) ) \ \Rightarrow \ p_2(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_8(f_{12}(f_{13}(c_{23}),x_{43}),f_{12}(f_{13}(c_{26}),f_{16}(c_{27},x_{43})))\qquad\mbox{cnf}(\mbox{p8}_{24},\mbox{negated\_conjecture})$

$p_{20}(f_{10}(f_{12}(f_{13}(c_{26}),c_{27})),f_{10}(f_{12}(f_{13}(c_{26}),x_{16})))\qquad\mbox{cnf}(\mbox{p20}_{25},\mbox{negated\_conjecture})$

$\neg\,p_8(f_{12}(f_{13}(c_{23}),f_{17}(c_{28},c_{27})),f_{12}(f_{13}(c_{26}),c_{28}))\qquad\mbox{cnf}(\mbox{not\_p8}_{26},\mbox{negated\_conjecture})$

$( p_3(x_7,x_5)\mbox{ and }p_3(x_8,x_6)\mbox{ and }p_{19}(x_7,x_8) ) \ \Rightarrow \ p_{19}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p19}_{27},\mbox{negated\_conjecture})$

$( p_3(x_{23},x_{21})\mbox{ and }p_3(x_{24},x_{22})\mbox{ and }p_{22}(x_{23},x_{24}) ) \ \Rightarrow \ p_{22}(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p22}_{28},\mbox{negated\_conjecture})$

$( p_9(x_{14},x_{12})\mbox{ and }p_9(x_{15},x_{13})\mbox{ and }p_{20}(x_{14},x_{15}) ) \ \Rightarrow \ p_{20}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p20}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{57},x_{59})\mbox{ and }p_3(x_{56},x_{58}) ) \ \Rightarrow \ p_8(f_{18}(x_{56},x_{57}),f_{18}(x_{58},x_{59}))\qquad\mbox{cnf}(\mbox{p8}_{30},\mbox{negated\_conjecture})$

$( p_{11}(x_{39},x_{41})\mbox{ and }p_8(x_{40},x_{42}) ) \ \Rightarrow \ p_8(f_{12}(x_{39},x_{40}),f_{12}(x_{41},x_{42}))\qquad\mbox{cnf}(\mbox{p8}_{31},\mbox{negated\_conjecture})$

$( p_8(x_{46},x_{48})\mbox{ and }p_8(x_{47},x_{49}) ) \ \Rightarrow \ p_8(f_{16}(x_{46},x_{47}),f_{16}(x_{48},x_{49}))\qquad\mbox{cnf}(\mbox{p8}_{32},\mbox{negated\_conjecture})$

$( p_8(x_{52},x_{54})\mbox{ and }p_8(x_{53},x_{55}) ) \ \Rightarrow \ p_8(f_{17}(x_{52},x_{53}),f_{17}(x_{54},x_{55}))\qquad\mbox{cnf}(\mbox{p8}_{33},\mbox{negated\_conjecture})$

$( p_{22}(x_{20},c_{24})\mbox{ and }p_3(f_4(x_{19}),x_{20}) ) \ \Rightarrow \ ( p_{21}(f_{13}(x_{19}))\mbox{ or }p_8(f_{12}(f_{13}(x_{19}),f_{18}(x_{20},x_{19})),f_{14}(f_{15}(f_5(c_{25})))) )\qquad\mbox{cnf}(\mbox{p21}_{34},\mbox{negated\_conjecture})$

\medskip\textbf{SYN013-1.p} A problem in quantification theory






$m\ne n\qquad\mbox{cnf}(c_1,\mbox{negated\_conjecture})$

$n\ne k\qquad\mbox{cnf}(c_2,\mbox{negated\_conjecture})$

$k\ne m\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = m ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = y ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,f(y)) )\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(f(y),y) )\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$( \mbox{element}(y,v)\mbox{ and }\mbox{element}(v,y) ) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,m)\mbox{ or }v = m\mbox{ or }v = y )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$g(y) = n \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$g(y) = y \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(y,g(y))\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(g(y),y)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( \mbox{element}(y,n)\mbox{ and }\mbox{element}(y,v)\mbox{ and }\mbox{element}(v,y) ) \ \Rightarrow \ ( y = n\mbox{ or }v = n\mbox{ or }v = y )\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$y = m \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$y = n \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$\mbox{element}(y,k) \ \Rightarrow \ ( y = k\mbox{ or }y = m\mbox{ or }y = n )\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{SYN378+1.p} Peter Andrews Problem X2130






$\forall x{:}\;\mbox{big\_p}(x) \ \Rightarrow \ ( \neg\,\exists y{:}\;\mbox{big\_q}(y)\mbox{ or }\exists z{:}\;( \mbox{big\_p}(z) \ \Rightarrow \ \mbox{big\_q}(z) ) )\qquad\mbox{fof}(x_{2130},\mbox{conjecture})$

\medskip\textbf{SYN930+1.p} From Smullyan






$\forall x{:}\;( p(x)\mbox{ or }c )\iff ( \forall x{:}\;p(x)\mbox{ or }c )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN987$\wedge$1.p} All things are true






$\forall x{:}\;\mbox{\$o}{:}\;x\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SYN073+1.p} Pelletier Problem 50






$\forall x{:}\;( \mbox{big\_f}(a,x)\mbox{ or }\forall y{:}\;\mbox{big\_f}(x,y) ) \ \Rightarrow \ \exists x_1{:}\;\forall y_1{:}\;\mbox{big\_f}(x_1,y_1)\qquad\mbox{fof}(\mbox{pel}_{50},\mbox{conjecture})$

\medskip\textbf{SYN240-1.p} RPT63 synthetic problem 25 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(x,e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN166-1.p} RPT63 synthetic problem 64 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(a,b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN196-1.p} RPT63 synthetic problem 94 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_1(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN321-1.p} Church problem 46.3 (2)






$f(a,z) \ \Rightarrow \ g(a,u)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$g(u,u) \ \Rightarrow \ f(u,b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,b)\mbox{ or }g(a,u)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(a,b) \ \Rightarrow \ \neg\,g(a,d)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN050-1.p} Pelletier Problem 20






$( \mbox{big\_p}(y)\mbox{ and }\mbox{big\_q}(z) ) \ \Rightarrow \ \mbox{big\_r}(f(y,z))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(y)\mbox{ and }\mbox{big\_q}(z) ) \ \Rightarrow \ \mbox{big\_s}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{big\_q}(b)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_r}(w)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN364$\wedge$5.p} TPS problem X2115






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\mbox{cM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xu}@ \mbox{xv}\mbox{ or }( \mbox{cM}@ \mbox{xu}\mbox{ and }\mbox{cQ}@ ( f@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xw} ) \ \Rightarrow \ \neg\,\mbox{cM}@ ( g@ \mbox{xw} ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( g@ \mbox{xu} )@ \mbox{xv}\mbox{ and }\mbox{cP}@ \mbox{xu}@ \mbox{xu} )\qquad\mbox{thf}(\mbox{cX}_{2115},\mbox{conjecture})$

\medskip\textbf{SYN227-1.p} RPT63 synthetic problem 12 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN287-1.p} RPT63 synthetic problem 72 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(b,x,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN612-1.p} Harrison problem 3832






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{28},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_4(f_6(f_7(f_8(c_{20})))))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN068-1.p} Pelletier Problem 44






$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(f(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_h}(x,f(x))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(g(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,\mbox{big\_h}(x,g(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_j}(a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_g}(x) \ \Rightarrow \ \mbox{big\_h}(a,x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{big\_j}(x) \ \Rightarrow \ \mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN044+1.p} Pelletier Problem 10

A reasonably simple probloem designed to see whether 'natural'
 systems correctly manipulate premises.




$q \ \Rightarrow \ r\qquad\mbox{fof}(\mbox{pel10}_1,\mbox{axiom})$

$r \ \Rightarrow \ ( p\mbox{ and }q )\qquad\mbox{fof}(\mbox{pel10}_2,\mbox{axiom})$

$p \ \Rightarrow \ ( q\mbox{ or }r )\qquad\mbox{fof}(\mbox{pel10}_3,\mbox{axiom})$

$p\iff q\qquad\mbox{fof}(\mbox{pel}_{10},\mbox{conjecture})$

\medskip\textbf{SYN303-1.p} Problem for testing satisfiability






$p(x,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(x,y) \ \Rightarrow \ p(y,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(f(x),f(y)) \ \Rightarrow \ p(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(x,y) \ \Rightarrow \ p(f(x),f(y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,p(a,f(x))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN381$\wedge$5.p} TPS problem X2133






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i}{:}\;\exists \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xu}@ \mbox{xv} )\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xw}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{xz}@ \mbox{xw}\mbox{ or }\mbox{cQ}@ \mbox{xz}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cX}_{2133},\mbox{conjecture})$

\medskip\textbf{SYN369+1.p} Peter Andrews Problem X2120






$\forall u,v,w{:}\;( \mbox{big\_p}(u,v)\mbox{ or }\mbox{big\_p}(v,w) ) \ \Rightarrow \ \exists x{:}\;\forall y{:}\;\mbox{big\_p}(x,y)\qquad\mbox{fof}(x_{2120},\mbox{conjecture})$

\medskip\textbf{SYN400+1.p} Kalish and Montague Problem 227

Laws of vacuous quantification.




$\forall x{:}\;p\iff p\qquad\mbox{fof}(\mbox{kalish}_{227},\mbox{conjecture})$

\medskip\textbf{SYN112-1.p} RPT63 synthetic problem 10 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN138-1.p} RPT63 synthetic problem 36 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_3(c,e,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN416$\wedge$4.p} Pelletier Problem 16






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ p ) ) )\qquad\mbox{thf}(\mbox{pel}_{16},\mbox{conjecture})$

\medskip\textbf{SYN071-1.p} Pelletier Problem 48






$a = b\mbox{ or }c = d\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$a = c\mbox{ or }b = d\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$a\ne d\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$b\ne c\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN029-1.p} EW2






$q\mbox{ or }p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$p \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( r\mbox{ and }q ) \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$r\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$p \ \Rightarrow \ \neg\,q\qquad\mbox{cnf}(\mbox{prove\_something},\mbox{negated\_conjecture})$

\medskip\textbf{SYN270-1.p} RPT63 synthetic problem 55 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(b,b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN118-1.p} RPT63 synthetic problem 16 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_0(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN041-1.p} Pelletier Problem 3, 16

3: The hardest theorm proved by a breadth-first logic
 theorist.
 16:A surprising theorem of propositional logic.




$p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,q\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN720-1.p} Synthetic domain theory for EBL






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

\medskip\textbf{SYN410+1.p} Kalish and Montague Problem 249






$\forall x,y{:}\;f(x,y) \ \Rightarrow \ \exists u,v{:}\;f(u,v)\qquad\mbox{fof}(\mbox{kalish}_{249},\mbox{conjecture})$

\medskip\textbf{SYN074-1.p} Pelletier Problem 51






$\mbox{big\_f}(x,y) \ \Rightarrow \ x = a\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(x,y) \ \Rightarrow \ y = b\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( x = a\mbox{ and }y = b ) \ \Rightarrow \ \mbox{big\_f}(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_f}(f(x),y) \ \Rightarrow \ ( y = g(x)\mbox{ or }f(x) = x )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(f(x),y) \ \Rightarrow \ ( y = g(x)\mbox{ or }\mbox{big\_f}(f(x),h(x,z))\mbox{ or }h(x,z) = z )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(f(x),y)\mbox{ and }h(x,z) = z\mbox{ and }\mbox{big\_f}(f(x),h(x,z)) ) \ \Rightarrow \ y = g(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$y = g(x) \ \Rightarrow \ ( \mbox{big\_f}(f(x),y)\mbox{ or }\mbox{big\_f}(f(x),h(x,z))\mbox{ or }h(x,z) = z )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$y = g(x) \ \Rightarrow \ ( \mbox{big\_f}(f(x),y)\mbox{ or }f(x) = x )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( y = g(x)\mbox{ and }h(x,z) = z\mbox{ and }\mbox{big\_f}(f(x),h(x,z)) ) \ \Rightarrow \ \mbox{big\_f}(f(x),y)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$f(x) = x \ \Rightarrow \ ( \mbox{big\_f}(f(x),h(x,z))\mbox{ or }h(x,z) = z )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( f(x) = x\mbox{ and }h(x,z) = z ) \ \Rightarrow \ \neg\,\mbox{big\_f}(f(x),h(x,z))\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{SYN383+1.p} Peter Andrews Problem X2135






$\exists x{:}\;\forall y{:}\;( ( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(y) ) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_p}(y) ) )\qquad\mbox{fof}(x_{2135},\mbox{conjecture})$

\medskip\textbf{SYN407$\wedge$7.p} Kalish and Montague Problem 241






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$h{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{h\_type},\mbox{type})$

$g{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( f@ x ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( g@ x ) )@ ( \mbox{mbox\_s}_4@ ( h@ x ) ) ) ) ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( f@ y ) )@ ( \mbox{mbox\_s}_4@ ( g@ y ) ) ) ) ) )@ ( \mbox{mexists\_ind}@ \lambda z{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( f@ z ) )@ ( \mbox{mbox\_s}_4@ ( h@ z ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kalish}_{241},\mbox{conjecture})$

\medskip\textbf{SYN364+1.p} Peter Andrews Problem X2115






$( \forall x{:}\;( \exists y{:}\;\mbox{big\_p}(x,y) \ \Rightarrow \ \forall z{:}\;\mbox{big\_p}(z,z) )\mbox{ and }\forall u{:}\;\exists v{:}\;( \mbox{big\_p}(u,v)\mbox{ or }( \mbox{big\_m}(u)\mbox{ and }\mbox{big\_q}(f(u,v)) ) )\mbox{ and }\forall w{:}\;( \mbox{big\_q}(w) \ \Rightarrow \ \neg\,\mbox{big\_m}(g(w)) ) ) \ \Rightarrow \ \forall u{:}\;\exists v{:}\;( \mbox{big\_p}(g(u),v)\mbox{ and }\mbox{big\_p}(u,u) )\qquad\mbox{fof}(x_{2115},\mbox{conjecture})$

\medskip\textbf{SYN620-1.p} Harrison problem 3996






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_5(x_{56},x_{56})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_3(x_{45},x_{45})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{11}(x_{20},x_{20})\qquad\mbox{cnf}(\mbox{p11}_5,\mbox{negated\_conjecture})$

$\neg\,p_{18}(f_6(c_{24}))\qquad\mbox{cnf}(\mbox{f6\_is\_not\_p18}_6,\mbox{negated\_conjecture})$

$\neg\,p_{18}(f_6(c_{22}))\qquad\mbox{cnf}(\mbox{f6\_is\_not\_p18}_7,\mbox{negated\_conjecture})$

$p_3(f_4(c_{22}),c_{23})\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_{19}(f_7(f_8(f_9(c_{25}))),c_{23})\qquad\mbox{cnf}(\mbox{p19}_9,\mbox{negated\_conjecture})$

$( p_{18}(x_{28})\mbox{ and }p_5(x_{28},x_{27}) ) \ \Rightarrow \ p_{18}(x_{27})\qquad\mbox{cnf}(\mbox{p18}_{10},\mbox{negated\_conjecture})$

$p_{11}(x_{10},x_{11}) \ \Rightarrow \ p_{10}(f_{14}(x_{10}),f_{14}(x_{11}))\qquad\mbox{cnf}(\mbox{p10}_{11},\mbox{negated\_conjecture})$

$p_2(x_{59},x_{60}) \ \Rightarrow \ p_5(f_6(x_{59}),f_6(x_{60}))\qquad\mbox{cnf}(\mbox{p5}_{12},\mbox{negated\_conjecture})$

$p_3(x_{54},x_{55}) \ \Rightarrow \ p_3(f_9(x_{54}),f_9(x_{55}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{52},x_{53}) \ \Rightarrow \ p_3(f_8(x_{52}),f_8(x_{53}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{50},x_{51}) \ \Rightarrow \ p_3(f_7(x_{50}),f_7(x_{51}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_3(f_4(x_{48}),f_4(x_{49}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_3(x_{25},x_{26}) \ \Rightarrow \ p_{11}(f_{15}(x_{25}),f_{15}(x_{26}))\qquad\mbox{cnf}(\mbox{p11}_{17},\mbox{negated\_conjecture})$

$p_{10}(x_{23},x_{24}) \ \Rightarrow \ p_{11}(f_{12}(x_{23}),f_{12}(x_{24}))\qquad\mbox{cnf}(\mbox{p11}_{18},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{19},\mbox{negated\_conjecture})$

$( p_5(x_{56},x_{57})\mbox{ and }p_5(x_{56},x_{58}) ) \ \Rightarrow \ p_5(x_{57},x_{58})\qquad\mbox{cnf}(\mbox{p5}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{45},x_{46})\mbox{ and }p_3(x_{45},x_{47}) ) \ \Rightarrow \ p_3(x_{46},x_{47})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{34})\mbox{ and }p_2(x_{33},x_{35}) ) \ \Rightarrow \ p_2(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_{11}(x_{20},x_{21})\mbox{ and }p_{11}(x_{20},x_{22}) ) \ \Rightarrow \ p_{11}(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p11}_{23},\mbox{negated\_conjecture})$

$p_{10}(f_{13}(f_6(c_{22}),x_9),f_{13}(f_6(c_{24}),f_{16}(c_{26},x_9)))\qquad\mbox{cnf}(\mbox{p10}_{24},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_{13}(f_6(c_{22}),f_{14}(f_{15}(f_7(c_{25})))),f_{14}(f_{15}(f_7(c_{25}))))\qquad\mbox{cnf}(\mbox{not\_p10}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{43},x_{41})\mbox{ and }p_3(x_{44},x_{42})\mbox{ and }p_{21}(x_{43},x_{44}) ) \ \Rightarrow \ p_{21}(x_{41},x_{42})\qquad\mbox{cnf}(\mbox{p21}_{26},\mbox{negated\_conjecture})$

$( p_3(x_{31},x_{29})\mbox{ and }p_3(x_{32},x_{30})\mbox{ and }p_{19}(x_{31},x_{32}) ) \ \Rightarrow \ p_{19}(x_{29},x_{30})\qquad\mbox{cnf}(\mbox{p19}_{27},\mbox{negated\_conjecture})$

$( p_{11}(x_{39},x_{37})\mbox{ and }p_{20}(x_{38},x_{39})\mbox{ and }p_{11}(x_{38},x_{36}) ) \ \Rightarrow \ p_{20}(x_{36},x_{37})\qquad\mbox{cnf}(\mbox{p20}_{28},\mbox{negated\_conjecture})$

$( p_{10}(x_4,x_6)\mbox{ and }p_5(x_3,x_5) ) \ \Rightarrow \ p_{10}(f_{13}(x_3,x_4),f_{13}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p10}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{17},x_{19})\mbox{ and }p_3(x_{16},x_{18}) ) \ \Rightarrow \ p_{10}(f_{17}(x_{16},x_{17}),f_{17}(x_{18},x_{19}))\qquad\mbox{cnf}(\mbox{p10}_{30},\mbox{negated\_conjecture})$

$( p_{10}(x_{12},x_{14})\mbox{ and }p_{10}(x_{13},x_{15}) ) \ \Rightarrow \ p_{10}(f_{16}(x_{12},x_{13}),f_{16}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p10}_{31},\mbox{negated\_conjecture})$

$p_{20}(f_{12}(f_{13}(f_6(c_{22}),f_{14}(f_{15}(f_7(c_{25}))))),f_{12}(f_{13}(f_6(c_{24}),x_{40})))\qquad\mbox{cnf}(\mbox{p20}_{32},\mbox{negated\_conjecture})$

$\neg\,p_{20}(f_{12}(f_{13}(f_6(c_{22}),f_{14}(f_{15}(f_7(c_{25}))))),f_{12}(f_{13}(f_6(c_{22}),c_{27})))\qquad\mbox{cnf}(\mbox{not\_p20}_{33},\mbox{negated\_conjecture})$

$( p_{21}(x_8,c_{23})\mbox{ and }p_3(f_4(x_7),x_8) ) \ \Rightarrow \ ( p_{18}(f_6(x_7))\mbox{ or }p_{10}(f_{13}(f_6(x_7),f_{17}(x_8,x_7)),f_{14}(f_{15}(f_7(c_{25})))) )\qquad\mbox{cnf}(\mbox{p18}_{34},\mbox{negated\_conjecture})$

\medskip\textbf{SYN188-1.p} RPT63 synthetic problem 86 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_1(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN551+3.p} Cute Little Problem






$\exists x{:}\;( x = f(g(x))\mbox{ and }\forall y{:}\;( y = f(g(y)) \ \Rightarrow \ y = x ) )\iff \exists x{:}\;( x = g(f(x))\mbox{ and }\forall y{:}\;( y = g(f(y)) \ \Rightarrow \ y = x ) )\qquad\mbox{fof}(\mbox{prove\_this\_cute\_thing},\mbox{conjecture})$

\medskip\textbf{SYN208-1.p} RPT63 synthetic problem 106 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(d,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN980+1.p} Syntactic from Shults






$\forall b{:}\;( \forall y{:}\;( ( r(b) \ \Rightarrow \ r(y) ) \ \Rightarrow \ p(f(y),y) ) \ \Rightarrow \ \exists x,y{:}\;( p(x,y)\mbox{ and }( q(f(b),b) \ \Rightarrow \ q(x,y) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN914-1.p} Show that 11 $>$ 10

Having at least eleven elements but allowing at most 10 is
 a contradiction.




$x = a_1\mbox{ or }x = a_2\mbox{ or }x = a_3\mbox{ or }x = a_4\mbox{ or }x = a_5\mbox{ or }x = a_6\mbox{ or }x = a_7\mbox{ or }x = a_8\mbox{ or }x = a_9\mbox{ or }x = a_{10}\qquad\mbox{cnf}(\mbox{at\_most}_{10},\mbox{axiom})$

$b_1\ne b_2\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_2,\mbox{negated\_conjecture})$

$b_1\ne b_3\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_3,\mbox{negated\_conjecture})$

$b_1\ne b_4\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_4,\mbox{negated\_conjecture})$

$b_1\ne b_5\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_5,\mbox{negated\_conjecture})$

$b_1\ne b_6\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_6,\mbox{negated\_conjecture})$

$b_1\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_7,\mbox{negated\_conjecture})$

$b_1\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_8,\mbox{negated\_conjecture})$

$b_1\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_9,\mbox{negated\_conjecture})$

$b_1\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_{10},\mbox{negated\_conjecture})$

$b_1\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_1}_{11},\mbox{negated\_conjecture})$

$b_2\ne b_3\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_3,\mbox{negated\_conjecture})$

$b_2\ne b_4\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_4,\mbox{negated\_conjecture})$

$b_2\ne b_5\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_5,\mbox{negated\_conjecture})$

$b_2\ne b_6\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_6,\mbox{negated\_conjecture})$

$b_2\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_7,\mbox{negated\_conjecture})$

$b_2\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_8,\mbox{negated\_conjecture})$

$b_2\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_9,\mbox{negated\_conjecture})$

$b_2\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_{10},\mbox{negated\_conjecture})$

$b_2\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_2}_{11},\mbox{negated\_conjecture})$

$b_3\ne b_4\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_4,\mbox{negated\_conjecture})$

$b_3\ne b_5\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_5,\mbox{negated\_conjecture})$

$b_3\ne b_6\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_6,\mbox{negated\_conjecture})$

$b_3\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_7,\mbox{negated\_conjecture})$

$b_3\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_8,\mbox{negated\_conjecture})$

$b_3\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_9,\mbox{negated\_conjecture})$

$b_3\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_{10},\mbox{negated\_conjecture})$

$b_3\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_3}_{11},\mbox{negated\_conjecture})$

$b_4\ne b_5\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_5,\mbox{negated\_conjecture})$

$b_4\ne b_6\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_6,\mbox{negated\_conjecture})$

$b_4\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_7,\mbox{negated\_conjecture})$

$b_4\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_8,\mbox{negated\_conjecture})$

$b_4\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_9,\mbox{negated\_conjecture})$

$b_4\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_{10},\mbox{negated\_conjecture})$

$b_4\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_4}_{11},\mbox{negated\_conjecture})$

$b_5\ne b_6\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_6,\mbox{negated\_conjecture})$

$b_5\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_7,\mbox{negated\_conjecture})$

$b_5\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_8,\mbox{negated\_conjecture})$

$b_5\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_9,\mbox{negated\_conjecture})$

$b_5\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_{10},\mbox{negated\_conjecture})$

$b_5\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_5}_{11},\mbox{negated\_conjecture})$

$b_6\ne b_7\qquad\mbox{cnf}(\mbox{at\_least\_11\_6}_7,\mbox{negated\_conjecture})$

$b_6\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_6}_8,\mbox{negated\_conjecture})$

$b_6\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_6}_9,\mbox{negated\_conjecture})$

$b_6\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_6}_{10},\mbox{negated\_conjecture})$

$b_6\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_6}_{11},\mbox{negated\_conjecture})$

$b_7\ne b_8\qquad\mbox{cnf}(\mbox{at\_least\_11\_7}_8,\mbox{negated\_conjecture})$

$b_7\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_7}_9,\mbox{negated\_conjecture})$

$b_7\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_7}_{10},\mbox{negated\_conjecture})$

$b_7\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_7}_{11},\mbox{negated\_conjecture})$

$b_8\ne b_9\qquad\mbox{cnf}(\mbox{at\_least\_11\_8}_9,\mbox{negated\_conjecture})$

$b_8\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_8}_{10},\mbox{negated\_conjecture})$

$b_8\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_8}_{11},\mbox{negated\_conjecture})$

$b_9\ne b_{10}\qquad\mbox{cnf}(\mbox{at\_least\_11\_9}_{10},\mbox{negated\_conjecture})$

$b_9\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_9}_{11},\mbox{negated\_conjecture})$

$b_{10}\ne b_{11}\qquad\mbox{cnf}(\mbox{at\_least\_11\_10}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{SYN285-1.p} RPT63 synthetic problem 70 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(a,d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN932+1.p} From Smullyan






$\exists x{:}\;c\iff c\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN316-1.p} Church problem 46.2 (2)






$f(x) \ \Rightarrow \ \neg\,f(y(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x)\mbox{ or }f(y(x))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN064+1.p} Pelletier Problem 35






$\exists x,y{:}\;( \mbox{big\_p}(x,y) \ \Rightarrow \ \forall z,w{:}\;\mbox{big\_p}(z,w) )\qquad\mbox{fof}(\mbox{pel}_{35},\mbox{conjecture})$

\medskip\textbf{SYN292-1.p} RPT63 synthetic problem 77 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(d,b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN349+1.p} Church problem 46.17 (5)






$\exists x_1{:}\;\forall x_2{:}\;\exists x_3{:}\;\forall x_4{:}\;( ( \mbox{big\_f}(x_1,x_4)\iff \mbox{big\_f}(x_2,x_4) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(x_1,x_4)\iff \mbox{big\_f}(x_4,x_3) )\iff \mbox{big\_f}(x_3,x_4) )\iff \mbox{big\_f}(x_4,x_2) ) )\qquad\mbox{fof}(\mbox{church\_46\_17}_5,\mbox{conjecture})$

\medskip\textbf{SYN000+2.p} Advanced TPTP FOF syntax

Advanced TPTP FOF syntax that you will encounter some time.




$\mbox{"An Apple"}\ne \mbox{"A \"Microsoft \\ escape\""}\qquad\mbox{fof}(\mbox{distinct\_object},\mbox{axiom})$

$p(12)\mbox{ or }p(-12)\qquad\mbox{fof}(\mbox{integers},\mbox{axiom})$

$p(123/456)\mbox{ or }p(-123/456)\mbox{ or }p(+123/456)\qquad\mbox{fof}(\mbox{rationals},\mbox{axiom})$

$p(\mbox{123.}_{456})\mbox{ or }p(-\mbox{123.}_{456})\mbox{ or }p(\mbox{123.456E}_{78})\mbox{ or }p(\mbox{123.456e}_{78})\mbox{ or }p(-\mbox{123.456E}_{78})\mbox{ or }p(\mbox{123.456E-}_{78})\mbox{ or }p(-\mbox{123.456E-}_{78})\qquad\mbox{fof}(\mbox{reals},\mbox{axiom})$

$\forall x{:}\;( p(x)\mbox{ nor }\neg\,q(x,a) )\mbox{ nand }p(x)\qquad\mbox{fof}(\mbox{never\_used\_connectives},\mbox{axiom})$

$\forall x{:}\;f(d) = f(x)\qquad\mbox{fof}(\mbox{role\_definition},\mbox{definition})$

$p(a)\qquad\mbox{fof}(\mbox{role\_assumption},\mbox{assumption})$

$p(l)\qquad\mbox{fof}(\mbox{role\_lemma},\mbox{lemma})$

$p(t)\qquad\mbox{fof}(\mbox{role\_theorem},\mbox{theorem})$

$p(u)\qquad\mbox{fof}(\mbox{role\_unknown},\mbox{unknown})$

$\mbox{include}(\mbox{'Axioms/SYN000+0.ax'},[\mbox{ia}_1,\mbox{ia}_3])$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{source\_unknown},\mbox{axiom})$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{source},\mbox{axiom})$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{source\_name},\mbox{axiom})$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{source\_copy},\mbox{axiom})$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{source\_introduced\_assumption},\mbox{axiom})$

$p(a)\qquad\mbox{fof}(\mbox{source\_inference},\mbox{axiom})$

$p(a)\qquad\mbox{fof}(\mbox{source\_inference\_with\_bind},\mbox{axiom})$

$\forall x{:}\;p(x)\qquad\mbox{fof}(\mbox{useful\_info},\mbox{axiom})$

\medskip\textbf{SYN986+1.003.p} Orevkov formula - size 3

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_3,z_2,z_1,z_0{:}\;( r(0,0,z_3)\mbox{ and }r(0,z_3,z_2)\mbox{ and }r(0,z_2,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN224-1.p} RPT63 synthetic problem 9 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(a,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN345-1.p} Church problem 46.16 (4)






$f(a,a,b) \ \Rightarrow \ ( f(a,b,y_1)\mbox{ or }f(a,b,b) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(y_2,y_1,z(y_1,y_2))\mbox{ and }f(a,a,b) ) \ \Rightarrow \ f(a,b,b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,b,b) \ \Rightarrow \ ( f(b,y_1,y_2)\mbox{ or }f(a,a,b) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(y_1,z(y_1,y_2),z(y_1,y_2))\mbox{ and }f(a,b,b) ) \ \Rightarrow \ f(a,a,b)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(y_1,y_2,z(y_1,y_2))\mbox{ or }f(a,a,b)\mbox{ or }f(a,b,b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(a,a,b)\mbox{ and }f(a,b,b) ) \ \Rightarrow \ f(y_1,y_2,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$f(b,b,y_1) \ \Rightarrow \ ( f(a,a,b)\mbox{ or }f(a,b,b) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( f(b,b,y_1)\mbox{ and }f(a,a,b) ) \ \Rightarrow \ \neg\,f(a,b,b)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN265-1.p} RPT63 synthetic problem 50 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_3(x,x,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN135-1.p} RPT63 synthetic problem 33 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_3(x,y,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN005-1.010.p} Disjunctions that form a contradiction

~p1(X1,X2) $\vee$ ~p2(X2,X3) $\vee$ ... $\vee$ ~p10(X10,X1).
 p1(a,a)      p2(a,a) ...        p10(a,a)




$( p_1(x_1,x_2)\mbox{ and }p_2(x_2,x_3)\mbox{ and }p_3(x_3,x_4)\mbox{ and }p_4(x_4,x_5)\mbox{ and }p_5(x_5,x_6)\mbox{ and }p_6(x_6,x_7)\mbox{ and }p_7(x_7,x_8)\mbox{ and }p_8(x_8,x_9)\mbox{ and }p_9(x_9,x_{10}) ) \ \Rightarrow \ \neg\,p_{10}(x_{10},x_1)\qquad\mbox{cnf}(\mbox{disjunction},\mbox{negated\_conjecture})$

$p_1(a,a)\qquad\mbox{cnf}(p_1,\mbox{negated\_conjecture})$

$p_2(a,a)\qquad\mbox{cnf}(p_2,\mbox{negated\_conjecture})$

$p_3(a,a)\qquad\mbox{cnf}(p_3,\mbox{negated\_conjecture})$

$p_4(a,a)\qquad\mbox{cnf}(p_4,\mbox{negated\_conjecture})$

$p_5(a,a)\qquad\mbox{cnf}(p_5,\mbox{negated\_conjecture})$

$p_6(a,a)\qquad\mbox{cnf}(p_6,\mbox{negated\_conjecture})$

$p_7(a,a)\qquad\mbox{cnf}(p_7,\mbox{negated\_conjecture})$

$p_8(a,a)\qquad\mbox{cnf}(p_8,\mbox{negated\_conjecture})$

$p_9(a,a)\qquad\mbox{cnf}(p_9,\mbox{negated\_conjecture})$

$p_{10}(a,a)\qquad\mbox{cnf}(p_{10},\mbox{negated\_conjecture})$

\medskip\textbf{SYN311-1.p} Problem for testing satisfiability






$p(x,x_1,x_2,x_3) \ \Rightarrow \ p(x_3,x,x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(x,x_1,x_2,x_3) \ \Rightarrow \ p(x_3,x_2,x_1,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(x,x_1,x_2,x_3) \ \Rightarrow \ p(x,x_1,x_2,g(x_3))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$p(x,x_1,x_2,x_3) \ \Rightarrow \ p(f(x),x_1,x_2,x_3)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$p(a,b,c,d)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\neg\,p(f(g(d)),f(g(c)),f(g(b)),f(g(a)))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN237-1.p} RPT63 synthetic problem 22 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(x,x,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN323-1.p} Church problem 46.4 (2)






$f(x,a)\mbox{ or }g(x,a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(a,x) \ \Rightarrow \ g(x,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$g(x,a) \ \Rightarrow \ f(x,a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(a,x) \ \Rightarrow \ \neg\,g(x,a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN329-1.p} Church problem 46.14 (1)






$f(x,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,f(z(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN370+1.p} Peter Andrews Problem X2121






$\exists v{:}\;\forall y{:}\;\exists z{:}\;( ( \mbox{big\_p}(a,y,h(y))\mbox{ or }\mbox{big\_p}(v,y,f(y)) ) \ \Rightarrow \ \mbox{big\_p}(v,y,z) )\qquad\mbox{fof}(x_{2121},\mbox{conjecture})$

\medskip\textbf{SYN772-1.p} PSAT inverse problem - K=4 C=20 V=8 D=1.6






$\mbox{ssRr}(\mbox{skf}_3(u),u)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{ssRr}(u,\mbox{skf}_2(u))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{ssRr}(u,v) \ \Rightarrow \ ( \mbox{ssPv}_3(v)\mbox{ or }\mbox{ssPv}_3(u)\mbox{ or }\mbox{ssPv}_5(u)\mbox{ or }\mbox{ssPv}_8(u) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(u) ) \ \Rightarrow \ ( \mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_5(u)\mbox{ or }\mbox{ssPv}_8(u) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(v)\mbox{ or }\mbox{ssPv}_6(v)\mbox{ or }\mbox{ssPv}_7(v) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_8(v)\mbox{ or }\mbox{ssPv}_5(u) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_5(u)\mbox{ or }\mbox{ssPv}_6(u) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssPv}_8(v) ) \ \Rightarrow \ \mbox{ssPv}_2(u)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_8(u) ) \ \Rightarrow \ \mbox{ssPv}_2(u)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w) ) \ \Rightarrow \ ( \mbox{ssPv}_8(u)\mbox{ or }\mbox{ssPv}_6(w)\mbox{ or }\mbox{ssPv}_4(v)\mbox{ or }\mbox{ssPv}_8(v) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_8(v)\mbox{ or }\mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_6(u) ) \ \Rightarrow \ ( \mbox{ssPv}_5(w)\mbox{ or }\mbox{ssPv}_8(u) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_5(u) ) \ \Rightarrow \ ( \mbox{ssPv}_7(w)\mbox{ or }\mbox{ssPv}_8(u) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_5(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_7(u) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_8(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_8(v) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_6(v) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_1(v) ) \ \Rightarrow \ ( \mbox{ssPv}_7(w)\mbox{ or }\mbox{ssPv}_3(v) )\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_3(v) ) \ \Rightarrow \ ( \mbox{ssPv}_5(w)\mbox{ or }\mbox{ssPv}_2(v) )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_7(w)\mbox{ and }\mbox{ssPv}_7(v) ) \ \Rightarrow \ ( \mbox{ssPv}_8(u)\mbox{ or }\mbox{ssPv}_1(v) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_5(u)\mbox{ and }\mbox{ssPv}_7(u) ) \ \Rightarrow \ \mbox{ssPv}_8(w)\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_3(w)\mbox{ and }\mbox{ssPv}_3(v) ) \ \Rightarrow \ \mbox{ssPv}_5(v)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_3(w)\mbox{ and }\mbox{ssRr}(v,x) ) \ \Rightarrow \ ( \mbox{ssPv}_4(u)\mbox{ or }\mbox{ssPv}_6(x)\mbox{ or }\mbox{ssPv}_7(v) )\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_6(u) ) \ \Rightarrow \ \mbox{ssPv}_7(x)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN393$\wedge$4.002.p} ILTP Problem SYJ206+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_2 )@ ( \mbox{iatom}@ a_1 ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYN387$\wedge$7.p} The Law of Excluded Middle






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ p ) ) ) )\qquad\mbox{thf}(\mbox{pel}_6,\mbox{conjecture})$

\medskip\textbf{SYN397+1.p} Kalish and Montague Problem 204

Laws of quantifier negation.




$\neg\,\exists x{:}\;f(x)\iff \forall y{:}\;\neg\,f(y)\qquad\mbox{fof}(\mbox{kalish}_{204},\mbox{conjecture})$

\medskip\textbf{SYN567-1.p} Harrison problem 3022






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_5(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_5(f_8(f_6(x_{17})),x_{17})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$\neg\,p_2(f_6(c_9),f_3(f_4(c_{10})))\qquad\mbox{cnf}(\mbox{not\_p2}_4,\mbox{negated\_conjecture})$

$\neg\,p_2(f_6(c_9),f_3(f_7(f_4(c_{10}))))\qquad\mbox{cnf}(\mbox{not\_p2}_5,\mbox{negated\_conjecture})$

$\neg\,p_2(f_6(c_9),f_3(f_7(f_7(f_4(c_{10})))))\qquad\mbox{cnf}(\mbox{not\_p2}_6,\mbox{negated\_conjecture})$

$p_2(x_4,x_5) \ \Rightarrow \ p_2(f_3(x_4),f_3(x_5))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_5(f_8(x_{15}),f_8(x_{16}))\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_7(x_{10}),f_7(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_5(x_8,x_9) \ \Rightarrow \ p_2(f_6(x_8),f_6(x_9))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_6,x_7) \ \Rightarrow \ p_2(f_4(x_6),f_4(x_7))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$\neg\,p_2(f_6(c_9),f_3(f_7(f_7(f_7(f_4(c_{10}))))))\qquad\mbox{cnf}(\mbox{not\_p2}_{12},\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_5(x_{12},x_{13})\mbox{ and }p_5(x_{12},x_{14}) ) \ \Rightarrow \ p_5(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$p_2(x_3,f_3(f_4(c_{10}))) \ \Rightarrow \ p_2(f_6(f_8(x_3)),x_3)\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_2(x_3,f_3(f_7(f_4(c_{10})))) \ \Rightarrow \ p_2(f_6(f_8(x_3)),x_3)\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$p_2(x_3,f_3(f_7(f_7(f_4(c_{10}))))) \ \Rightarrow \ p_2(f_6(f_8(x_3)),x_3)\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$p_2(x_3,f_3(f_7(f_7(f_7(f_4(c_{10})))))) \ \Rightarrow \ p_2(f_6(f_8(x_3)),x_3)\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$p_2(f_6(f_8(x_3)),x_3) \ \Rightarrow \ ( p_2(x_3,f_3(f_4(c_{10})))\mbox{ or }p_2(x_3,f_3(f_7(f_4(c_{10}))))\mbox{ or }p_2(x_3,f_3(f_7(f_7(f_4(c_{10})))))\mbox{ or }p_2(x_3,f_3(f_7(f_7(f_7(f_4(c_{10})))))) )\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{SYN307-1.p} Problem for testing satisfiability






$( p(x,y,u)\mbox{ and }p(y,z,u) ) \ \Rightarrow \ p(x,z,u)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(x,x,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(x,z,u) \ \Rightarrow \ ( p(x,y,u)\mbox{ or }p(y,z,u) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,p(x,x,b)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN061-1.p} Pelletier Problem 31






$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,\mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(x) \ \Rightarrow \ \neg\,\mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_i}(a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_h}(x)\mbox{ or }\mbox{big\_j}(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_i}(x) \ \Rightarrow \ \neg\,\mbox{big\_j}(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN998$\wedge$1.p} Leibniz equality is reflexive






$\mbox{leibeq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}@ x@ x )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN243-1.p} RPT63 synthetic problem 28 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(a,d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN387+1.p} The Law of Excluded Middle

The Law of Excluded Middle: can be quite difficult for
 'natural' systems.




$p\mbox{ or }\neg\,p\qquad\mbox{fof}(\mbox{pel}_6,\mbox{conjecture})$

\medskip\textbf{SYN039-1.p} A challenge to resolution programs






$( s(x,f_1(x,x_1))\mbox{ and }q(x_1,f_1(x,x_1)) ) \ \Rightarrow \ p(f_1(x,x_1),f_1(x,x_1))\qquad\mbox{cnf}(c_1,\mbox{negated\_conjecture})$

$s(x,f_1(x,x_1)) \ \Rightarrow \ ( p(f_1(x,x_1),f_1(x,x_1))\mbox{ or }q(f_1(x,x_1),z_1) )\qquad\mbox{cnf}(c_2,\mbox{negated\_conjecture})$

$( s(x,f_1(x,x_1))\mbox{ and }s(x_1,x_1) ) \ \Rightarrow \ p(f_1(x,x_1),f_1(x,x_1))\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$q(x_1,f_1(x,x_1)) \ \Rightarrow \ ( p(f_1(x,x_1),f_1(x,x_1))\mbox{ or }s(f_1(x,x_1),z) )\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$p(f_1(x,x_1),f_1(x,x_1))\mbox{ or }s(f_1(x,x_1),z)\mbox{ or }q(f_1(x,x_1),z_1)\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$s(x_1,x_1) \ \Rightarrow \ ( p(f_1(x,x_1),f_1(x,x_1))\mbox{ or }s(f_1(x,x_1),z) )\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$( q(z_1,z_1)\mbox{ and }q(x_1,f_1(x,x_1)) ) \ \Rightarrow \ p(f_1(x,x_1),f_1(x,x_1))\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$q(z_1,z_1) \ \Rightarrow \ ( p(f_1(x,x_1),f_1(x,x_1))\mbox{ or }q(f_1(x,x_1),z_1) )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$( q(z_1,z_1)\mbox{ and }s(x_1,x_1) ) \ \Rightarrow \ p(f_1(x,x_1),f_1(x,x_1))\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }s(x,f_1(x,x_1)) ) \ \Rightarrow \ \neg\,q(x_1,f_1(x,x_1))\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }s(x,f_1(x,x_1)) ) \ \Rightarrow \ q(f_1(x,x_1),z_1)\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }s(x,f_1(x,x_1)) ) \ \Rightarrow \ \neg\,s(x_1,x_1)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }q(x_1,f_1(x,x_1)) ) \ \Rightarrow \ s(f_1(x,x_1),z)\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$p(x,x) \ \Rightarrow \ ( s(f_1(x,x_1),z)\mbox{ or }q(f_1(x,x_1),z_1) )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }s(x_1,x_1) ) \ \Rightarrow \ s(f_1(x,x_1),z)\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }q(z_1,z_1) ) \ \Rightarrow \ \neg\,q(x_1,f_1(x,x_1))\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }q(z_1,z_1) ) \ \Rightarrow \ q(f_1(x,x_1),z_1)\qquad\mbox{cnf}(c_{17},\mbox{negated\_conjecture})$

$( p(x,x)\mbox{ and }q(z_1,z_1) ) \ \Rightarrow \ \neg\,s(x_1,x_1)\qquad\mbox{cnf}(c_{18},\mbox{negated\_conjecture})$

$( s(x,f_1(x,x_1))\mbox{ and }q(x_1,f_1(x,x_1)) ) \ \Rightarrow \ s(z,x)\qquad\mbox{cnf}(c_{19},\mbox{negated\_conjecture})$

$s(x,f_1(x,x_1)) \ \Rightarrow \ ( s(z,x)\mbox{ or }q(f_1(x,x_1),z_1) )\qquad\mbox{cnf}(c_{20},\mbox{negated\_conjecture})$

$( s(x,f_1(x,x_1))\mbox{ and }s(x_1,x_1) ) \ \Rightarrow \ s(z,x)\qquad\mbox{cnf}(c_{21},\mbox{negated\_conjecture})$

$q(x_1,f_1(x,x_1)) \ \Rightarrow \ ( s(z,x)\mbox{ or }s(f_1(x,x_1),z) )\qquad\mbox{cnf}(c_{22},\mbox{negated\_conjecture})$

$s(z,x)\mbox{ or }s(f_1(x,x_1),z)\mbox{ or }q(f_1(x,x_1),z_1)\qquad\mbox{cnf}(c_{23},\mbox{negated\_conjecture})$

$s(x_1,x_1) \ \Rightarrow \ ( s(z,x)\mbox{ or }s(f_1(x,x_1),z) )\qquad\mbox{cnf}(c_{24},\mbox{negated\_conjecture})$

$( q(z_1,z_1)\mbox{ and }q(x_1,f_1(x,x_1)) ) \ \Rightarrow \ s(z,x)\qquad\mbox{cnf}(c_{25},\mbox{negated\_conjecture})$

$q(z_1,z_1) \ \Rightarrow \ ( s(z,x)\mbox{ or }q(f_1(x,x_1),z_1) )\qquad\mbox{cnf}(c_{26},\mbox{negated\_conjecture})$

$( q(z_1,z_1)\mbox{ and }s(x_1,x_1) ) \ \Rightarrow \ s(z,x)\qquad\mbox{cnf}(c_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN343-1.p} Church problem 46.16 (2)






$f(a,y_1)\mbox{ or }f(a,a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(z(y_1,y_2),a) \ \Rightarrow \ f(a,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,a) \ \Rightarrow \ \neg\,f(y_1,y_2)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.006.p} Orevkov formula - size 6

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_6,z_5,z_4,z_3,z_2,z_1,z_0{:}\;( r(0,0,z_6)\mbox{ and }r(0,z_6,z_5)\mbox{ and }r(0,z_5,z_4)\mbox{ and }r(0,z_4,z_3)\mbox{ and }r(0,z_3,z_2)\mbox{ and }r(0,z_2,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN928+1.p} From Smullyan






$\forall x{:}\;p(x) \ \Rightarrow \ \exists x{:}\;p(x)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN397$\wedge$7.p} Kalish and Montague Problem 204






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( f@ x ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( f@ y ) ) ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( f@ y ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( f@ x ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kalish}_{204},\mbox{conjecture})$

\medskip\textbf{SYN197-1.p} RPT63 synthetic problem 95 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_1(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN334-1.p} Church problem 46.14 (6)






$( f(x,y)\mbox{ and }g(y,z(x,y)) ) \ \Rightarrow \ f(x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(x,y)\mbox{ and }f(x,z(x,y)) ) \ \Rightarrow \ g(y,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(z(x,y),z(x,y))\mbox{ and }f(x,y) ) \ \Rightarrow \ g(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(z(x,y),z(x,y))\mbox{ or }f(x,y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$g(z(x,y),z(x,y)) \ \Rightarrow \ f(x,y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$g(x,y)\mbox{ or }g(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$g(x,y) \ \Rightarrow \ \neg\,g(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN233-1.p} RPT63 synthetic problem 18 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN251-1.p} RPT63 synthetic problem 36 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(e,b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN376+1.p} Peter Andrews Problem X2127






$\exists x{:}\;\forall y{:}\;( \mbox{big\_p}(y)\iff \mbox{big\_p}(x) ) \ \Rightarrow \ ( \forall x{:}\;\mbox{big\_p}(x)\mbox{ or }\forall x{:}\;\neg\,\mbox{big\_p}(x) )\qquad\mbox{fof}(x_{2127},\mbox{conjecture})$

\medskip\textbf{SYN559-1.p} Harrison problem 2418






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_8(c_{13},c_9)\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_7(c_{10},c_{13})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(f_6(c_9,c_{13}),c_{14})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(f_5(c_{14},c_{10}),c_{15})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(f_4(c_{11}),c_{12}),c_{15})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_4(x_7),f_4(x_8))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$( p_2(x_{24},x_{22})\mbox{ and }p_8(x_{23},x_{24})\mbox{ and }p_2(x_{23},x_{21}) ) \ \Rightarrow \ p_8(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p8}_9,\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{18})\mbox{ and }p_7(x_{19},x_{20})\mbox{ and }p_2(x_{19},x_{17}) ) \ \Rightarrow \ p_7(x_{17},x_{18})\qquad\mbox{cnf}(\mbox{p7}_{10},\mbox{negated\_conjecture})$

$( p_2(x_3,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(f_3(x_3,x_4),f_3(x_5,x_6))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{15})\mbox{ and }p_2(x_{14},x_{16}) ) \ \Rightarrow \ p_2(f_6(x_{13},x_{14}),f_6(x_{15},x_{16}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_2(x_9,x_{11})\mbox{ and }p_2(x_{10},x_{12}) ) \ \Rightarrow \ p_2(f_5(x_9,x_{10}),f_5(x_{11},x_{12}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_8(x_{26},c_9)\mbox{ and }p_7(c_{10},x_{26})\mbox{ and }p_2(f_6(c_9,x_{26}),x_{25}) ) \ \Rightarrow \ \neg\,p_2(f_5(x_{25},c_{10}),f_3(f_4(c_{11}),c_{12}))\qquad\mbox{cnf}(\mbox{not\_p8}_{14},\mbox{negated\_conjecture})$

\medskip\textbf{SYN393$\wedge$4.003.p} Pelletier 12






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{12},\mbox{conjecture})$

\medskip\textbf{SYN941+1.p} Syntactic from Shults






$\forall b,c{:}\;( q(f(b)) \ \Rightarrow \ \exists x,y{:}\;( ( p(f(y)) \ \Rightarrow \ ( p(x)\mbox{ and }( r(y) \ \Rightarrow \ ( r(b)\mbox{ and }r(c) ) ) ) )\mbox{ and }q(x) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN198-1.p} RPT63 synthetic problem 96 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_1(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN375$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) )\mbox{ and }( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cX2126\_BUG},\mbox{conjecture})$

\medskip\textbf{SYN115-1.p} RPT63 synthetic problem 13 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_4(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN591-1.p} Harrison problem 3583






$p_9(c_{14})\qquad\mbox{cnf}(\mbox{c14\_is\_p9}_1,\mbox{negated\_conjecture})$

$p_8(c_{13})\qquad\mbox{cnf}(\mbox{c13\_is\_p8}_2,\mbox{negated\_conjecture})$

$p_7(x_{30},x_{30})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(x_{15},x_{15})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(x_{23},x_{23})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_{10}(c_{14},x_4)\qquad\mbox{cnf}(\mbox{p10}_6,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{15})\qquad\mbox{cnf}(\mbox{p10}_7,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{17})\qquad\mbox{cnf}(\mbox{p10}_8,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{16})\qquad\mbox{cnf}(\mbox{p10}_9,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{15},c_{17})\qquad\mbox{cnf}(\mbox{not\_p2}_{10},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(c_{18},c_{17}))\qquad\mbox{cnf}(\mbox{p12}_{11},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{13},f_4(c_{17},f_6(c_{18})))\qquad\mbox{cnf}(\mbox{not\_p11}_{12},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{13},f_4(f_6(c_{18}),c_{17}))\qquad\mbox{cnf}(\mbox{not\_p11}_{13},\mbox{negated\_conjecture})$

$( p_8(x_{34})\mbox{ and }p_7(x_{34},x_{33}) ) \ \Rightarrow \ p_8(x_{33})\qquad\mbox{cnf}(\mbox{p8}_{14},\mbox{negated\_conjecture})$

$( p_9(x_{36})\mbox{ and }p_7(x_{36},x_{35}) ) \ \Rightarrow \ p_9(x_{35})\qquad\mbox{cnf}(\mbox{p9}_{15},\mbox{negated\_conjecture})$

$p_{10}(c_{13},x_{10}) \ \Rightarrow \ p_{11}(c_{14},f_4(c_{15},x_{10}))\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_2(f_5(x_{18}),f_5(x_{19}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_6(x_{21}),f_6(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{30},x_{31})\mbox{ and }p_7(x_{30},x_{32}) ) \ \Rightarrow \ p_7(x_{31},x_{32})\qquad\mbox{cnf}(\mbox{p7}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{16})\mbox{ and }p_2(x_{15},x_{17}) ) \ \Rightarrow \ p_2(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{23},x_{24})\mbox{ and }p_3(x_{23},x_{25}) ) \ \Rightarrow \ p_3(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(x_{20},c_{17})) \ \Rightarrow \ p_2(f_6(x_{20}),f_5(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_2(x_3,x_1)\mbox{ and }p_7(x_2,x_0)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{14},x_{12})\mbox{ and }p_7(x_{13},x_{11})\mbox{ and }p_{12}(x_{13},x_{14}) ) \ \Rightarrow \ p_{12}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p12}_{24},\mbox{negated\_conjecture})$

$( p_3(x_8,x_6)\mbox{ and }p_7(x_7,x_5)\mbox{ and }p_{11}(x_7,x_8) ) \ \Rightarrow \ p_{11}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_3(f_4(x_{26},x_{27}),f_4(x_{28},x_{29}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(x_9,c_{17})) \ \Rightarrow \ ( p_{11}(c_{13},f_4(c_{17},f_5(x_9)))\mbox{ or }p_{11}(c_{13},f_4(f_5(x_9),c_{17})) )\qquad\mbox{cnf}(\mbox{p11}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN417+1.p} Harrison's cute problem






$\exists x{:}\;( x = f(g(x))\mbox{ and }\forall y{:}\;( y = f(g(y)) \ \Rightarrow \ x = y ) )\iff \exists x{:}\;( x = g(f(x))\mbox{ and }\forall y{:}\;( y = g(f(y)) \ \Rightarrow \ x = y ) )\qquad\mbox{fof}(\mbox{cute},\mbox{conjecture})$

\medskip\textbf{SYN075-1.p} Pelletier Problem 52






$\mbox{big\_f}(x,y) \ \Rightarrow \ x = a\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(x,y) \ \Rightarrow \ y = b\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( x = a\mbox{ and }y = b ) \ \Rightarrow \ \mbox{big\_f}(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( \mbox{big\_f}(y,f(x))\mbox{ and }y = g(x) ) \ \Rightarrow \ f(x) = x\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(y,f(x))\mbox{ and }\mbox{big\_f}(h(x,z),f(x)) ) \ \Rightarrow \ ( y = g(x)\mbox{ or }\mbox{big\_f}(h(x,z),f(x)) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$y = g(x) \ \Rightarrow \ ( \mbox{big\_f}(y,f(x))\mbox{ or }f(x) = x )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$y = g(x) \ \Rightarrow \ ( \mbox{big\_f}(y,f(x))\mbox{ or }\mbox{big\_f}(h(x,z),f(x))\mbox{ or }h(x,z) = z )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( y = g(x)\mbox{ and }h(x,z) = z\mbox{ and }\mbox{big\_f}(h(x,z),f(x)) ) \ \Rightarrow \ \mbox{big\_f}(y,f(x))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$f(x) = x \ \Rightarrow \ ( \mbox{big\_f}(h(x,z),f(x))\mbox{ or }h(x,z) = z )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( f(x) = x\mbox{ and }h(x,z) = z ) \ \Rightarrow \ \neg\,\mbox{big\_f}(h(x,z),f(x))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{SYN556-1.p} Harrison problem 2167






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_3(x_{15},x_{15})\qquad\mbox{cnf}(\mbox{p3}_2,\mbox{negated\_conjecture})$

$p_2(c_{10},c_{13})\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_2(x_9,x_{10}) \ \Rightarrow \ p_2(f_7(x_9),f_7(x_{10}))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_6(x_7),f_6(x_8))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$( p_3(x_{15},x_{16})\mbox{ and }p_3(x_{15},x_{17}) ) \ \Rightarrow \ p_3(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_3(c_9,f_4(c_{13},f_5(f_8(c_{11},c_{13}),f_6(f_7(c_{12}))),c_{11}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$\neg\,p_3(c_9,f_4(c_{10},f_5(f_8(c_{11},c_{10}),f_6(f_7(c_{12}))),c_{11}))\qquad\mbox{cnf}(\mbox{not\_p3}_9,\mbox{negated\_conjecture})$

$( p_2(x_3,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(f_5(x_3,x_4),f_5(x_5,x_6))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{14})\mbox{ and }p_3(x_{11},x_{13}) ) \ \Rightarrow \ p_2(f_8(x_{11},x_{12}),f_8(x_{13},x_{14}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_2(x_{19},x_{22})\mbox{ and }p_3(x_{20},x_{23})\mbox{ and }p_2(x_{18},x_{21}) ) \ \Rightarrow \ p_3(f_4(x_{18},x_{19},x_{20}),f_4(x_{21},x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{SYN592-1.p} Harrison problem 3584






$p_9(c_{14})\qquad\mbox{cnf}(\mbox{c14\_is\_p9}_1,\mbox{negated\_conjecture})$

$p_8(c_{13})\qquad\mbox{cnf}(\mbox{c13\_is\_p8}_2,\mbox{negated\_conjecture})$

$p_7(x_{30},x_{30})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_2(x_{15},x_{15})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(x_{23},x_{23})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_{10}(c_{14},x_4)\qquad\mbox{cnf}(\mbox{p10}_6,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{15})\qquad\mbox{cnf}(\mbox{p10}_7,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{17})\qquad\mbox{cnf}(\mbox{p10}_8,\mbox{negated\_conjecture})$

$p_{10}(c_{13},c_{16})\qquad\mbox{cnf}(\mbox{p10}_9,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{17},c_{15})\qquad\mbox{cnf}(\mbox{not\_p2}_{10},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(c_{18},c_{17}))\qquad\mbox{cnf}(\mbox{p12}_{11},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{13},f_4(c_{17},f_5(c_{18})))\qquad\mbox{cnf}(\mbox{not\_p11}_{12},\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{13},f_4(f_5(c_{18}),c_{17}))\qquad\mbox{cnf}(\mbox{not\_p11}_{13},\mbox{negated\_conjecture})$

$( p_8(x_{34})\mbox{ and }p_7(x_{34},x_{33}) ) \ \Rightarrow \ p_8(x_{33})\qquad\mbox{cnf}(\mbox{p8}_{14},\mbox{negated\_conjecture})$

$( p_9(x_{36})\mbox{ and }p_7(x_{36},x_{35}) ) \ \Rightarrow \ p_9(x_{35})\qquad\mbox{cnf}(\mbox{p9}_{15},\mbox{negated\_conjecture})$

$p_{10}(c_{13},x_{10}) \ \Rightarrow \ p_{11}(c_{14},f_4(c_{15},x_{10}))\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_2(f_5(x_{18}),f_5(x_{19}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_6(x_{21}),f_6(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{30},x_{31})\mbox{ and }p_7(x_{30},x_{32}) ) \ \Rightarrow \ p_7(x_{31},x_{32})\qquad\mbox{cnf}(\mbox{p7}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{16})\mbox{ and }p_2(x_{15},x_{17}) ) \ \Rightarrow \ p_2(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_3(x_{23},x_{24})\mbox{ and }p_3(x_{23},x_{25}) ) \ \Rightarrow \ p_3(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(x_{20},c_{17})) \ \Rightarrow \ p_2(f_6(x_{20}),f_5(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_2(x_3,x_1)\mbox{ and }p_7(x_2,x_0)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{14},x_{12})\mbox{ and }p_7(x_{13},x_{11})\mbox{ and }p_{12}(x_{13},x_{14}) ) \ \Rightarrow \ p_{12}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p12}_{24},\mbox{negated\_conjecture})$

$( p_3(x_8,x_6)\mbox{ and }p_7(x_7,x_5)\mbox{ and }p_{11}(x_7,x_8) ) \ \Rightarrow \ p_{11}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_3(f_4(x_{26},x_{27}),f_4(x_{28},x_{29}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_{12}(c_{14},f_4(x_9,c_{17})) \ \Rightarrow \ ( p_{11}(c_{13},f_4(c_{17},f_6(x_9)))\mbox{ or }p_{11}(c_{13},f_4(f_6(x_9),c_{17})) )\qquad\mbox{cnf}(\mbox{p11}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN316+1.p} Church problem 46.2 (2)






$\exists x{:}\;\forall y{:}\;( \mbox{big\_f}(x)\iff \mbox{big\_f}(y) )\qquad\mbox{fof}(\mbox{church\_46\_2}_2,\mbox{conjecture})$

\medskip\textbf{SYN361+1.p} Peter Andrews Problem X2112






$( \exists v{:}\;\forall x{:}\;\mbox{big\_p}(x,v)\mbox{ and }\forall x{:}\;( \mbox{big\_s}(x) \ \Rightarrow \ \exists y{:}\;\mbox{big\_q}(y,x) )\mbox{ and }\forall x,y{:}\;( \mbox{big\_p}(x,y) \ \Rightarrow \ \neg\,\mbox{big\_q}(x,y) ) ) \ \Rightarrow \ \exists u{:}\;\neg\,\mbox{big\_s}(u)\qquad\mbox{fof}(x_{2112},\mbox{conjecture})$

\medskip\textbf{SYN352+1.p} Church problem 46.18 (4)






$\forall x_1,x_2{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( \mbox{big\_f}(x_1,x_2) \ \Rightarrow \ ( ( \mbox{big\_f}(y_1,y_2) \ \Rightarrow \ ( \mbox{big\_f}(x_2,z)\mbox{ or }\mbox{big\_f}(y_2,z) ) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(y_1,y_2) \ \Rightarrow \ ( \mbox{big\_f}(x_2,z)\iff \mbox{big\_f}(y_1,z) ) ) \ \Rightarrow \ \mbox{big\_f}(z,z) ) \ \Rightarrow \ ( \mbox{big\_f}(y_1,y_2)\mbox{ and }( \mbox{big\_f}(y_1,z)\iff \mbox{big\_f}(y_2,z) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_18}_4,\mbox{conjecture})$

\medskip\textbf{SYN167-1.p} RPT63 synthetic problem 65 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(d,x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN036+1.p} Andrews Challenge Problem






$( \exists x{:}\;\forall y{:}\;( \mbox{big\_p}(x)\iff \mbox{big\_p}(y) )\iff ( \exists u{:}\;\mbox{big\_q}(u)\iff \forall w{:}\;\mbox{big\_q}(w) ) )\iff ( \exists x_1{:}\;\forall y_1{:}\;( \mbox{big\_q}(x_1)\iff \mbox{big\_q}(y_1) )\iff ( \exists u_1{:}\;\mbox{big\_p}(u_1)\iff \forall w_1{:}\;\mbox{big\_p}(w_1) ) )\qquad\mbox{fof}(\mbox{pel}_{34},\mbox{conjecture})$

\medskip\textbf{SYN084-1.p} Pelletier Problem 62






$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(y))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(f(f(x)))\mbox{ or }\mbox{big\_p}(x) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{big\_p}(a) \ \Rightarrow \ ( \mbox{big\_p}(y)\mbox{ or }\mbox{big\_p}(f(f(x)))\mbox{ or }\mbox{big\_p}(x) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(c)\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(f(c))\mbox{ or }\mbox{big\_p}(f(f(x)))\mbox{ or }\mbox{big\_p}(x) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(f(c)))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(f(f(x)))\mbox{ or }\mbox{big\_p}(x) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(f(c)))\mbox{ and }\mbox{big\_p}(f(x))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ \mbox{big\_p}(f(f(x)))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(c)\mbox{ and }\mbox{big\_p}(f(x))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(f(c))\mbox{ or }\mbox{big\_p}(f(f(x))) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(x))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ ( \mbox{big\_p}(y)\mbox{ or }\mbox{big\_p}(f(f(x))) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(y))\mbox{ and }\mbox{big\_p}(f(x))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ \mbox{big\_p}(f(f(x)))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(f(c)))\mbox{ and }\mbox{big\_p}(b) ) \ \Rightarrow \ \mbox{big\_p}(f(b))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{big\_p}(c)\mbox{ and }\mbox{big\_p}(b) ) \ \Rightarrow \ ( \mbox{big\_p}(f(c))\mbox{ or }\mbox{big\_p}(f(b)) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{big\_p}(a)\mbox{ and }\mbox{big\_p}(b) ) \ \Rightarrow \ ( \mbox{big\_p}(y)\mbox{ or }\mbox{big\_p}(f(b)) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{big\_p}(a)\mbox{ and }\mbox{big\_p}(f(y))\mbox{ and }\mbox{big\_p}(b) ) \ \Rightarrow \ \mbox{big\_p}(f(b))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

\medskip\textbf{SYN402+1.p} Kalish and Montague Problem 230

Laws of vacuous quantification.




$\forall x{:}\;( f(x) \ \Rightarrow \ \exists y{:}\;f(y) )\qquad\mbox{fof}(\mbox{kalish}_{230},\mbox{conjecture})$

\medskip\textbf{SYN728-1.p} Peter Andrews Problem THM69






$p(a,b) \ \Rightarrow \ p(c,c)\qquad\mbox{cnf}(\mbox{thm69}_1,\mbox{negated\_conjecture})$

$m(a)\mbox{ or }p(a,\mbox{sk}_1(a))\qquad\mbox{cnf}(\mbox{thm69}_2,\mbox{negated\_conjecture})$

$q(f(a,\mbox{sk}_1(a)))\mbox{ or }p(a,\mbox{sk}_1(a))\qquad\mbox{cnf}(\mbox{thm69}_3,\mbox{negated\_conjecture})$

$m(g(a)) \ \Rightarrow \ \neg\,q(a)\qquad\mbox{cnf}(\mbox{thm69}_4,\mbox{negated\_conjecture})$

$p(\mbox{sk}_2,\mbox{sk}_2) \ \Rightarrow \ \neg\,p(g(\mbox{sk}_2),a)\qquad\mbox{cnf}(\mbox{thm69}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN273-1.p} RPT63 synthetic problem 58 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(e,c,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN555-1.p} Harrison problem 2162






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_4(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$p_3(x_3,x_3)\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_4(f_5(c_7,c_{10},c_9),c_9)\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$\neg\,p_4(f_5(c_7,c_{10},c_8),c_8)\qquad\mbox{cnf}(\mbox{not\_p4}_5,\mbox{negated\_conjecture})$

$p_4(f_5(c_7,x_{24},c_8),f_5(c_7,x_{24},c_9))\qquad\mbox{cnf}(\mbox{p4}_6,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$( p_4(x_{12},x_{13})\mbox{ and }p_4(x_{12},x_{14}) ) \ \Rightarrow \ p_4(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p4}_8,\mbox{negated\_conjecture})$

$( p_3(x_3,x_4)\mbox{ and }p_3(x_3,x_5) ) \ \Rightarrow \ p_3(x_4,x_5)\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$( p_4(x_8,x_{11})\mbox{ and }p_4(x_7,x_{10})\mbox{ and }p_3(x_6,x_9) ) \ \Rightarrow \ p_3(f_6(x_6,x_7,x_8),f_6(x_9,x_{10},x_{11}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$( p_3(x_{16},x_{19})\mbox{ and }p_4(x_{17},x_{20})\mbox{ and }p_2(x_{15},x_{18}) ) \ \Rightarrow \ p_4(f_5(x_{15},x_{16},x_{17}),f_5(x_{18},x_{19},x_{20}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$( p_4(f_5(c_7,x_{21},x_{23}),x_{23})\mbox{ and }p_4(f_5(c_7,f_6(x_{21},x_{23},x_{22}),x_{23}),f_5(c_7,f_6(x_{21},x_{23},x_{22}),x_{22})) ) \ \Rightarrow \ p_4(f_5(c_7,x_{21},x_{22}),x_{22})\qquad\mbox{cnf}(\mbox{p4}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{SYN136-1.p} RPT63 synthetic problem 34 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_3(x,a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN610-1.p} Harrison problem 3830






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{28},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN159-1.p} RPT63 synthetic problem 57 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(a,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN046+1.p} Pelletier Problem 15






$( p \ \Rightarrow \ q )\iff ( \neg\,p\mbox{ or }q )\qquad\mbox{fof}(\mbox{pel}_{15},\mbox{conjecture})$

\medskip\textbf{SYN926+1.p} From Smullyan






$\exists x{:}\;( p(x)\mbox{ and }q(x) ) \ \Rightarrow \ ( \exists x{:}\;p(x)\mbox{ and }\exists x{:}\;q(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN044$\wedge$4.p} Pelletier Problem 10






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ r ) )\qquad\mbox{thf}(\mbox{pel10}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ r )@ ( \mbox{iand}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) ) )\qquad\mbox{thf}(\mbox{pel10}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{ior}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ r ) ) )\qquad\mbox{thf}(\mbox{pel10}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )\qquad\mbox{thf}(\mbox{pel}_{10},\mbox{conjecture})$

\medskip\textbf{SYN203-1.p} RPT63 synthetic problem 101 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(x,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN078-1.p} Pelletier Problem 56






$( \mbox{big\_f}(x)\mbox{ and }y = f(y)\mbox{ and }\mbox{big\_f}(z) ) \ \Rightarrow \ ( \mbox{big\_f}(y)\mbox{ or }\mbox{big\_f}(f(z)) )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }y = f(y) ) \ \Rightarrow \ ( \mbox{big\_f}(y)\mbox{ or }\mbox{big\_f}(a) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }y = f(y) ) \ \Rightarrow \ ( \mbox{big\_f}(y)\mbox{ or }b = f(b) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }y = f(y)\mbox{ and }\mbox{big\_f}(b) ) \ \Rightarrow \ \mbox{big\_f}(y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_f}(c)\mbox{ or }\mbox{big\_f}(f(x)) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\mbox{big\_f}(c)\mbox{ or }\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\mbox{big\_f}(c)\mbox{ or }b = f(b)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$\mbox{big\_f}(b) \ \Rightarrow \ \mbox{big\_f}(c)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(f(c))\mbox{ and }\mbox{big\_f}(x) ) \ \Rightarrow \ \mbox{big\_f}(f(x))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$\mbox{big\_f}(f(c)) \ \Rightarrow \ \mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$\mbox{big\_f}(f(c)) \ \Rightarrow \ b = f(b)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$\mbox{big\_f}(f(c)) \ \Rightarrow \ \neg\,\mbox{big\_f}(b)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{SYN571-1.p} Harrison problem 3108






$p_4(x_{26},x_{26})\qquad\mbox{cnf}(\mbox{p4}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_{11}(f_5(x_{10}),c_{14})\qquad\mbox{cnf}(\mbox{p11}_3,\mbox{negated\_conjecture})$

$\neg\,p_{11}(c_{13},f_5(x_{40}))\qquad\mbox{cnf}(\mbox{not\_p11}_4,\mbox{negated\_conjecture})$

$p_{11}(x_4,x_5)\mbox{ or }p_{11}(x_5,x_4)\qquad\mbox{cnf}(\mbox{p11}_5,\mbox{negated\_conjecture})$

$\neg\,p_{11}(x_{39},f_3(x_{39},f_5(f_6(f_7(c_{12})))))\qquad\mbox{cnf}(\mbox{not\_p11}_6,\mbox{negated\_conjecture})$

$p_{11}(f_5(f_{10}(x_9)),x_9) \ \Rightarrow \ p_{11}(c_{14},x_9)\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_4(x_{33},x_{34}) \ \Rightarrow \ p_4(f_7(x_{33}),f_7(x_{34}))\qquad\mbox{cnf}(\mbox{p4}_8,\mbox{negated\_conjecture})$

$p_4(x_{18},x_{19}) \ \Rightarrow \ p_2(f_5(x_{18}),f_5(x_{19}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_4(f_{10}(x_{29}),f_{10}(x_{30}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_4(x_{31},x_{32}) \ \Rightarrow \ p_4(f_6(x_{31}),f_6(x_{32}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_4(x_{26},x_{27})\mbox{ and }p_4(x_{26},x_{28}) ) \ \Rightarrow \ p_4(x_{27},x_{28})\qquad\mbox{cnf}(\mbox{p4}_{13},\mbox{negated\_conjecture})$

$p_2(f_9(f_5(x_{24}),f_5(x_{25})),f_5(f_8(x_{24},x_{25})))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_{11}(f_9(x_6,x_8),x_7) \ \Rightarrow \ p_{11}(x_6,f_3(x_7,x_8))\qquad\mbox{cnf}(\mbox{p11}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_6,f_3(x_7,x_8)) \ \Rightarrow \ p_{11}(f_9(x_6,x_8),x_7)\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{11}(x_2,x_3) ) \ \Rightarrow \ p_{11}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p11}_{17},\mbox{negated\_conjecture})$

$( p_4(x_{35},x_{37})\mbox{ and }p_4(x_{36},x_{38}) ) \ \Rightarrow \ p_4(f_8(x_{35},x_{36}),f_8(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p4}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{14},x_{16})\mbox{ and }p_2(x_{15},x_{17}) ) \ \Rightarrow \ p_2(f_3(x_{14},x_{15}),f_3(x_{16},x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{22})\mbox{ and }p_2(x_{21},x_{23}) ) \ \Rightarrow \ p_2(f_9(x_{20},x_{21}),f_9(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{SYN992$\wedge$1.p} There exists a reflexive relation






$\mbox{refl}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{refl\_type},\mbox{type})$

$\mbox{refl} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( r@ x@ x ) )\qquad\mbox{thf}(\mbox{refl},\mbox{definition})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{refl}@ r )\qquad\mbox{thf}(\mbox{ax},\mbox{conjecture})$

\medskip\textbf{SYN624-1.p} Harrison problem 4028






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_5(x_{57},x_{57})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_3(x_{46},x_{46})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{11}(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p11}_5,\mbox{negated\_conjecture})$

$\neg\,p_{18}(f_6(c_{24}))\qquad\mbox{cnf}(\mbox{f6\_is\_not\_p18}_6,\mbox{negated\_conjecture})$

$\neg\,p_{18}(f_6(c_{22}))\qquad\mbox{cnf}(\mbox{f6\_is\_not\_p18}_7,\mbox{negated\_conjecture})$

$p_3(f_4(c_{22}),c_{23})\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_{19}(f_7(f_8(f_9(c_{25}))),c_{23})\qquad\mbox{cnf}(\mbox{p19}_9,\mbox{negated\_conjecture})$

$( p_{18}(x_{29})\mbox{ and }p_5(x_{29},x_{28}) ) \ \Rightarrow \ p_{18}(x_{28})\qquad\mbox{cnf}(\mbox{p18}_{10},\mbox{negated\_conjecture})$

$p_{10}(f_{16}(x_{16},f_{14}(f_{15}(f_7(c_{25})))),x_{16})\qquad\mbox{cnf}(\mbox{p10}_{11},\mbox{negated\_conjecture})$

$p_{11}(x_{10},x_{11}) \ \Rightarrow \ p_{10}(f_{14}(x_{10}),f_{14}(x_{11}))\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$p_2(x_{60},x_{61}) \ \Rightarrow \ p_5(f_6(x_{60}),f_6(x_{61}))\qquad\mbox{cnf}(\mbox{p5}_{13},\mbox{negated\_conjecture})$

$p_3(x_{55},x_{56}) \ \Rightarrow \ p_3(f_9(x_{55}),f_9(x_{56}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{53},x_{54}) \ \Rightarrow \ p_3(f_8(x_{53}),f_8(x_{54}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_3(x_{51},x_{52}) \ \Rightarrow \ p_3(f_7(x_{51}),f_7(x_{52}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$p_2(x_{49},x_{50}) \ \Rightarrow \ p_3(f_4(x_{49}),f_4(x_{50}))\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$p_3(x_{26},x_{27}) \ \Rightarrow \ p_{11}(f_{15}(x_{26}),f_{15}(x_{27}))\qquad\mbox{cnf}(\mbox{p11}_{18},\mbox{negated\_conjecture})$

$p_{10}(x_{24},x_{25}) \ \Rightarrow \ p_{11}(f_{12}(x_{24}),f_{12}(x_{25}))\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_{13}(f_6(c_{24}),c_{26}),f_{14}(f_{15}(f_7(c_{25}))))\qquad\mbox{cnf}(\mbox{not\_p10}_{20},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{21},\mbox{negated\_conjecture})$

$( p_5(x_{57},x_{58})\mbox{ and }p_5(x_{57},x_{59}) ) \ \Rightarrow \ p_5(x_{58},x_{59})\qquad\mbox{cnf}(\mbox{p5}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{46},x_{47})\mbox{ and }p_3(x_{46},x_{48}) ) \ \Rightarrow \ p_3(x_{47},x_{48})\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{34},x_{35})\mbox{ and }p_2(x_{34},x_{36}) ) \ \Rightarrow \ p_2(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$( p_{11}(x_{21},x_{22})\mbox{ and }p_{11}(x_{21},x_{23}) ) \ \Rightarrow \ p_{11}(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

$p_{10}(f_{13}(f_6(c_{22}),x_9),f_{13}(f_6(c_{24}),f_{16}(c_{26},x_9)))\qquad\mbox{cnf}(\mbox{p10}_{26},\mbox{negated\_conjecture})$

$p_{20}(f_{12}(f_{13}(f_6(c_{24}),c_{26})),f_{12}(f_{13}(f_6(c_{24}),x_{41})))\qquad\mbox{cnf}(\mbox{p20}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_{13}(f_6(c_{24}),c_{26}),f_{13}(f_6(c_{22}),f_{14}(f_{15}(f_7(c_{25})))))\qquad\mbox{cnf}(\mbox{not\_p10}_{28},\mbox{negated\_conjecture})$

$( p_3(x_{44},x_{42})\mbox{ and }p_3(x_{45},x_{43})\mbox{ and }p_{21}(x_{44},x_{45}) ) \ \Rightarrow \ p_{21}(x_{42},x_{43})\qquad\mbox{cnf}(\mbox{p21}_{29},\mbox{negated\_conjecture})$

$( p_3(x_{32},x_{30})\mbox{ and }p_3(x_{33},x_{31})\mbox{ and }p_{19}(x_{32},x_{33}) ) \ \Rightarrow \ p_{19}(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p19}_{30},\mbox{negated\_conjecture})$

$( p_{11}(x_{40},x_{38})\mbox{ and }p_{20}(x_{39},x_{40})\mbox{ and }p_{11}(x_{39},x_{37}) ) \ \Rightarrow \ p_{20}(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p20}_{31},\mbox{negated\_conjecture})$

$( p_{10}(x_4,x_6)\mbox{ and }p_5(x_3,x_5) ) \ \Rightarrow \ p_{10}(f_{13}(x_3,x_4),f_{13}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p10}_{32},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{20})\mbox{ and }p_3(x_{17},x_{19}) ) \ \Rightarrow \ p_{10}(f_{17}(x_{17},x_{18}),f_{17}(x_{19},x_{20}))\qquad\mbox{cnf}(\mbox{p10}_{33},\mbox{negated\_conjecture})$

$( p_{10}(x_{12},x_{14})\mbox{ and }p_{10}(x_{13},x_{15}) ) \ \Rightarrow \ p_{10}(f_{16}(x_{12},x_{13}),f_{16}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p10}_{34},\mbox{negated\_conjecture})$

$( p_{21}(x_8,c_{23})\mbox{ and }p_3(f_4(x_7),x_8) ) \ \Rightarrow \ ( p_{18}(f_6(x_7))\mbox{ or }p_{10}(f_{13}(f_6(x_7),f_{17}(x_8,x_7)),f_{14}(f_{15}(f_7(c_{25})))) )\qquad\mbox{cnf}(\mbox{p18}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{SYN253-1.p} RPT63 synthetic problem 38 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_4(e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN231-1.p} RPT63 synthetic problem 16 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN344-1.p} Church problem 46.16 (3)






$f(a,z(y_1,y_2))\mbox{ or }f(a,a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y_1,z(y_1,y_2))\mbox{ or }f(a,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y_2,a) \ \Rightarrow \ f(a,a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(a,a) \ \Rightarrow \ f(y_1,y_2)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(a,a) \ \Rightarrow \ \neg\,f(z(y_1,y_2),z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN338-1.p} Church problem 46.15 (3)






$f(x,y(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(z,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,f(y(x),y(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN317-1.p} Church problem 46.2 (3)






$f(z)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,g(z)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(c) \ \Rightarrow \ ( g(c)\mbox{ or }f(a)\mbox{ or }g(b) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN063-2.p} Pelletier Problem 33






$\mbox{big\_p}(a) \ \Rightarrow \ \mbox{big\_p}(c)\qquad\mbox{cnf}(\mbox{pel33}_1,\mbox{negated\_conjecture})$

$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{pel33}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_p}(c)\qquad\mbox{cnf}(\mbox{pel33}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN137-1.p} RPT63 synthetic problem 35 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_3(b,c,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN145-1.p} RPT63 synthetic problem 43 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_0(d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN030-1.p} EW3






$( t\mbox{ and }s ) \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$r \ \Rightarrow \ s\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$t \ \Rightarrow \ r\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( r\mbox{ and }q ) \ \Rightarrow \ t\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$q \ \Rightarrow \ r\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$q\mbox{ or }r\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$r \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( s\mbox{ and }p ) \ \Rightarrow \ q\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$( p\mbox{ and }q ) \ \Rightarrow \ \neg\,r\qquad\mbox{cnf}(\mbox{theorem},\mbox{negated\_conjecture})$

\medskip\textbf{SYN001$\wedge$4.002.p} ILTP Problem SYJ212+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ a_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_2 )@ ( \mbox{iatom}@ a_1 ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYN254-1.p} RPT63 synthetic problem 39 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_4(e,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN572-1.p} Harrison problem 3112






$p_2(x_6,x_6)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_6(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$p_4(x_{22},x_{22})\qquad\mbox{cnf}(\mbox{p4}_3,\mbox{negated\_conjecture})$

$p_4(f_{10}(f_9(x_{27})),f_{10}(x_{27}))\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_2(f_3(x_{13},f_5(f_7(f_8(c_{13})))),x_{13})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_6(x_{37},x_{38}) \ \Rightarrow \ p_6(f_8(x_{37}),f_8(x_{38}))\qquad\mbox{cnf}(\mbox{p6}_6,\mbox{negated\_conjecture})$

$p_4(x_{18},x_{19}) \ \Rightarrow \ p_2(f_5(x_{18}),f_5(x_{19}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_2(x_{20},x_{21}) \ \Rightarrow \ p_2(f_9(x_{20}),f_9(x_{21}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_2(x_{25},x_{26}) \ \Rightarrow \ p_4(f_{10}(x_{25}),f_{10}(x_{26}))\qquad\mbox{cnf}(\mbox{p4}_9,\mbox{negated\_conjecture})$

$p_6(x_{32},x_{33}) \ \Rightarrow \ p_4(f_7(x_{32}),f_7(x_{33}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_2(f_3(x_{14},f_9(x_{14})),f_5(f_7(f_8(c_{13}))))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_2(x_6,x_7)\mbox{ and }p_2(x_6,x_8) ) \ \Rightarrow \ p_2(x_7,x_8)\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_6(x_{34},x_{35})\mbox{ and }p_6(x_{34},x_{36}) ) \ \Rightarrow \ p_6(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p6}_{13},\mbox{negated\_conjecture})$

$( p_4(x_{22},x_{23})\mbox{ and }p_4(x_{22},x_{24}) ) \ \Rightarrow \ p_4(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p4}_{14},\mbox{negated\_conjecture})$

$p_{12}(f_{10}(f_3(x_4,x_5)),f_{11}(f_{10}(x_4),f_{10}(x_5)))\qquad\mbox{cnf}(\mbox{p12}_{15},\mbox{negated\_conjecture})$

$\neg\,p_{12}(f_{10}(c_{14}),f_{11}(f_{10}(f_3(c_{14},c_{15})),f_{10}(c_{15})))\qquad\mbox{cnf}(\mbox{not\_p12}_{16},\mbox{negated\_conjecture})$

$p_2(f_3(x_{15},f_3(x_{16},x_{17})),f_3(f_3(x_{15},x_{16}),x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_4(x_2,x_0)\mbox{ and }p_4(x_3,x_1)\mbox{ and }p_{12}(x_2,x_3) ) \ \Rightarrow \ p_{12}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p12}_{18},\mbox{negated\_conjecture})$

$( p_4(x_{28},x_{30})\mbox{ and }p_4(x_{29},x_{31}) ) \ \Rightarrow \ p_4(f_{11}(x_{28},x_{29}),f_{11}(x_{30},x_{31}))\qquad\mbox{cnf}(\mbox{p4}_{19},\mbox{negated\_conjecture})$

$( p_2(x_9,x_{11})\mbox{ and }p_2(x_{10},x_{12}) ) \ \Rightarrow \ p_2(f_3(x_9,x_{10}),f_3(x_{11},x_{12}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{SYN187-1.p} RPT63 synthetic problem 85 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_1(d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN191-1.p} RPT63 synthetic problem 89 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_3(c,b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN165-1.p} RPT63 synthetic problem 63 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(x,a,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN643-1.p} Harrison problem 4298






$p_{15}(f_3(c_{19}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_1,\mbox{negated\_conjecture})$

$p_{15}(f_3(c_{18}))\qquad\mbox{cnf}(\mbox{f3\_is\_p15}_2,\mbox{negated\_conjecture})$

$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_3,\mbox{negated\_conjecture})$

$p_6(x_{45},x_{45})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_4(x_{36},x_{36})\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{16}(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p16}_7,\mbox{negated\_conjecture})$

$p_{12}(x_3,x_3)\qquad\mbox{cnf}(\mbox{p12}_8,\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{19})),f_8(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_9,\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{18})),f_5(c_{18}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_4(f_5(f_3(c_{19})),f_5(c_{19}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{18})),f_7(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{12},\mbox{negated\_conjecture})$

$p_6(f_7(f_3(c_{19})),f_7(c_{19}))\qquad\mbox{cnf}(\mbox{p6}_{13},\mbox{negated\_conjecture})$

$p_6(f_8(f_3(c_{18})),f_8(c_{18}))\qquad\mbox{cnf}(\mbox{p6}_{14},\mbox{negated\_conjecture})$

$( p_{15}(x_{17})\mbox{ and }p_2(x_{17},x_{16}) ) \ \Rightarrow \ p_{15}(x_{16})\qquad\mbox{cnf}(\mbox{p15}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_6,x_7) \ \Rightarrow \ p_{12}(f_{13}(x_6),f_{13}(x_7))\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$p_2(x_{54},x_{55}) \ \Rightarrow \ p_6(f_8(x_{54}),f_8(x_{55}))\qquad\mbox{cnf}(\mbox{p6}_{17},\mbox{negated\_conjecture})$

$p_2(x_{52},x_{53}) \ \Rightarrow \ p_6(f_7(x_{52}),f_7(x_{53}))\qquad\mbox{cnf}(\mbox{p6}_{18},\mbox{negated\_conjecture})$

$p_2(x_{39},x_{40}) \ \Rightarrow \ p_4(f_5(x_{39}),f_5(x_{40}))\qquad\mbox{cnf}(\mbox{p4}_{19},\mbox{negated\_conjecture})$

$p_2(x_{34},x_{35}) \ \Rightarrow \ p_2(f_3(x_{34}),f_3(x_{35}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_{15}(f_{10}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{22},\mbox{negated\_conjecture})$

$( p_6(x_{45},x_{46})\mbox{ and }p_6(x_{45},x_{47}) ) \ \Rightarrow \ p_6(x_{46},x_{47})\qquad\mbox{cnf}(\mbox{p6}_{23},\mbox{negated\_conjecture})$

$( p_4(x_{36},x_{37})\mbox{ and }p_4(x_{36},x_{38}) ) \ \Rightarrow \ p_4(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p4}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_2(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_{16}(x_{18},x_{19})\mbox{ and }p_{16}(x_{18},x_{20}) ) \ \Rightarrow \ p_{16}(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p16}_{26},\mbox{negated\_conjecture})$

$( p_{12}(x_3,x_4)\mbox{ and }p_{12}(x_3,x_5) ) \ \Rightarrow \ p_{12}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

$\neg\,p_4(f_5(f_{10}(f_3(c_{19}),f_3(c_{18}))),f_9(f_5(c_{19}),f_5(c_{18})))\qquad\mbox{cnf}(\mbox{not\_p4}_{28},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,f_3(c_{19})) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,c_{19}) )\qquad\mbox{cnf}(\mbox{p12}_{29},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},f_3(c_{18})) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},c_{18}) )\qquad\mbox{cnf}(\mbox{p12}_{30},\mbox{negated\_conjecture})$

$p_{17}(x_8,x_9,c_{19}) \ \Rightarrow \ ( p_{12}(f_{13}(x_8),c_{20})\mbox{ or }p_{17}(x_8,x_9,f_3(c_{19})) )\qquad\mbox{cnf}(\mbox{p12}_{31},\mbox{negated\_conjecture})$

$p_{17}(x_{10},x_{11},c_{18}) \ \Rightarrow \ ( p_{12}(f_{13}(x_{10}),c_{20})\mbox{ or }p_{17}(x_{10},x_{11},f_3(c_{18})) )\qquad\mbox{cnf}(\mbox{p12}_{32},\mbox{negated\_conjecture})$

$( p_6(x_{48},x_{50})\mbox{ and }p_6(x_{49},x_{51}) ) \ \Rightarrow \ p_6(f_{14}(x_{48},x_{49}),f_{14}(x_{50},x_{51}))\qquad\mbox{cnf}(\mbox{p6}_{33},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{32})\mbox{ and }p_2(x_{31},x_{33}) ) \ \Rightarrow \ p_2(f_{10}(x_{30},x_{31}),f_{10}(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p2}_{34},\mbox{negated\_conjecture})$

$( p_4(x_{42},x_{44})\mbox{ and }p_4(x_{41},x_{43}) ) \ \Rightarrow \ p_4(f_9(x_{41},x_{42}),f_9(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p4}_{35},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_8(f_{10}(x_{14},x_{15})),f_{14}(f_8(x_{14}),f_8(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{36},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_4(f_5(f_{10}(x_{14},x_{15})),f_9(f_5(x_{14}),f_5(x_{15})))\qquad\mbox{cnf}(\mbox{p4}_{37},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ p_6(f_7(f_{10}(x_{14},x_{15})),f_{14}(f_7(x_{14}),f_7(x_{15})))\qquad\mbox{cnf}(\mbox{p6}_{38},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{23})\mbox{ and }p_{16}(x_{25},x_{22})\mbox{ and }p_{11}(x_{24},x_{21})\mbox{ and }p_{17}(x_{24},x_{25},x_{26}) ) \ \Rightarrow \ p_{17}(x_{21},x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p17}_{39},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{14})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{40},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{15}) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) )\qquad\mbox{cnf}(\mbox{p12}_{41},\mbox{negated\_conjecture})$

$( p_{15}(x_{14})\mbox{ and }p_{15}(x_{15})\mbox{ and }p_{17}(x_{12},x_{13},x_{14})\mbox{ and }p_{17}(x_{12},x_{13},f_{10}(x_{14},x_{15})) ) \ \Rightarrow \ ( p_{12}(f_{13}(x_{12}),c_{20})\mbox{ or }p_{17}(x_{12},x_{13},x_{15}) )\qquad\mbox{cnf}(\mbox{p12}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{SYN244-1.p} RPT63 synthetic problem 29 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(c,x,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN377$\wedge$5.p} TPS problem X2128






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cX}_{2128},\mbox{conjecture})$

\medskip\textbf{SYN339-1.p} Church problem 46.15 (4)






$f(x,y(x),z)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,f(y(x),z,z)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN347+1.p} Church problem 46.17 (3)






$\forall z_1,z_2{:}\;\exists x_1,x_2{:}\;\forall y{:}\;( ( ( \mbox{big\_f}(x_1,y)\iff \mbox{big\_f}(x_2,y) )\iff \mbox{big\_f}(z_1,z_2) )\mbox{ or }( \mbox{big\_f}(z_1,y)\iff \mbox{big\_f}(z_2,y) ) )\qquad\mbox{fof}(\mbox{church\_46\_17}_3,\mbox{conjecture})$

\medskip\textbf{SYN618-1.p} Harrison problem 3944






$p_9(c_{16})\qquad\mbox{cnf}(\mbox{c16\_is\_p9}_1,\mbox{negated\_conjecture})$

$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_2,\mbox{negated\_conjecture})$

$p_5(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_4(x_{29},x_{29})\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_2(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{14}(x_{17},x_{17})\qquad\mbox{cnf}(\mbox{p14}_6,\mbox{negated\_conjecture})$

$p_{12}(c_{19},c_{20})\qquad\mbox{cnf}(\mbox{p12}_7,\mbox{negated\_conjecture})$

$p_{13}(c_{23},c_{21})\qquad\mbox{cnf}(\mbox{p13}_8,\mbox{negated\_conjecture})$

$p_{13}(c_{19},c_{23})\qquad\mbox{cnf}(\mbox{p13}_9,\mbox{negated\_conjecture})$

$p_{13}(c_{19},c_{21})\qquad\mbox{cnf}(\mbox{p13}_{10},\mbox{negated\_conjecture})$

$p_{12}(c_{20},c_{21})\qquad\mbox{cnf}(\mbox{p12}_{11},\mbox{negated\_conjecture})$

$p_2(c_{17},f_3(c_{16}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_{11}(c_{16},c_{17},c_{18})\qquad\mbox{cnf}(\mbox{p11}_{13},\mbox{negated\_conjecture})$

$p_5(f_8(c_{16},c_{20}),f_6(f_7(c_{22})))\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_9(x_{45})\mbox{ and }p_2(x_{45},x_{44}) ) \ \Rightarrow \ p_9(x_{44})\qquad\mbox{cnf}(\mbox{p9}_{15},\mbox{negated\_conjecture})$

$p_5(f_8(f_3(c_{16}),c_{23}),f_6(f_7(c_{22})))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$\neg\,p_5(f_8(f_3(c_{16}),c_{20}),f_6(f_7(c_{22})))\qquad\mbox{cnf}(\mbox{not\_p5}_{17},\mbox{negated\_conjecture})$

$p_4(x_{38},x_{39}) \ \Rightarrow \ p_5(f_6(x_{38}),f_6(x_{39}))\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$p_2(x_{27},x_{28}) \ \Rightarrow \ p_2(f_3(x_{27}),f_3(x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$p_4(x_{32},x_{33}) \ \Rightarrow \ p_4(f_7(x_{32}),f_7(x_{33}))\qquad\mbox{cnf}(\mbox{p4}_{20},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{21},\mbox{negated\_conjecture})$

$( p_5(x_{34},x_{35})\mbox{ and }p_5(x_{34},x_{36}) ) \ \Rightarrow \ p_5(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p5}_{22},\mbox{negated\_conjecture})$

$( p_4(x_{29},x_{30})\mbox{ and }p_4(x_{29},x_{31}) ) \ \Rightarrow \ p_4(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p4}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{24},x_{25})\mbox{ and }p_2(x_{24},x_{26}) ) \ \Rightarrow \ p_2(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$( p_{14}(x_{17},x_{18})\mbox{ and }p_{14}(x_{17},x_{19}) ) \ \Rightarrow \ p_{14}(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p14}_{25},\mbox{negated\_conjecture})$

$( p_5(x_{11},x_9)\mbox{ and }p_5(x_{12},x_{10})\mbox{ and }p_{12}(x_{11},x_{12}) ) \ \Rightarrow \ p_{12}(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p12}_{26},\mbox{negated\_conjecture})$

$( p_{14}(x_{23},x_{20})\mbox{ and }p_{15}(x_{23},x_{22})\mbox{ and }p_{10}(x_{22},x_{21}) ) \ \Rightarrow \ p_{15}(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p15}_{27},\mbox{negated\_conjecture})$

$( p_5(x_{15},x_{13})\mbox{ and }p_5(x_{16},x_{14})\mbox{ and }p_{13}(x_{15},x_{16}) ) \ \Rightarrow \ p_{13}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p13}_{28},\mbox{negated\_conjecture})$

$( p_{13}(x_{37},c_{21})\mbox{ and }p_{13}(c_{19},x_{37})\mbox{ and }p_{15}(c_{24},c_{18}) ) \ \Rightarrow \ p_5(x_{37},c_{20})\qquad\mbox{cnf}(\mbox{p5}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{40},x_{42})\mbox{ and }p_5(x_{41},x_{43}) ) \ \Rightarrow \ p_5(f_8(x_{40},x_{41}),f_8(x_{42},x_{43}))\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

$( p_{13}(x_{37},c_{21})\mbox{ and }p_{13}(c_{19},x_{37})\mbox{ and }p_5(f_8(c_{17},x_{37}),f_6(f_7(c_{22}))) ) \ \Rightarrow \ p_5(x_{37},c_{20})\qquad\mbox{cnf}(\mbox{p5}_{31},\mbox{negated\_conjecture})$

$( p_{13}(x_{37},c_{21})\mbox{ and }p_{13}(c_{19},x_{37})\mbox{ and }p_5(f_8(c_{16},x_{37}),f_6(f_7(c_{22}))) ) \ \Rightarrow \ p_5(x_{37},c_{20})\qquad\mbox{cnf}(\mbox{p5}_{32},\mbox{negated\_conjecture})$

$( p_2(x_8,x_4)\mbox{ and }p_2(x_7,x_3)\mbox{ and }p_{10}(x_6,x_5)\mbox{ and }p_{11}(x_7,x_8,x_6) ) \ \Rightarrow \ p_{11}(x_3,x_4,x_5)\qquad\mbox{cnf}(\mbox{p11}_{33},\mbox{negated\_conjecture})$

\medskip\textbf{SYN357$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ ( \mbox{cP}@ y ) )\qquad\mbox{thf}(\mbox{cLX}_{2108},\mbox{conjecture})$

\medskip\textbf{SYN264-1.p} RPT63 synthetic problem 49 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(e,c,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN616-1.p} Harrison problem 3901






$p_{17}(c_{28})\qquad\mbox{cnf}(\mbox{c28\_is\_p17}_1,\mbox{negated\_conjecture})$

$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_2,\mbox{negated\_conjecture})$

$p_9(x_{60},x_{60})\qquad\mbox{cnf}(\mbox{p9}_3,\mbox{negated\_conjecture})$

$p_2(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{18}(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p18}_5,\mbox{negated\_conjecture})$

$p_{19}(c_{26},c_{27})\qquad\mbox{cnf}(\mbox{p19}_6,\mbox{negated\_conjecture})$

$p_2(c_{25},c_{28})\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$\neg\,p_{15}(c_{25},c_{27})\qquad\mbox{cnf}(\mbox{not\_p15}_8,\mbox{negated\_conjecture})$

$p_{13}(f_3(f_4(f_5(c_{20}))),c_{21})\qquad\mbox{cnf}(\mbox{p13}_9,\mbox{negated\_conjecture})$

$( p_{17}(x_{23})\mbox{ and }p_2(x_{23},x_{22}) ) \ \Rightarrow \ p_{17}(x_{22})\qquad\mbox{cnf}(\mbox{p17}_{10},\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_2(f_6(x_{50}),f_6(x_{51}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{44},x_{45}) \ \Rightarrow \ p_2(f_3(x_{44}),f_3(x_{45}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{46},x_{47}) \ \Rightarrow \ p_2(f_4(x_{46}),f_4(x_{47}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_2(f_5(x_{48}),f_5(x_{49}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$( p_9(x_{60},x_{61})\mbox{ and }p_9(x_{60},x_{62}) ) \ \Rightarrow \ p_9(x_{61},x_{62})\qquad\mbox{cnf}(\mbox{p9}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{31},x_{32})\mbox{ and }p_2(x_{31},x_{33}) ) \ \Rightarrow \ p_2(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_{18}(x_{24},x_{25})\mbox{ and }p_{18}(x_{24},x_{26}) ) \ \Rightarrow \ p_{18}(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p18}_{18},\mbox{negated\_conjecture})$

$p_2(f_{11}(c_{23},c_{24},f_3(f_5(c_{20}))),f_8(c_{21},f_3(f_5(c_{20}))))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_{17}(x_{17})\mbox{ and }p_{16}(x_{17},f_8(c_{21},f_3(f_5(c_{20})))) ) \ \Rightarrow \ p_{15}(x_{17},c_{24})\qquad\mbox{cnf}(\mbox{p15}_{20},\mbox{negated\_conjecture})$

$( p_2(x_5,x_3)\mbox{ and }p_2(x_6,x_4)\mbox{ and }p_{13}(x_5,x_6) ) \ \Rightarrow \ p_{13}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p13}_{21},\mbox{negated\_conjecture})$

$( p_{18}(x_{30},x_{27})\mbox{ and }p_{19}(x_{30},x_{29})\mbox{ and }p_{10}(x_{29},x_{28}) ) \ \Rightarrow \ p_{19}(x_{27},x_{28})\qquad\mbox{cnf}(\mbox{p19}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{20},x_{18})\mbox{ and }p_2(x_{21},x_{19})\mbox{ and }p_{16}(x_{20},x_{21}) ) \ \Rightarrow \ p_{16}(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p16}_{23},\mbox{negated\_conjecture})$

$( p_{15}(x_{16},x_{15})\mbox{ and }p_2(x_{16},x_{13})\mbox{ and }p_{10}(x_{15},x_{14}) ) \ \Rightarrow \ p_{15}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p15}_{24},\mbox{negated\_conjecture})$

$p_{14}(f_7(c_{22},f_8(c_{21},f_3(f_5(c_{20})))),f_3(f_5(c_{20})),f_6(c_{21}))\qquad\mbox{cnf}(\mbox{p14}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{56},x_{58})\mbox{ and }p_2(x_{57},x_{59}) ) \ \Rightarrow \ p_2(f_8(x_{56},x_{57}),f_8(x_{58},x_{59}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$( p_2(x_{40},x_{42})\mbox{ and }p_2(x_{41},x_{43}) ) \ \Rightarrow \ p_2(f_{12}(x_{40},x_{41}),f_{12}(x_{42},x_{43}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_2(x_{53},x_{55})\mbox{ and }p_2(x_{52},x_{54}) ) \ \Rightarrow \ p_2(f_7(x_{52},x_{53}),f_7(x_{54},x_{55}))\qquad\mbox{cnf}(\mbox{p2}_{28},\mbox{negated\_conjecture})$

$p_{14}(f_7(c_{22},f_{12}(f_8(c_{21},f_3(f_5(c_{20}))),c_{25})),f_3(f_5(c_{20})),f_6(c_{21}))\qquad\mbox{cnf}(\mbox{p14}_{29},\mbox{negated\_conjecture})$

$\neg\,p_{14}(f_7(c_{22},f_{12}(f_8(c_{21},f_3(f_5(c_{20}))),c_{28})),f_3(f_5(c_{20})),f_6(c_{21}))\qquad\mbox{cnf}(\mbox{not\_p14}_{30},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_9)\mbox{ and }p_2(x_{11},x_8)\mbox{ and }p_2(x_{10},x_7)\mbox{ and }p_{14}(x_{10},x_{11},x_{12}) ) \ \Rightarrow \ p_{14}(x_7,x_8,x_9)\qquad\mbox{cnf}(\mbox{p14}_{31},\mbox{negated\_conjecture})$

$( p_2(x_{36},x_{39})\mbox{ and }p_9(x_{34},x_{37})\mbox{ and }p_{10}(x_{35},x_{38}) ) \ \Rightarrow \ p_2(f_{11}(x_{34},x_{35},x_{36}),f_{11}(x_{37},x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p2}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{SYN040+1.p} Pelletier Problem 1

A biconditional version of the 'most difficult' theorem
 proved by the original Logic Theorist.




$( p \ \Rightarrow \ q )\iff ( \neg\,q \ \Rightarrow \ \neg\,p )\qquad\mbox{fof}(\mbox{pel}_1,\mbox{conjecture})$

\medskip\textbf{SYN065+1.p} Pelletier Problem 36






$\forall x{:}\;\exists y{:}\;\mbox{big\_f}(x,y)\qquad\mbox{fof}(\mbox{pel36}_1,\mbox{axiom})$

$\forall x{:}\;\exists y{:}\;\mbox{big\_g}(x,y)\qquad\mbox{fof}(\mbox{pel36}_2,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{big\_f}(x,y)\mbox{ or }\mbox{big\_g}(x,y) ) \ \Rightarrow \ \forall z{:}\;( ( \mbox{big\_f}(y,z)\mbox{ or }\mbox{big\_g}(y,z) ) \ \Rightarrow \ \mbox{big\_h}(x,z) ) )\qquad\mbox{fof}(\mbox{pel36}_3,\mbox{axiom})$

$\forall x{:}\;\exists y{:}\;\mbox{big\_h}(x,y)\qquad\mbox{fof}(\mbox{pel}_{36},\mbox{conjecture})$

\medskip\textbf{SYN554-1.p} Harrison problem 1667






$p_2(x_2,x_2)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$\neg\,p_4(x_{15},x_{15})\qquad\mbox{cnf}(\mbox{not\_p4}_2,\mbox{negated\_conjecture})$

$\neg\,p_2(c_6,c_7)\qquad\mbox{cnf}(\mbox{not\_p2}_3,\mbox{negated\_conjecture})$

$p_2(f_3(c_5,c_6),f_3(c_5,c_7))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(x_0,x_1)\mbox{ or }p_4(x_0,x_1)\mbox{ or }p_4(x_1,x_0)\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$( p_2(x_2,x_4)\mbox{ and }p_2(x_2,x_3) ) \ \Rightarrow \ p_2(x_3,x_4)\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_4(x_{13},x_{14}) \ \Rightarrow \ p_4(f_3(c_5,x_{13}),f_3(c_5,x_{14}))\qquad\mbox{cnf}(\mbox{p4}_7,\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{10})\mbox{ and }p_4(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_9) ) \ \Rightarrow \ p_4(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p4}_8,\mbox{negated\_conjecture})$

$( p_2(x_5,x_7)\mbox{ and }p_2(x_6,x_8) ) \ \Rightarrow \ p_2(f_3(x_5,x_6),f_3(x_7,x_8))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN920+1.p} From Smullyan






$( ( \forall x{:}\;( ( f(x)\mbox{ and }g(x) ) \ \Rightarrow \ h(x) ) \ \Rightarrow \ \exists x{:}\;( f(x)\mbox{ and }\neg\,g(x) ) )\mbox{ and }( \forall w{:}\;( f(w) \ \Rightarrow \ g(w) )\mbox{ or }\forall z{:}\;( f(z) \ \Rightarrow \ h(z) ) ) ) \ \Rightarrow \ ( \forall r{:}\;( ( f(r)\mbox{ and }h(r) ) \ \Rightarrow \ g(r) ) \ \Rightarrow \ \exists v{:}\;( f(v)\mbox{ and }g(v)\mbox{ and }\neg\,h(v) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN937+1.p} From Smullyan






$\forall x{:}\;( p(x) \ \Rightarrow \ c )\iff ( \exists x{:}\;p(x) \ \Rightarrow \ c )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN360+1.p} Peter Andrews Problem X2111






$( \forall x{:}\;( \exists y{:}\;\mbox{big\_p}(x,y) \ \Rightarrow \ \forall y{:}\;\mbox{big\_q}(x,y) )\mbox{ and }\forall z{:}\;\exists y{:}\;\mbox{big\_p}(z,y) ) \ \Rightarrow \ \forall y,x{:}\;\mbox{big\_q}(x,y)\qquad\mbox{fof}(x_{2111},\mbox{conjecture})$

\medskip\textbf{SYN093-1.002.p} Plaisted problem u(t(2,2))






$\mbox{p\_1}_2 \ \Rightarrow \ u_1\qquad\mbox{cnf}(\mbox{u\_s2\_goal}_1,\mbox{negated\_conjecture})$

$\neg\,u_1\qquad\mbox{cnf}(\mbox{u\_1\_s2\_goal}_1,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ ( u_2\mbox{ or }\mbox{p\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type11}_1,\mbox{axiom})$

$u_2 \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{u\_2\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ ( u_3\mbox{ or }\mbox{p\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type11}_2,\mbox{axiom})$

$u_3 \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{u\_3\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ ( u_4\mbox{ or }\mbox{q\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type12}_1,\mbox{axiom})$

$u_4 \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{u\_4\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ ( u_5\mbox{ or }\mbox{q\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type12}_2,\mbox{axiom})$

$u_5 \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{u\_5\_s2\_type12}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ ( u_6\mbox{ or }\mbox{p\_1}_1 )\qquad\mbox{cnf}(\mbox{u\_s2\_type21}_1,\mbox{axiom})$

$u_6 \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_6\_s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ ( u_7\mbox{ or }\mbox{q\_1}_1 )\qquad\mbox{cnf}(\mbox{u\_s2\_type21}_2,\mbox{axiom})$

$u_7 \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_7\_s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ ( u_8\mbox{ or }\mbox{p\_2}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type22}_1,\mbox{axiom})$

$u_8 \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_8\_s2\_type22}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ ( u_9\mbox{ or }\mbox{q\_2}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type22}_2,\mbox{axiom})$

$u_9 \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_9\_s2\_type22}_2,\mbox{axiom})$

$u_{10}\mbox{ or }\mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_t2}_1,\mbox{axiom})$

$u_{10} \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_10\_t2}_1,\mbox{axiom})$

$u_{11}\mbox{ or }\mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_t2}_2,\mbox{axiom})$

$u_{11} \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_11\_t2}_2,\mbox{axiom})$

$u_{12}\mbox{ or }\mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_t2}_3,\mbox{axiom})$

$u_{12} \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_12\_t2}_3,\mbox{axiom})$

$u_{13}\mbox{ or }\mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_t2}_4,\mbox{axiom})$

$u_{13} \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_13\_t2}_4,\mbox{axiom})$

\medskip\textbf{SYN144-1.p} RPT63 synthetic problem 42 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_5(e,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN308-1.p} Problem for testing satisfiability






$p(f(f(a)),\mbox{gf}(c,\mbox{gf}(y,x)))\mbox{ or }p(u,f(u))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( p(f(h(x,y,z)),z)\mbox{ and }g(u) ) \ \Rightarrow \ ( g(b)\mbox{ or }g(v) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$e(\mbox{gf}(h(z,x,y),f(b)))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$r(\mbox{gf}(v,\mbox{gf}(v,u)),\mbox{gf}(u,v))\mbox{ or }r(f(b),\mbox{gf}(f(f(a)),\mbox{gf}(u,v)))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN300-1.p} RPT63 synthetic problem 85 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(b,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN248-1.p} RPT63 synthetic problem 33 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(d,c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN393$\wedge$4.004.p} ILTP Problem SYJ206+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$a_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_4 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_3 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_2 )@ ( \mbox{iatom}@ a_1 ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYN550+1.p} dia box p $\leq$$>$ dia box dia box p






$\forall x{:}\;\mbox{reachable}(x,x)\qquad\mbox{fof}(\mbox{reflexivity\_of\_reachable},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{reachable}(x,y)\mbox{ and }\mbox{reachable}(y,z) ) \ \Rightarrow \ \mbox{reachable}(x,z) )\qquad\mbox{fof}(\mbox{transitivity\_of\_reachable},\mbox{axiom})$

$\exists x{:}\;( \mbox{reachable}(\mbox{initial\_world},x)\mbox{ and }\forall y{:}\;( \mbox{reachable}(x,y) \ \Rightarrow \ p(y) ) )\iff \exists x{:}\;( \mbox{reachable}(\mbox{initial\_world},x)\mbox{ and }\forall y{:}\;( \mbox{reachable}(x,y) \ \Rightarrow \ \exists z{:}\;( \mbox{reachable}(y,z)\mbox{ and }\forall w{:}\;( \mbox{reachable}(z,w) \ \Rightarrow \ p(w) ) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN007+1.014.p} Pelletier Problem 71

Clausal forms of statements of the form :
 (p1 $<$$\to$ (p2 $<$$\to$...(pN $<$$\to$ (p1 $<$$\to$ (p2 $<$$\to$...$<$$\to$ pN)...)




$p_1\iff ( p_2\iff ( p_3\iff ( p_4\iff ( p_5\iff ( p_6\iff ( p_7\iff ( p_8\iff ( p_9\iff ( p_{10}\iff ( p_{11}\iff ( p_{12}\iff ( p_{13}\iff ( p_{14}\iff ( p_1\iff ( p_2\iff ( p_3\iff ( p_4\iff ( p_5\iff ( p_6\iff ( p_7\iff ( p_8\iff ( p_9\iff ( p_{10}\iff ( p_{11}\iff ( p_{12}\iff ( p_{13}\iff p_{14} ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN207-1.p} RPT63 synthetic problem 105 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN382$\wedge$5.p} TPS problem X2134






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ or }\mbox{cQ}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cQ}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cX}_{2134},\mbox{conjecture})$

\medskip\textbf{SYN961+1.p} Syntactic from Shults






$\forall a,b{:}\;( \forall x{:}\;p(x) \ \Rightarrow \ ( p(a)\mbox{ and }p(b) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN061+1.p} Pelletier Problem 31






$\neg\,\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }( \mbox{big\_g}(x)\mbox{ or }\mbox{big\_h}(x) ) )\qquad\mbox{fof}(\mbox{pel31}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_i}(x)\mbox{ and }\mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel31}_2,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{big\_h}(x) \ \Rightarrow \ \mbox{big\_j}(x) )\qquad\mbox{fof}(\mbox{pel31}_3,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_i}(x)\mbox{ and }\mbox{big\_j}(x) )\qquad\mbox{fof}(\mbox{pel}_{31},\mbox{conjecture})$

\medskip\textbf{SYN035-1.p} ROB1






$p(a,b)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$( p(f(a,b),f(a,b))\mbox{ and }p(b,f(a,b)) ) \ \Rightarrow \ q(a,b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( q(f(a,b),f(a,b))\mbox{ and }q(a,f(a,b))\mbox{ and }p(f(a,b),f(a,b)) ) \ \Rightarrow \ \neg\,p(b,f(a,b))\qquad\mbox{cnf}(\mbox{theorem},\mbox{negated\_conjecture})$

\medskip\textbf{SYN391+1.p} Pelletier 9






$( ( p_1\mbox{ or }p_2 )\mbox{ and }( \neg\,p_1\mbox{ or }p_2 )\mbox{ and }( p_1\mbox{ or }\neg\,p_2 ) ) \ \Rightarrow \ \neg\,\neg\,p_1\mbox{ or }\neg\,p_2\qquad\mbox{fof}(\mbox{pel}_9,\mbox{conjecture})$

\medskip\textbf{SYN348+1.p} Church problem 46.17 (4)






$\exists x_1{:}\;\forall x_2{:}\;\exists x_3{:}\;\forall x_4{:}\;( ( ( ( \mbox{big\_f}(x_1,x_4)\iff \mbox{big\_f}(x_4,x_3) )\iff \mbox{big\_f}(x_3,x_4) )\iff \mbox{big\_f}(x_4,x_1) )\mbox{ and }( ( ( \mbox{big\_f}(x_2,x_4)\iff \mbox{big\_f}(x_4,x_3) )\iff \mbox{big\_f}(x_3,x_4) )\iff \mbox{big\_f}(x_4,x_2) ) )\qquad\mbox{fof}(\mbox{church\_46\_17}_4,\mbox{conjecture})$

\medskip\textbf{SYN606-1.p} Harrison problem 3826






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),c_{22})),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{28},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN051+1.p} Pelletier Problem 21

A moderately tricky problem, especially for 'natural' systems
 with 'strong' restrictions on variables generated from
 existential quantifiers.




$\exists x{:}\;( p \ \Rightarrow \ \mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel21}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ p )\qquad\mbox{fof}(\mbox{pel21}_2,\mbox{axiom})$

$\exists x{:}\;( p\iff \mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel}_{21},\mbox{conjecture})$

\medskip\textbf{SYN349-1.p} Church problem 46.17 (5)






$f(w(x),g(x,y)) \ \Rightarrow \ f(x,g(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,g(x,y)) \ \Rightarrow \ f(w(x),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(y,g(x,y)) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(y,g(x,y)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(x,g(x,y))\mbox{ or }f(g(x,y),y)\mbox{ or }f(y,g(x,y))\mbox{ or }f(g(x,y),w(x))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y) ) \ \Rightarrow \ ( f(y,g(x,y))\mbox{ or }f(g(x,y),w(x)) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(g(x,y),y)\mbox{ or }f(y,g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( f(g(x,y),y)\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(y,g(x,y)) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( f(y,g(x,y))\mbox{ and }f(g(x,y),w(x)) ) \ \Rightarrow \ ( f(x,g(x,y))\mbox{ or }f(g(x,y),y) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( f(x,g(x,y))\mbox{ and }f(g(x,y),y)\mbox{ and }f(y,g(x,y)) ) \ \Rightarrow \ \neg\,f(g(x,y),w(x))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{SYN222-1.p} RPT63 synthetic problem 7 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN721+1.p} Peter Andrews Problem LX1






$( r(a,b)\mbox{ and }\forall x{:}\;( \exists y{:}\;r(x,y) \ \Rightarrow \ q(x,x) )\mbox{ and }\forall u,v{:}\;( q(u,v) \ \Rightarrow \ \forall z{:}\;r(z,v) ) ) \ \Rightarrow \ \exists w{:}\;( r(b,w)\mbox{ and }q(w,a) )\qquad\mbox{fof}(\mbox{lx}_1,\mbox{conjecture})$

\medskip\textbf{SYN321+1.p} Church problem 46.3 (2)






$\exists x,y{:}\;( ( \exists z{:}\;\mbox{big\_f}(x,z) \ \Rightarrow \ \forall z{:}\;\mbox{big\_g}(x,z) ) \ \Rightarrow \ ( \forall z{:}\;( \mbox{big\_g}(z,z) \ \Rightarrow \ \mbox{big\_f}(z,y) ) \ \Rightarrow \ ( \mbox{big\_f}(x,y)\iff \forall z{:}\;\mbox{big\_g}(x,z) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_3}_2,\mbox{conjecture})$

\medskip\textbf{SYN117-1.p} RPT63 synthetic problem 15 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_5(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN993$\wedge$1.p} Skolemization test 1






$q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_decl},\mbox{type})$

$\exists x{:}\;\mbox{\$i}{:}\;\forall y{:}\;\mbox{\$i}{:}\;( q@ x@ y )\mbox{ or }\exists u{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;\neg\,q@ v@ u\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN979+1.p} Syntactic from Shults






$\forall a,b{:}\;\exists x,y{:}\;( ( ( q(x) \ \Rightarrow \ p(x,a) )\mbox{ and }q(a)\mbox{ and }q(b)\mbox{ and }( r(y) \ \Rightarrow \ p(b,y) )\mbox{ and }r(a)\mbox{ and }r(b)\mbox{ and }( s(a) \ \Rightarrow \ p(x,y) )\mbox{ and }s(a) ) \ \Rightarrow \ p(a,b) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN724+1.p} Peter Andrews Problem THM31






$\forall x{:}\;( r(x) \ \Rightarrow \ s(x) )\iff \forall x{:}\;( ( r(x)\mbox{ and }s(x) )\iff r(x) )\qquad\mbox{fof}(\mbox{thm}_{31},\mbox{conjecture})$

\medskip\textbf{SYN991$\wedge$1.p} Inconsistency of axioms that says all relations are reflexive






$\mbox{refl}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{refl\_type},\mbox{type})$

$\mbox{refl} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( r@ x@ x ) )\qquad\mbox{thf}(\mbox{refl},\mbox{definition})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{refl}@ r )\qquad\mbox{thf}(\mbox{ax},\mbox{axiom})$

\medskip\textbf{SYN284-1.p} RPT63 synthetic problem 69 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(a,c,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN336-1.p} Church problem 46.15 (1)






$f(a,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y_1,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(y_2,z(y_1,y_2))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(y_1,a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,f(z(y_1,y_2),y_2)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN149-1.p} RPT63 synthetic problem 47 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_1(e,e,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN077+1.p} Pelletier Problem 54

Montegue's paradox of grounded classes




$\forall y{:}\;\exists z{:}\;\forall x{:}\;( \mbox{big\_f}(x,z)\iff x = y )\qquad\mbox{fof}(\mbox{pel54}_1,\mbox{axiom})$

$\neg\,\exists w{:}\;\forall x{:}\;( \mbox{big\_f}(x,w)\iff \forall u{:}\;( \mbox{big\_f}(x,u) \ \Rightarrow \ \exists y{:}\;( \mbox{big\_f}(y,u)\mbox{ and }\neg\,\exists z{:}\;( \mbox{big\_f}(z,u)\mbox{ and }\mbox{big\_f}(z,y) ) ) ) )\qquad\mbox{fof}(\mbox{pel}_{54},\mbox{conjecture})$

\medskip\textbf{SYN576-1.p} Harrison problem 3280






$p_8(x_{42},x_{42})\qquad\mbox{cnf}(\mbox{p8}_1,\mbox{negated\_conjecture})$

$p_2(x_{13},x_{13})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_4(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p4}_3,\mbox{negated\_conjecture})$

$p_5(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_{12}(c_{15},f_6(c_{13}))\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{negated\_conjecture})$

$p_{12}(c_{16},f_6(c_{13}))\qquad\mbox{cnf}(\mbox{p12}_6,\mbox{negated\_conjecture})$

$p_{11}(x_4,x_5)\mbox{ or }p_{11}(x_5,x_4)\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_2(f_3(x_{22},x_{23}),f_3(x_{23},x_{22}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_4(x_{40},x_{41}) \ \Rightarrow \ p_5(f_6(x_{40}),f_6(x_{41}))\qquad\mbox{cnf}(\mbox{p5}_9,\mbox{negated\_conjecture})$

$p_2(x_{16},x_{17}) \ \Rightarrow \ p_2(f_{10}(x_{16}),f_{10}(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$( p_8(x_{42},x_{43})\mbox{ and }p_8(x_{42},x_{44}) ) \ \Rightarrow \ p_8(x_{43},x_{44})\qquad\mbox{cnf}(\mbox{p8}_{11},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_4(x_{34},x_{35})\mbox{ and }p_4(x_{34},x_{36}) ) \ \Rightarrow \ p_4(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p4}_{13},\mbox{negated\_conjecture})$

$( p_5(x_{37},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_5(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{11}(x_2,x_3) ) \ \Rightarrow \ p_{11}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p11}_{15},\mbox{negated\_conjecture})$

$( p_2(x_8,x_6)\mbox{ and }p_5(x_9,x_7)\mbox{ and }p_{12}(x_8,x_9) ) \ \Rightarrow \ p_{12}(x_6,x_7)\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$\neg\,p_{12}(f_7(f_9(c_{13},c_{14},f_3(c_{15},c_{16})),f_3(c_{15},c_{16})),f_6(c_{13}))\qquad\mbox{cnf}(\mbox{not\_p12}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{20})\mbox{ and }p_2(x_{19},x_{21}) ) \ \Rightarrow \ p_2(f_3(x_{18},x_{19}),f_3(x_{20},x_{21}))\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{24},x_{26})\mbox{ and }p_2(x_{25},x_{27}) ) \ \Rightarrow \ p_2(f_7(x_{24},x_{25}),f_7(x_{26},x_{27}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_4(x_{28},x_{31})\mbox{ and }p_8(x_{29},x_{32})\mbox{ and }p_2(x_{30},x_{33}) ) \ \Rightarrow \ p_2(f_9(x_{28},x_{29},x_{30}),f_9(x_{31},x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{12}(x_{12},f_6(x_{10}))\mbox{ and }p_{12}(x_{11},f_6(x_{10}))\mbox{ and }p_{11}(f_{10}(x_{11}),f_{10}(x_{12})) ) \ \Rightarrow \ p_{12}(f_7(f_9(x_{10},c_{14},f_3(x_{11},x_{12})),x_{12}),f_6(x_{10}))\qquad\mbox{cnf}(\mbox{p12}_{21},\mbox{negated\_conjecture})$

$( p_{12}(x_{12},f_6(x_{10}))\mbox{ and }p_{12}(x_{11},f_6(x_{10}))\mbox{ and }p_{11}(f_{10}(x_{11}),f_{10}(x_{12})) ) \ \Rightarrow \ p_{12}(f_7(f_9(x_{10},c_{14},f_3(x_{11},x_{12})),f_3(x_{11},x_{12})),f_6(x_{10}))\qquad\mbox{cnf}(\mbox{p12}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN083+1.p} Pelletier Problem 61






$\forall x,y,z{:}\;f(x,f(y,z)) = f(f(x,y),z)\qquad\mbox{fof}(\mbox{p61}_1,\mbox{axiom})$

$\forall x,y,z,w{:}\;f(x,f(y,f(z,w))) = f(f(f(x,y),z),w)\qquad\mbox{fof}(\mbox{pel}_{61},\mbox{conjecture})$

\medskip\textbf{SYN396+1.p} Kalish and Montague Problem 203

Laws of quantifier negation.




$\neg\,\forall x{:}\;f(x)\iff \exists y{:}\;\neg\,f(y)\qquad\mbox{fof}(\mbox{kalish}_{203},\mbox{conjecture})$

\medskip\textbf{SYN725+1.p} Peter Andrews Problem THM39






$\forall x{:}\;( r(x)\mbox{ or }( s(x)\iff ( r(x)\mbox{ and }s(x) ) ) ) \ \Rightarrow \ \forall x{:}\;( r(x) \ \Rightarrow \ s(x) )\qquad\mbox{fof}(\mbox{thm}_{39},\mbox{conjecture})$

\medskip\textbf{SYN161-1.p} RPT63 synthetic problem 59 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN150-1.p} RPT63 synthetic problem 48 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_2(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN357+1.p} Peter Andrews Problem X2108






$\forall x{:}\;\exists y{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_p}(y) )\qquad\mbox{fof}(x_{2108},\mbox{conjecture})$

\medskip\textbf{SYN915$\wedge$4.p} TRUE






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$\mbox{ivalid}@ ( \mbox{iatom}@ \mbox{itrue} )\qquad\mbox{thf}(\mbox{truth},\mbox{conjecture})$

\medskip\textbf{SYN916-1.p} FALSE






$\mbox{\$true}\qquad\mbox{cnf}(\mbox{true\_clause},\mbox{negated\_conjecture})$

\medskip\textbf{SYN084-2.p} Pelletier Problem 62






$\mbox{big\_p}(a)\qquad\mbox{cnf}(\mbox{pel62}_1,\mbox{negated\_conjecture})$

$\mbox{big\_p}(a) \ \Rightarrow \ ( \mbox{big\_p}(f(f(a)))\mbox{ or }\mbox{big\_p}(a) )\qquad\mbox{cnf}(\mbox{pel62}_2,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(f(a))\mbox{ and }\mbox{big\_p}(a) ) \ \Rightarrow \ \mbox{big\_p}(f(f(a)))\qquad\mbox{cnf}(\mbox{pel62}_3,\mbox{negated\_conjecture})$

$\mbox{big\_p}(f(f(\mbox{sk}_1))) \ \Rightarrow \ \neg\,\mbox{big\_p}(f(f(\mbox{sk}_2)))\qquad\mbox{cnf}(\mbox{pel62}_4,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(\mbox{sk}_1)\mbox{ and }\mbox{big\_p}(\mbox{sk}_2) ) \ \Rightarrow \ ( \mbox{big\_p}(f(\mbox{sk}_1))\mbox{ or }\mbox{big\_p}(f(\mbox{sk}_2)) )\qquad\mbox{cnf}(\mbox{pel62}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(\mbox{sk}_1)\mbox{ and }\mbox{big\_p}(f(f(\mbox{sk}_2))) ) \ \Rightarrow \ \mbox{big\_p}(f(\mbox{sk}_1))\qquad\mbox{cnf}(\mbox{pel62}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(\mbox{sk}_2)\mbox{ and }\mbox{big\_p}(f(f(\mbox{sk}_1))) ) \ \Rightarrow \ \mbox{big\_p}(f(\mbox{sk}_2))\qquad\mbox{cnf}(\mbox{pel62}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN304-1.p} Problem for testing satisfiability






$p(x,f(x)) \ \Rightarrow \ r(f(x),g(x),x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(a,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(f(a),g(a),a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(f(x),x,x)\mbox{ or }l(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\neg\,s(f(x),x,a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\neg\,s(f(x),x,b)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$l(a)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$l(b)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

\medskip\textbf{SYN056+1.p} Pelletier Problem 26






$\exists x{:}\;\mbox{big\_p}(x)\iff \exists y{:}\;\mbox{big\_q}(y)\qquad\mbox{fof}(\mbox{pel26}_1,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(y) ) \ \Rightarrow \ ( \mbox{big\_r}(x)\iff \mbox{big\_s}(y) ) )\qquad\mbox{fof}(\mbox{pel26}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_r}(x) )\iff \forall y{:}\;( \mbox{big\_q}(y) \ \Rightarrow \ \mbox{big\_s}(y) )\qquad\mbox{fof}(\mbox{pel}_{26},\mbox{conjecture})$

\medskip\textbf{SYN413+1.p} Kalish and Montague Problem 256






$\forall z{:}\;\exists y{:}\;\forall x{:}\;( f(x,y)\iff ( f(x,z)\mbox{ and }\neg\,f(x,x) ) ) \ \Rightarrow \ \neg\,\exists v{:}\;\forall u{:}\;f(u,v)\qquad\mbox{fof}(\mbox{kalish}_{256},\mbox{conjecture})$

\medskip\textbf{SYN271-1.p} RPT63 synthetic problem 56 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(d,x,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN575-1.p} Harrison problem 3265






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_3(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_9(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p9}_5,\mbox{negated\_conjecture})$

$p_{11}(c_{14},c_{15})\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_3(f_4(f_5(x_{17})),f_4(x_{17}))\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_3(f_6(f_5(x_{20})),f_6(x_{20}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_9(x_{32},f_6(c_{14})) \ \Rightarrow \ p_9(x_{32},c_{13})\qquad\mbox{cnf}(\mbox{p9}_9,\mbox{negated\_conjecture})$

$p_9(x_{32},c_{13}) \ \Rightarrow \ p_9(x_{32},f_6(c_{14}))\qquad\mbox{cnf}(\mbox{p9}_{10},\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_5(x_{10}),f_5(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_3(f_6(x_{18}),f_6(x_{19}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_3(f_4(x_{15}),f_4(x_{16}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{21},x_{22})\mbox{ and }p_7(x_{21},x_{23}) ) \ \Rightarrow \ p_7(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{13})\mbox{ and }p_3(x_{12},x_{14}) ) \ \Rightarrow \ p_3(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{31},x_{28})\mbox{ and }p_9(x_{31},x_{30})\mbox{ and }p_3(x_{30},x_{29}) ) \ \Rightarrow \ p_9(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_7(f_8(x_{24},x_{25}),f_8(x_{26},x_{27}))\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_{11}(x_{35},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_2(x_{34},f_5(x_{35})) ) \ \Rightarrow \ ( p_9(f_8(x_{33},x_{34}),x_{33})\mbox{ or }p_9(f_8(x_{33},x_{34}),f_6(x_{34})) )\qquad\mbox{cnf}(\mbox{p9}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_{35},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_2(x_{34},f_5(x_{35}))\mbox{ and }p_9(f_8(x_{33},x_{34}),x_{33}) ) \ \Rightarrow \ \neg\,p_9(f_8(x_{33},x_{34}),f_6(x_{34}))\qquad\mbox{cnf}(\mbox{not\_p11}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN603-1.p} Harrison problem 3812






$p_{14}(c_{20})\qquad\mbox{cnf}(\mbox{c20\_is\_p14}_1,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_{15}(c_{22},c_{21})\qquad\mbox{cnf}(\mbox{p15}_3,\mbox{negated\_conjecture})$

$p_2(f_3(c_{18}),c_{19})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(c_{22},f_9(c_{20},c_{23}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(f_4(c_{16})),c_{17})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$( p_{14}(x_{12})\mbox{ and }p_2(x_{12},x_{11}) ) \ \Rightarrow \ p_{14}(x_{11})\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_{15}(f_{10}(x_{17}),c_{21})\qquad\mbox{cnf}(\mbox{p15}_8,\mbox{negated\_conjecture})$

$p_2(x_{35},x_{36}) \ \Rightarrow \ p_2(f_5(x_{35}),f_5(x_{36}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{28}) \ \Rightarrow \ p_2(f_{10}(x_{27}),f_{10}(x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_2(f_{11}(x_{29}),f_{11}(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{31},x_{32}) \ \Rightarrow \ p_2(f_3(x_{31}),f_3(x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{33},x_{34}) \ \Rightarrow \ p_2(f_4(x_{33}),f_4(x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_{13}(x_6,c_{18}) \ \Rightarrow \ p_{12}(x_6,f_3(x_6),f_3(f_5(x_6)))\qquad\mbox{cnf}(\mbox{p12}_{15},\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_2(f_{10}(x_{17}),f_9(c_{20},f_{11}(x_{17})))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_9,x_7)\mbox{ and }p_2(x_{10},x_8)\mbox{ and }p_{13}(x_9,x_{10}) ) \ \Rightarrow \ p_{13}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p13}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{13})\mbox{ and }p_2(x_{16},x_{14})\mbox{ and }p_{15}(x_{15},x_{16}) ) \ \Rightarrow \ p_{15}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$p_2(f_7(f_8(c_{21},c_{22}),c_{20}),f_6(f_5(f_4(c_{16})),f_3(f_4(c_{16}))))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{49},x_{51})\mbox{ and }p_2(x_{50},x_{52}) ) \ \Rightarrow \ p_2(f_9(x_{49},x_{50}),f_9(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{37},x_{39})\mbox{ and }p_2(x_{38},x_{40}) ) \ \Rightarrow \ p_2(f_6(x_{37},x_{38}),f_6(x_{39},x_{40}))\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{41},x_{43})\mbox{ and }p_2(x_{42},x_{44}) ) \ \Rightarrow \ p_2(f_7(x_{41},x_{42}),f_7(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{45},x_{47})\mbox{ and }p_2(x_{46},x_{48}) ) \ \Rightarrow \ p_2(f_8(x_{45},x_{46}),f_8(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_2(f_7(f_8(c_{21},f_{10}(x_{17})),c_{20}),f_6(f_5(x_{17}),f_3(x_{17})))\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$( p_2(x_5,x_2)\mbox{ and }p_2(x_4,x_1)\mbox{ and }p_2(x_3,x_0)\mbox{ and }p_{12}(x_3,x_4,x_5) ) \ \Rightarrow \ p_{12}(x_0,x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_{25},\mbox{negated\_conjecture})$

$( p_{15}(x_{53},c_{21})\mbox{ and }p_2(x_{53},f_9(c_{20},x_{54})) ) \ \Rightarrow \ \neg\,p_2(f_7(f_8(c_{21},x_{53}),c_{20}),f_6(f_5(f_4(c_{16})),c_{17}))\qquad\mbox{cnf}(\mbox{not\_p15}_{26},\mbox{negated\_conjecture})$

$( p_{15}(x_{21},c_{21})\mbox{ and }p_2(x_{21},f_9(c_{20},x_{22}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{21}),c_{20}),f_6(f_5(f_4(c_{16})),x_{23})) ) \ \Rightarrow \ p_2(x_{21},c_{22})\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_{15}(x_{21},c_{21})\mbox{ and }p_2(x_{21},f_9(c_{20},x_{22}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{21}),c_{20}),f_6(f_5(f_4(c_{16})),x_{23})) ) \ \Rightarrow \ p_2(x_{23},f_3(f_4(c_{16})))\qquad\mbox{cnf}(\mbox{p2}_{28},\mbox{negated\_conjecture})$

$( p_{15}(x_{24},c_{21})\mbox{ and }p_{13}(x_{17},f_5(c_{18}))\mbox{ and }p_2(x_{24},f_9(c_{20},x_{25}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{24}),c_{20}),f_6(f_5(x_{17}),x_{26})) ) \ \Rightarrow \ p_2(x_{26},f_3(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{29},\mbox{negated\_conjecture})$

$( p_{15}(x_{24},c_{21})\mbox{ and }p_{13}(x_{17},f_5(c_{18}))\mbox{ and }p_2(x_{24},f_9(c_{20},x_{25}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{24}),c_{20}),f_6(f_5(x_{17}),x_{26})) ) \ \Rightarrow \ p_2(x_{24},f_{10}(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN089-1.002.p} Plaisted problem t(2,2)






$\neg\,\mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_2,\mbox{axiom})$

$\mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{t2}_1,\mbox{axiom})$

$\mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{t2}_2,\mbox{axiom})$

$\mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{t2}_3,\mbox{axiom})$

$\mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{t2}_4,\mbox{axiom})$

\medskip\textbf{SYN210-1.p} RPT63 synthetic problem 108 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_4(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN108-1.p} RPT63 synthetic problem 6 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_2(a,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN009-1.p} A problem to demonstrate the usefulness of relevancy testing






$\neg\,p(c,x,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,q(x,c,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,r(x,y,c)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$s(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$s(c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( s(x)\mbox{ and }s(y)\mbox{ and }s(z) ) \ \Rightarrow \ ( p(x,y,z)\mbox{ or }q(x,y,z)\mbox{ or }r(x,y,z) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN333-1.p} Church problem 46.14 (5)






$f(x,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( f(y,z(x,y))\mbox{ and }f(z(x,y),z(x,y)) ) \ \Rightarrow \ g(x,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( f(y,z(x,y))\mbox{ and }f(z(x,y),z(x,y))\mbox{ and }g(x,z(x,y)) ) \ \Rightarrow \ \neg\,g(z(x,y),z(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN206-1.p} RPT63 synthetic problem 104 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(b,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN084+1.p} Pelletier Problem 62






$\forall x{:}\;( ( \mbox{big\_p}(a)\mbox{ and }( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_p}(f(x)) ) ) \ \Rightarrow \ \mbox{big\_p}(f(f(x))) )\iff \forall x_1{:}\;( ( \neg\,\mbox{big\_p}(a)\mbox{ or }\mbox{big\_p}(x_1)\mbox{ or }\mbox{big\_p}(f(f(x_1))) )\mbox{ and }( \neg\,\mbox{big\_p}(a)\mbox{ or }\neg\,\mbox{big\_p}(f(x_1))\mbox{ or }\mbox{big\_p}(f(f(x_1))) ) )\qquad\mbox{fof}(\mbox{pel}_{62},\mbox{conjecture})$

\medskip\textbf{SYN729-1.p} Peter Andrews Problem THM72






$p(a) \ \Rightarrow \ l(a,g(h(\mbox{sk}_1(a))))\qquad\mbox{cnf}(\mbox{thm72}_1,\mbox{negated\_conjecture})$

$p(a) \ \Rightarrow \ p(\mbox{sk}_1(a))\qquad\mbox{cnf}(\mbox{thm72}_2,\mbox{negated\_conjecture})$

$p(a) \ \Rightarrow \ p(g(a))\qquad\mbox{cnf}(\mbox{thm72}_3,\mbox{negated\_conjecture})$

$p(a) \ \Rightarrow \ p(h(a))\qquad\mbox{cnf}(\mbox{thm72}_4,\mbox{negated\_conjecture})$

$p(\mbox{sk}_2)\qquad\mbox{cnf}(\mbox{thm72}_5,\mbox{negated\_conjecture})$

$p(a) \ \Rightarrow \ \neg\,l(\mbox{sk}_2,a)\qquad\mbox{cnf}(\mbox{thm72}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN169-1.p} RPT63 synthetic problem 67 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_3(x,d,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN164-1.p} RPT63 synthetic problem 62 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_0(b,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN729+1.p} Peter Andrews Problem THM72






$( \forall x{:}\;\exists y{:}\;( p(x) \ \Rightarrow \ ( l(x,g(h(y)))\mbox{ and }p(y) ) )\mbox{ and }\forall w{:}\;( p(w) \ \Rightarrow \ ( p(g(w))\mbox{ and }p(h(w)) ) ) ) \ \Rightarrow \ \forall x{:}\;( p(x) \ \Rightarrow \ \exists y{:}\;( l(x,y)\mbox{ and }p(y) ) )\qquad\mbox{fof}(\mbox{thm}_{72},\mbox{conjecture})$

\medskip\textbf{SYN301-1.p} RPT63 synthetic problem 86 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_4(a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN553-1.p} Harrison problem 1630






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_2(c_3,c_4)\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$\neg\,p_2(c_4,f_5(f_6(c_7)))\qquad\mbox{cnf}(\mbox{not\_p2}_3,\mbox{negated\_conjecture})$

$\neg\,p_2(c_3,f_5(f_6(c_7)))\qquad\mbox{cnf}(\mbox{not\_p2}_4,\mbox{negated\_conjecture})$

$p_2(f_8(x_{11},x_{12}),f_8(x_{12},x_{11}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$( p_2(x_1,x_3)\mbox{ and }p_2(x_3,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_2(f_8(x_8,f_8(x_9,x_{10})),f_8(f_8(x_8,x_9),x_{10}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$( p_2(x_4,x_6)\mbox{ and }p_2(x_5,x_7) ) \ \Rightarrow \ p_2(f_8(x_4,x_5),f_8(x_6,x_7))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$\neg\,p_2(f_8(f_8(f_9(c_4),c_4),f_9(c_3)),f_8(f_9(c_4),f_8(f_9(c_3),c_4)))\qquad\mbox{cnf}(\mbox{not\_p2}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN412+1.p} Kalish and Montague Problem 255






$\neg\,\exists x{:}\;\forall y{:}\;( f(x,y)\iff \neg\,f(x,x) )\qquad\mbox{fof}(\mbox{kalish}_{255},\mbox{conjecture})$

\medskip\textbf{SYN587-1.p} Harrison problem 3572






$p_{12}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{negated\_conjecture})$

$p_5(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p5}_2,\mbox{negated\_conjecture})$

$p_3(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{13},x_{13})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_3(c_{21},c_{19})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$\neg\,p_{12}(f_{13}(c_{20}),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{not\_p12}_6,\mbox{negated\_conjecture})$

$p_2(x_3,x_4) \ \Rightarrow \ p_{12}(f_{13}(x_3),f_{13}(x_4))\qquad\mbox{cnf}(\mbox{p12}_7,\mbox{negated\_conjecture})$

$p_3(x_{42},x_{43}) \ \Rightarrow \ p_5(f_7(x_{42}),f_7(x_{43}))\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$p_5(x_{40},x_{41}) \ \Rightarrow \ p_5(f_{10}(x_{40}),f_{10}(x_{41}))\qquad\mbox{cnf}(\mbox{p5}_9,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{36}) \ \Rightarrow \ p_3(f_9(x_{35}),f_9(x_{36}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_3(x_{33},x_{34}) \ \Rightarrow \ p_3(f_8(x_{33}),f_8(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_3(x_{27},x_{28}) \ \Rightarrow \ p_3(f_{11}(x_{27}),f_{11}(x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$( p_{12}(x_0,x_1)\mbox{ and }p_{12}(x_0,x_2) ) \ \Rightarrow \ p_{12}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_{13},\mbox{negated\_conjecture})$

$( p_5(x_{37},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_5(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_3(x_{24},x_{25})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_3(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{13},x_{14})\mbox{ and }p_2(x_{13},x_{15}) ) \ \Rightarrow \ p_2(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_9)\mbox{ and }p_2(x_{12},x_{10})\mbox{ and }p_{15}(x_{11},x_{12}) ) \ \Rightarrow \ p_{15}(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p15}_{17},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),c_{19}),c_{20})\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),c_{21}),c_{20})\qquad\mbox{cnf}(\mbox{not\_p15}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{18})\mbox{ and }p_3(x_{17},x_{19}) ) \ \Rightarrow \ p_2(f_4(x_{16},x_{17}),f_4(x_{18},x_{19}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{14}(x_{29},x_{30}),f_{14}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{23})\mbox{ and }p_5(x_{20},x_{22}) ) \ \Rightarrow \ p_2(f_6(x_{20},x_{21}),f_6(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$\neg\,p_{15}(f_4(f_6(f_{10}(c_{16}),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(c_{19})),c_{20})\qquad\mbox{cnf}(\mbox{not\_p15}_{23},\mbox{negated\_conjecture})$

$p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{14}(x_5,x_6)),x_5)\qquad\mbox{cnf}(\mbox{p12}_{24},\mbox{negated\_conjecture})$

$p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(f_{14}(x_5,x_6))),x_5) \ \Rightarrow \ p_{12}(f_{13}(x_5),f_{13}(c_{18}))\qquad\mbox{cnf}(\mbox{p12}_{25},\mbox{negated\_conjecture})$

$( p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_7),x_5)\mbox{ and }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),x_8),x_5) ) \ \Rightarrow \ ( p_3(x_7,x_8)\mbox{ or }p_{12}(f_{13}(x_5),f_{13}(c_{18}))\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_8)),x_5)\mbox{ or }p_{15}(f_4(f_6(f_{10}(x_6),f_6(f_7(f_8(f_9(c_{17}))),c_{18})),f_{11}(x_7)),x_5) )\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{SYN000+1.p} Basic TPTP FOF syntax

Basic TPTP FOF syntax that you can't survive without parsing.




$( p_0\mbox{ and }\neg\,q_0 ) \ \Rightarrow \ ( r_0\mbox{ or }\neg\,s_0 )\qquad\mbox{fof}(\mbox{propositional},\mbox{axiom})$

$\forall x{:}\;( ( p(x)\mbox{ or }\neg\,q(x,a) ) \ \Rightarrow \ \exists y,z{:}\;( r(x,f(y),g(x,f(y),z))\mbox{ and }\neg\,s(f(f(f(b)))) ) )\qquad\mbox{fof}(\mbox{first\_order},\mbox{axiom})$

$\exists y{:}\;\forall x,z{:}\;( f(y) = g(x,f(y),z)\mbox{ or }f(f(f(b)))\ne a\mbox{ or }x = f(y) )\qquad\mbox{fof}(\mbox{equality},\mbox{axiom})$

$\mbox{\$true}\mbox{ or }\mbox{\$false}\qquad\mbox{fof}(\mbox{true\_false},\mbox{axiom})$

$\mbox{'A proposition'}\mbox{ or }\mbox{'A predicate'}(a)\mbox{ or }p(\mbox{'A constant'})\mbox{ or }p(\mbox{'A function'}(a))\mbox{ or }p(\mbox{'A \'quoted \\ escape\''})\qquad\mbox{fof}(\mbox{single\_quoted},\mbox{axiom})$

$\forall x{:}\;( ( p(x) \ \Leftarrow \ \neg\,q(x,a) )\iff \exists y,z{:}\;r(x,f(y),g(x,f(y),z)) <~> \neg\,s(f(f(f(b)))) )\qquad\mbox{fof}(\mbox{useful\_connectives},\mbox{axiom})$

$\forall x{:}\;( ( p(x)\mbox{ or }\neg\,q(x,a) ) \ \Rightarrow \ \exists y,z{:}\;( r(x,f(y),g(x,f(y),z))\mbox{ and }\neg\,s(f(f(f(b)))) ) )\qquad\mbox{fof}(123,\mbox{axiom})$

$p(h)\qquad\mbox{fof}(\mbox{role\_hypothesis},\mbox{hypothesis})$

$\exists x{:}\;p(x)\qquad\mbox{fof}(\mbox{role\_conjecture},\mbox{conjecture})$

$\mbox{include}(\mbox{'Axioms/SYN000+0.ax'})$

\medskip\textbf{SYN155-1.p} RPT63 synthetic problem 53 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_4(x,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN563-1.p} Harrison problem 2755






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_6(x_{22},x_{22})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$\neg\,p_6(f_4(c_7),c_8)\qquad\mbox{cnf}(\mbox{not\_p6}_3,\mbox{negated\_conjecture})$

$p_2(f_4(c_7),f_4(c_8))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$\neg\,p_6(f_4(c_7),f_3(f_4(c_8)))\qquad\mbox{cnf}(\mbox{not\_p6}_5,\mbox{negated\_conjecture})$

$p_5(x_{20},x_{21}) \ \Rightarrow \ p_6(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p6}_6,\mbox{negated\_conjecture})$

$p_5(f_3(f_3(x_{16})),f_3(f_4(x_{16})))\qquad\mbox{cnf}(\mbox{p5}_7,\mbox{negated\_conjecture})$

$p_6(f_4(x_{10}),x_{11}) \ \Rightarrow \ p_5(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p5}_8,\mbox{negated\_conjecture})$

$p_6(c_7,x_{27}) \ \Rightarrow \ p_6(c_7,f_3(x_{27}))\qquad\mbox{cnf}(\mbox{p6}_9,\mbox{negated\_conjecture})$

$p_5(x_{10},x_{11}) \ \Rightarrow \ p_6(f_4(x_{10}),x_{11})\qquad\mbox{cnf}(\mbox{p6}_{10},\mbox{negated\_conjecture})$

$p_2(x_3,x_4) \ \Rightarrow \ p_2(f_3(x_3),f_3(x_4))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_5,x_6) \ \Rightarrow \ p_2(f_4(x_5),f_4(x_6))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_6(x_{17},x_{19})\mbox{ and }p_6(x_{19},x_{18}) ) \ \Rightarrow \ p_6(x_{17},x_{18})\qquad\mbox{cnf}(\mbox{p6}_{14},\mbox{negated\_conjecture})$

$( p_5(x_9,x_8)\mbox{ and }p_6(x_7,x_9) ) \ \Rightarrow \ p_5(x_7,x_8)\qquad\mbox{cnf}(\mbox{p5}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{23})\mbox{ and }p_6(x_{26},x_{25})\mbox{ and }p_2(x_{25},x_{24}) ) \ \Rightarrow \ p_6(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p6}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{13})\mbox{ and }p_5(x_{14},x_{15})\mbox{ and }p_2(x_{14},x_{12}) ) \ \Rightarrow \ p_5(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

\medskip\textbf{SYN046$\wedge$4.p} Pelletier Problem 15






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) )@ ( \mbox{iatom}@ q ) ) )\qquad\mbox{thf}(\mbox{pel}_{15},\mbox{conjecture})$

\medskip\textbf{SYN048-1.p} Pelletier Problem 18






$\mbox{big\_f}(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_f}(f(x))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN194-1.p} RPT63 synthetic problem 92 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_5(c,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN028-1.p} EW1






$q\mbox{ or }p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$q \ \Rightarrow \ r\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$r \ \Rightarrow \ s\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( s\mbox{ and }r ) \ \Rightarrow \ t\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$t \ \Rightarrow \ p\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\neg\,p\qquad\mbox{cnf}(\mbox{prove\_something},\mbox{negated\_conjecture})$

\medskip\textbf{SYN086-1.003.p} Plaisted problem s(2,3)






$\neg\,\mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_goal}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_3\mbox{ and }\mbox{p\_1}_2 ) \ \Rightarrow \ \mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_3,\mbox{axiom})$

$( \mbox{q\_2}_3\mbox{ and }\mbox{q\_1}_2 ) \ \Rightarrow \ \mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_4,\mbox{axiom})$

$( \mbox{p\_3}_3\mbox{ and }\mbox{p\_2}_2 ) \ \Rightarrow \ \mbox{p\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_5,\mbox{axiom})$

$( \mbox{q\_3}_3\mbox{ and }\mbox{q\_2}_2 ) \ \Rightarrow \ \mbox{p\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type11}_6,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{s2\_type12}_2,\mbox{axiom})$

$( \mbox{p\_2}_3\mbox{ and }\mbox{q\_1}_2 ) \ \Rightarrow \ \mbox{q\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_3,\mbox{axiom})$

$( \mbox{q\_2}_3\mbox{ and }\mbox{p\_1}_2 ) \ \Rightarrow \ \mbox{q\_1}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_4,\mbox{axiom})$

$( \mbox{p\_3}_3\mbox{ and }\mbox{q\_2}_2 ) \ \Rightarrow \ \mbox{q\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_5,\mbox{axiom})$

$( \mbox{q\_3}_3\mbox{ and }\mbox{p\_2}_2 ) \ \Rightarrow \ \mbox{q\_2}_3\qquad\mbox{cnf}(\mbox{s2\_type12}_6,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_1,\mbox{axiom})$

$\mbox{p\_2}_3 \ \Rightarrow \ \mbox{p\_3}_3\qquad\mbox{cnf}(\mbox{s2\_type22}_2,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{s2\_type22}_3,\mbox{axiom})$

$\mbox{q\_2}_3 \ \Rightarrow \ \mbox{q\_3}_3\qquad\mbox{cnf}(\mbox{s2\_type22}_4,\mbox{axiom})$

\medskip\textbf{SYN922+1.p} From Smullyan






$\forall x{:}\;( p(x)\mbox{ and }q(x) )\iff ( \forall x{:}\;p(x)\mbox{ and }\forall x{:}\;q(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN125-1.p} RPT63 synthetic problem 23 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(e,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN075+1.p} Pelletier Problem 52






$\exists z,w{:}\;\forall x,y{:}\;( \mbox{big\_f}(x,y)\iff ( x = z\mbox{ and }y = w ) )\qquad\mbox{fof}(\mbox{pel52}_1,\mbox{axiom})$

$\exists w{:}\;\forall y{:}\;( \exists z{:}\;\forall x{:}\;( \mbox{big\_f}(x,y)\iff x = z )\iff y = w )\qquad\mbox{fof}(\mbox{pel}_{52},\mbox{conjecture})$

\medskip\textbf{SYN388+1.p} Expanded Law of Excluded Middle

Expanded Law of Excluded Middle. The strategies of the
 original Logic Theorist cannot prove this.




$p\mbox{ or }\neg\,\neg\,\neg\,p\qquad\mbox{fof}(\mbox{pel}_7,\mbox{conjecture})$

\medskip\textbf{SYN185-1.p} RPT63 synthetic problem 83 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_0(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN066+1.p} Pelletier Problem 37






$\forall z{:}\;\exists w{:}\;\forall x{:}\;\exists y{:}\;( ( \mbox{big\_p}(x,z) \ \Rightarrow \ \mbox{big\_p}(y,w) )\mbox{ and }\mbox{big\_p}(y,z)\mbox{ and }( \mbox{big\_p}(y,w) \ \Rightarrow \ \exists u{:}\;\mbox{big\_q}(u,w) ) )\qquad\mbox{fof}(\mbox{pel37}_1,\mbox{axiom})$

$\forall x,z{:}\;( \neg\,\mbox{big\_p}(x,z) \ \Rightarrow \ \exists y{:}\;\mbox{big\_q}(y,z) )\qquad\mbox{fof}(\mbox{pel37}_2,\mbox{axiom})$

$\exists x,y{:}\;\mbox{big\_q}(x,y) \ \Rightarrow \ \forall z{:}\;\mbox{big\_r}(z,z)\qquad\mbox{fof}(\mbox{pel37}_3,\mbox{axiom})$

$\forall x{:}\;\exists y{:}\;\mbox{big\_r}(x,y)\qquad\mbox{fof}(\mbox{pel}_{37},\mbox{conjecture})$

\medskip\textbf{SYN377$\wedge$7.p} Peter Andrews Problem X2128






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$\mbox{big\_p}{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{big\_p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) ) ) ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ y ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{big\_p}@ x ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(x_{2128},\mbox{conjecture})$

\medskip\textbf{SYN395+1.p} Kalish and Montague Problem 202

Laws of distribution.




$\forall x{:}\;( f(x) \ \Rightarrow \ g(x) ) \ \Rightarrow \ ( \exists y{:}\;f(y) \ \Rightarrow \ \exists z{:}\;g(z) )\qquad\mbox{fof}(\mbox{kalish}_{202},\mbox{conjecture})$

\medskip\textbf{SYN940+1.p} Syntactic from Shults






$\forall b,c{:}\;( \forall z{:}\;q(f(z)) \ \Rightarrow \ \exists x,y{:}\;( ( p(f(y)) \ \Rightarrow \ ( p(x)\mbox{ and }( r(y) \ \Rightarrow \ ( r(b)\mbox{ and }r(c) ) ) ) )\mbox{ and }q(x) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN059-1.p} Pelletier Problem 29






$\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_g}(b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_g}(z) ) \ \Rightarrow \ ( \mbox{big\_h}(x)\mbox{ or }\mbox{big\_h}(y) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_g}(z) ) \ \Rightarrow \ ( \mbox{big\_h}(x)\mbox{ or }\mbox{big\_j}(z) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x) \ \Rightarrow \ ( \mbox{big\_h}(x)\mbox{ or }\mbox{big\_f}(e)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ ( \mbox{big\_h}(x)\mbox{ or }\mbox{big\_f}(e) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_h}(e) ) \ \Rightarrow \ ( \mbox{big\_h}(x)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_h}(e)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ \mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_g}(z) ) \ \Rightarrow \ ( \mbox{big\_j}(x)\mbox{ or }\mbox{big\_h}(y) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_g}(z) ) \ \Rightarrow \ ( \mbox{big\_j}(x)\mbox{ or }\mbox{big\_j}(z) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$\mbox{big\_g}(x) \ \Rightarrow \ ( \mbox{big\_j}(x)\mbox{ or }\mbox{big\_f}(e)\mbox{ or }\mbox{big\_g}(j) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ ( \mbox{big\_j}(x)\mbox{ or }\mbox{big\_f}(e) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_h}(e) ) \ \Rightarrow \ ( \mbox{big\_j}(x)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{big\_g}(x)\mbox{ and }\mbox{big\_h}(e)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ \mbox{big\_j}(x)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ ( \mbox{big\_f}(c)\mbox{ or }\mbox{big\_h}(x) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ ( \mbox{big\_f}(c)\mbox{ or }\mbox{big\_j}(y) )\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$\mbox{big\_f}(c)\mbox{ or }\mbox{big\_f}(e)\mbox{ or }\mbox{big\_g}(f)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$\mbox{big\_j}(f) \ \Rightarrow \ ( \mbox{big\_f}(c)\mbox{ or }\mbox{big\_f}(e) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$\mbox{big\_h}(e) \ \Rightarrow \ ( \mbox{big\_f}(c)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(e)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ \mbox{big\_f}(c)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ ( \mbox{big\_g}(d)\mbox{ or }\mbox{big\_h}(x) )\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ ( \mbox{big\_g}(d)\mbox{ or }\mbox{big\_j}(y) )\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

$\mbox{big\_g}(d)\mbox{ or }\mbox{big\_f}(e)\mbox{ or }\mbox{big\_g}(f)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{negated\_conjecture})$

$\mbox{big\_j}(f) \ \Rightarrow \ ( \mbox{big\_g}(d)\mbox{ or }\mbox{big\_f}(e) )\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{negated\_conjecture})$

$\mbox{big\_h}(e) \ \Rightarrow \ ( \mbox{big\_g}(d)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(e)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ \mbox{big\_g}(d)\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_j}(d)\mbox{ and }\mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ \mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_j}(d)\mbox{ and }\mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ \mbox{big\_j}(y)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_j}(d) ) \ \Rightarrow \ ( \mbox{big\_f}(e)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_j}(d)\mbox{ and }\mbox{big\_j}(f) ) \ \Rightarrow \ \mbox{big\_f}(e)\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_h}(e) ) \ \Rightarrow \ ( \mbox{big\_j}(d)\mbox{ or }\mbox{big\_g}(f) )\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{negated\_conjecture})$

$( \mbox{big\_h}(c)\mbox{ and }\mbox{big\_j}(d)\mbox{ and }\mbox{big\_h}(e) ) \ \Rightarrow \ \neg\,\mbox{big\_j}(f)\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{SYN062-1.p} Pelletier Problem 32






$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_g}(x) ) \ \Rightarrow \ \mbox{big\_i}(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_h}(x) ) \ \Rightarrow \ \mbox{big\_i}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_i}(x)\mbox{ and }\mbox{big\_h}(x) ) \ \Rightarrow \ \mbox{big\_j}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_k}(x) \ \Rightarrow \ \mbox{big\_h}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_f}(a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\mbox{big\_k}(a)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_j}(a)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN614-1.p} Harrison problem 3878






$p_{13}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p13}_1,\mbox{negated\_conjecture})$

$p_7(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$p_{15}(x_7,x_7)\qquad\mbox{cnf}(\mbox{p15}_4,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{21},f_3(c_{22}))\qquad\mbox{cnf}(\mbox{not\_p2}_5,\mbox{negated\_conjecture})$

$p_{19}(c_{23},f_3(f_4(f_5(c_{22}))))\qquad\mbox{cnf}(\mbox{p19}_6,\mbox{negated\_conjecture})$

$p_{20}(c_{24},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\qquad\mbox{cnf}(\mbox{p20}_7,\mbox{negated\_conjecture})$

$p_7(x_{52},x_{53}) \ \Rightarrow \ p_7(f_{12}(x_{52}),f_{12}(x_{53}))\qquad\mbox{cnf}(\mbox{p7}_8,\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_2(f_3(x_{29}),f_3(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{31},x_{32}) \ \Rightarrow \ p_2(f_4(x_{31}),f_4(x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{33},x_{34}) \ \Rightarrow \ p_2(f_5(x_{33}),f_5(x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_7(f_{11}(x_{50}),f_{11}(x_{51}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_2(c_{26},f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_{13}(x_0,x_1)\mbox{ and }p_{13}(x_0,x_2) ) \ \Rightarrow \ p_{13}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p13}_{14},\mbox{negated\_conjecture})$

$( p_7(x_{43},x_{44})\mbox{ and }p_7(x_{43},x_{45}) ) \ \Rightarrow \ p_7(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_{15}(x_7,x_8)\mbox{ and }p_{15}(x_7,x_9) ) \ \Rightarrow \ p_{15}(x_8,x_9)\qquad\mbox{cnf}(\mbox{p15}_{17},\mbox{negated\_conjecture})$

$p_{19}(f_6(f_3(f_4(f_5(c_{22}))),f_{18}(c_{21},f_3(f_5(c_{22})))),c_{24})\qquad\mbox{cnf}(\mbox{p19}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{42},x_{40})\mbox{ and }p_{20}(x_{41},x_{42})\mbox{ and }p_2(x_{41},x_{39}) ) \ \Rightarrow \ p_{20}(x_{39},x_{40})\qquad\mbox{cnf}(\mbox{p20}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{14})\mbox{ and }p_2(x_{17},x_{15})\mbox{ and }p_{19}(x_{16},x_{17}) ) \ \Rightarrow \ p_{19}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p19}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_4,x_6)\mbox{ and }p_2(x_3,x_5) ) \ \Rightarrow \ p_{13}(f_{16}(x_3,x_4),f_{16}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p13}_{21},\mbox{negated\_conjecture})$

$( p_7(x_{58},x_{60})\mbox{ and }p_7(x_{59},x_{61}) ) \ \Rightarrow \ p_7(f_9(x_{58},x_{59}),f_9(x_{60},x_{61}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_7(x_{54},x_{56})\mbox{ and }p_7(x_{55},x_{57}) ) \ \Rightarrow \ p_7(f_8(x_{54},x_{55}),f_8(x_{56},x_{57}))\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_2(x_{47},x_{49})\mbox{ and }p_7(x_{46},x_{48}) ) \ \Rightarrow \ p_7(f_{10}(x_{46},x_{47}),f_{10}(x_{48},x_{49}))\qquad\mbox{cnf}(\mbox{p7}_{24},\mbox{negated\_conjecture})$

$( p_2(x_{35},x_{37})\mbox{ and }p_2(x_{36},x_{38}) ) \ \Rightarrow \ p_2(f_6(x_{35},x_{36}),f_6(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_2(x_{26},x_{28}) ) \ \Rightarrow \ p_2(f_{18}(x_{25},x_{26}),f_{18}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$( p_{13}(x_{21},x_{23})\mbox{ and }p_7(x_{22},x_{24}) ) \ \Rightarrow \ p_2(f_{14}(x_{21},x_{22}),f_{14}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_2(x_{10},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_{15}(f_{17}(x_{10},x_{11}),f_{17}(x_{12},x_{13}))\qquad\mbox{cnf}(\mbox{p15}_{28},\mbox{negated\_conjecture})$

$p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{26}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{p7}_{29},\mbox{negated\_conjecture})$

$( p_{19}(x_{62},f_3(f_4(f_5(c_{22}))))\mbox{ and }p_{20}(x_{63},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\mbox{ and }p_2(f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25}),f_3(c_{22})) ) \ \Rightarrow \ \neg\,p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),x_{62}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25})),f_9(f_{11}(x_{63}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{not\_p19}_{30},\mbox{negated\_conjecture})$

$( p_{19}(x_{62},f_3(f_4(f_5(c_{22}))))\mbox{ and }p_{20}(x_{63},f_6(f_3(f_4(f_5(c_{22}))),c_{21}))\mbox{ and }p_{19}(f_6(f_3(f_4(f_5(c_{22}))),f_{18}(c_{21},f_3(f_5(c_{22})))),x_{63}) ) \ \Rightarrow \ \neg\,p_7(c_{25},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{22})))),x_{62}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),f_{14}(f_{16}(c_{28},f_{17}(c_{21},c_{27})),c_{25})),f_9(f_{11}(x_{63}),f_{10}(f_{11}(f_3(f_4(f_5(c_{22})))),c_{27})))))\qquad\mbox{cnf}(\mbox{not\_p19}_{31},\mbox{negated\_conjecture})$

\medskip\textbf{SYN012-1.p} A problem to demonstrate Model Elimination






$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(x,g(x,y)) ) \ \Rightarrow \ \mbox{big\_g}(y,g(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_g}(y,g(x,y)) ) \ \Rightarrow \ \mbox{big\_f}(x,g(x,y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(g(x,y),g(x,y)) ) \ \Rightarrow \ \mbox{big\_g}(g(x,y),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x,y)\mbox{ or }\mbox{big\_f}(g(x,y),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_g}(g(x,y),g(x,y)) \ \Rightarrow \ \mbox{big\_f}(x,y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\mbox{big\_g}(x,y)\mbox{ or }\mbox{big\_g}(g(x,y),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\mbox{big\_g}(x,y) \ \Rightarrow \ \neg\,\mbox{big\_g}(g(x,y),g(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN258-1.p} RPT63 synthetic problem 43 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(x,y,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN058$\wedge$5.p} TPS problem PELL28






$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}_0 ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}\mbox{ or }\mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}\mbox{ and }\mbox{cS}@ \mbox{xx} ) )\mbox{ and }( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cG}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}\mbox{ and }\mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cG}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPELL}_{28},\mbox{conjecture})$

\medskip\textbf{SYN228-1.p} RPT63 synthetic problem 13 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(b,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN342+1.p} Church problem 46.15 (7)






$\forall x_1{:}\;\exists x_2{:}\;\forall x_3{:}\;\exists x_4{:}\;( \mbox{big\_f}(x_1,x_2,x_3) \ \Rightarrow \ \mbox{big\_f}(x_4,x_4,x_1) )\qquad\mbox{fof}(\mbox{church\_46\_15}_7,\mbox{conjecture})$

\medskip\textbf{SYN229-1.p} RPT63 synthetic problem 14 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN355$\wedge$5.p} TPS problem X2106






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}@ \mbox{xx} \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}\mbox{ or }\mbox{cQ}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cX}_{2106},\mbox{conjecture})$

\medskip\textbf{SYN619-1.p} Harrison problem 3992






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_8(x_{51},x_{51})\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_5(x_{46},x_{46})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_4(x_{39},x_{39})\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{17}(x_9,x_9)\qquad\mbox{cnf}(\mbox{p17}_6,\mbox{negated\_conjecture})$

$p_{18}(c_{24},f_6(c_{25}))\qquad\mbox{cnf}(\mbox{p18}_7,\mbox{negated\_conjecture})$

$\neg\,p_{18}(c_{24},f_6(f_7(c_{22},c_{23})))\qquad\mbox{cnf}(\mbox{not\_p18}_8,\mbox{negated\_conjecture})$

$p_{17}(f_3(x_{10}),x_{11}) \ \Rightarrow \ p_{17}(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p17}_9,\mbox{negated\_conjecture})$

$p_4(x_{49},x_{50}) \ \Rightarrow \ p_5(f_6(x_{49}),f_6(x_{50}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_4(x_3,x_4) \ \Rightarrow \ p_{10}(f_{12}(x_3),f_{12}(x_4))\qquad\mbox{cnf}(\mbox{p10}_{11},\mbox{negated\_conjecture})$

$p_{10}(x_{25},x_{26}) \ \Rightarrow \ p_2(f_{11}(x_{25}),f_{11}(x_{26}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_8(x_{31},x_{32}) \ \Rightarrow \ p_2(f_{14}(x_{31}),f_{14}(x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_8(x_{33},x_{34}) \ \Rightarrow \ p_2(f_{15}(x_{33}),f_{15}(x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_8(x_{35},x_{36}) \ \Rightarrow \ p_2(f_{16}(x_{35}),f_{16}(x_{36}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_2(x_{37},x_{38}) \ \Rightarrow \ p_2(f_3(x_{37}),f_3(x_{38}))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{17},\mbox{negated\_conjecture})$

$( p_8(x_{51},x_{52})\mbox{ and }p_8(x_{51},x_{53}) ) \ \Rightarrow \ p_8(x_{52},x_{53})\qquad\mbox{cnf}(\mbox{p8}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{46},x_{47})\mbox{ and }p_5(x_{46},x_{48}) ) \ \Rightarrow \ p_5(x_{47},x_{48})\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$( p_4(x_{39},x_{40})\mbox{ and }p_4(x_{39},x_{41}) ) \ \Rightarrow \ p_4(x_{40},x_{41})\qquad\mbox{cnf}(\mbox{p4}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$p_2(f_{13}(x_{21},x_{23}),f_{13}(x_{22},x_{24})) \ \Rightarrow \ p_2(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$p_2(f_{13}(x_{21},x_{23}),f_{13}(x_{22},x_{24})) \ \Rightarrow \ p_2(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_{18}(x_{12},f_6(c_{25})) \ \Rightarrow \ ( p_{18}(x_{12},f_6(c_{19}))\mbox{ or }p_2(f_{14}(x_{12}),c_{20}) )\qquad\mbox{cnf}(\mbox{p18}_{24},\mbox{negated\_conjecture})$

$( p_2(x_7,x_5)\mbox{ and }p_2(x_8,x_6)\mbox{ and }p_{17}(x_7,x_8) ) \ \Rightarrow \ p_{17}(x_5,x_6)\qquad\mbox{cnf}(\mbox{p17}_{25},\mbox{negated\_conjecture})$

$( p_5(x_{16},x_{14})\mbox{ and }p_8(x_{15},x_{13})\mbox{ and }p_{18}(x_{15},x_{16}) ) \ \Rightarrow \ p_{18}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p18}_{26},\mbox{negated\_conjecture})$

$p_{18}(x_{12},f_6(c_{25})) \ \Rightarrow \ ( p_{18}(x_{12},f_6(c_{19}))\mbox{ or }p_{17}(f_3(c_{21}),f_{15}(x_{12})) )\qquad\mbox{cnf}(\mbox{p18}_{27},\mbox{negated\_conjecture})$

$( p_2(x_{55},x_{57})\mbox{ and }p_2(x_{54},x_{56}) ) \ \Rightarrow \ p_8(f_9(x_{54},x_{55}),f_9(x_{56},x_{57}))\qquad\mbox{cnf}(\mbox{p8}_{28},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{29})\mbox{ and }p_2(x_{28},x_{30}) ) \ \Rightarrow \ p_2(f_{13}(x_{27},x_{28}),f_{13}(x_{29},x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{22})\mbox{ and }p_2(x_{23},x_{24}) ) \ \Rightarrow \ p_2(f_{13}(x_{21},x_{23}),f_{13}(x_{22},x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{30},\mbox{negated\_conjecture})$

$( p_2(x_{42},x_{44})\mbox{ and }p_4(x_{43},x_{45}) ) \ \Rightarrow \ p_4(f_7(x_{42},x_{43}),f_7(x_{44},x_{45}))\qquad\mbox{cnf}(\mbox{p4}_{31},\mbox{negated\_conjecture})$

$( p_2(x_{59},c_{20})\mbox{ and }p_{17}(c_{21},x_{58}) ) \ \Rightarrow \ \neg\,p_8(c_{24},f_9(f_{13}(x_{59},x_{58}),x_{60}))\qquad\mbox{cnf}(\mbox{not\_p2}_{32},\mbox{negated\_conjecture})$

$p_{18}(x_{12},f_6(c_{25})) \ \Rightarrow \ ( p_{18}(x_{12},f_6(c_{19}))\mbox{ or }p_8(x_{12},f_9(f_{13}(f_{14}(x_{12}),f_{15}(x_{12})),f_{16}(x_{12}))) )\qquad\mbox{cnf}(\mbox{p18}_{33},\mbox{negated\_conjecture})$

$p_{18}(x_{17},f_6(c_{19})) \ \Rightarrow \ ( p_{18}(x_{17},f_6(f_7(c_{22},c_{23})))\mbox{ or }p_8(x_{17},f_9(f_{13}(c_{20},c_{21}),f_{11}(f_{12}(f_7(c_{22},c_{23}))))) )\qquad\mbox{cnf}(\mbox{p18}_{34},\mbox{negated\_conjecture})$

\medskip\textbf{SYN558-1.p} Harrison problem 2379






$p_2(x_0,x_0)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_4(x_8,x_8)\qquad\mbox{cnf}(\mbox{p4}_2,\mbox{negated\_conjecture})$

$p_6(c_7,f_3(x_{21}))\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_5(c_7,c_{10},c_9)\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_5(c_7,c_{10},c_8)\qquad\mbox{cnf}(\mbox{p5}_5,\mbox{negated\_conjecture})$

$p_5(c_7,x_{21},f_3(x_{21}))\qquad\mbox{cnf}(\mbox{p5}_6,\mbox{negated\_conjecture})$

$p_2(x_6,x_7) \ \Rightarrow \ p_2(f_3(x_6),f_3(x_7))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_5(c_7,c_8,x_{26}) \ \Rightarrow \ \neg\,p_5(c_7,c_9,x_{26})\qquad\mbox{cnf}(\mbox{not\_p5}_8,\mbox{negated\_conjecture})$

$( p_2(x_0,x_1)\mbox{ and }p_2(x_0,x_2) ) \ \Rightarrow \ p_2(x_1,x_2)\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$( p_4(x_8,x_9)\mbox{ and }p_4(x_8,x_{10}) ) \ \Rightarrow \ p_4(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$( p_5(x_{11},x_{12},x_{14})\mbox{ and }p_5(x_{11},x_{14},x_{13}) ) \ \Rightarrow \ p_5(x_{11},x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p5}_{11},\mbox{negated\_conjecture})$

$( p_4(x_{25},x_{22})\mbox{ and }p_6(x_{25},x_{24})\mbox{ and }p_2(x_{24},x_{23}) ) \ \Rightarrow \ p_6(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p6}_{12},\mbox{negated\_conjecture})$

$( p_6(c_7,x_4)\mbox{ and }p_6(c_7,x_3)\mbox{ and }p_5(c_7,x_5,x_4)\mbox{ and }p_5(c_7,x_5,x_3) ) \ \Rightarrow \ p_2(x_3,x_4)\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{16})\mbox{ and }p_2(x_{19},x_{17})\mbox{ and }p_4(x_{20},x_{15})\mbox{ and }p_5(x_{20},x_{18},x_{19}) ) \ \Rightarrow \ p_5(x_{15},x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

\medskip\textbf{SYN068+1.p} Pelletier Problem 44






$\forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ ( \exists y{:}\;( \mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) )\mbox{ and }\exists y_1{:}\;( \mbox{big\_g}(y_1)\mbox{ and }\neg\,\mbox{big\_h}(x,y_1) ) ) )\qquad\mbox{fof}(\mbox{pel44}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_j}(x)\mbox{ and }\forall y{:}\;( \mbox{big\_g}(y) \ \Rightarrow \ \mbox{big\_h}(x,y) ) )\qquad\mbox{fof}(\mbox{pel44}_2,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_j}(x)\mbox{ and }\neg\,\mbox{big\_f}(x) )\qquad\mbox{fof}(\mbox{pel}_{44},\mbox{conjecture})$

\medskip\textbf{SYN014-2.p} A problem in quantification theory






$x{=}x\qquad\mbox{cnf}(\mbox{reflexivityish},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetryish},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivityish},\mbox{axiom})$

$( a{=}b\mbox{ and }\mbox{element}(c,a) ) \ \Rightarrow \ \mbox{element}(c,b)\qquad\mbox{cnf}(\mbox{element\_substitutionish}_1,\mbox{axiom})$

$( \mbox{element}(a,b)\mbox{ and }a{=}c ) \ \Rightarrow \ \mbox{element}(c,b)\qquad\mbox{cnf}(\mbox{element\_substitutionish}_2,\mbox{axiom})$

$\mbox{element}(a,a) \ \Rightarrow \ ( a{=}k\mbox{ or }a{=}a )\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$a{=}k \ \Rightarrow \ ( \mbox{element}(a,a)\mbox{ or }a{=}a )\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$( f(a){=}m\mbox{ and }\mbox{element}(a,m) ) \ \Rightarrow \ a{=}m\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$( f(a){=}a\mbox{ and }\mbox{element}(a,m) ) \ \Rightarrow \ a{=}m\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$\mbox{element}(a,m) \ \Rightarrow \ ( \mbox{element}(a,f(a))\mbox{ or }a{=}m )\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$\mbox{element}(a,m) \ \Rightarrow \ ( \mbox{element}(f(a),a)\mbox{ or }a{=}m )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$( \mbox{element}(a,b)\mbox{ and }\mbox{element}(b,a) ) \ \Rightarrow \ ( a{=}b\mbox{ or }a{=}m\mbox{ or }\mbox{element}(b,m)\mbox{ or }b{=}m )\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$g(a){=}n \ \Rightarrow \ ( \mbox{element}(a,n)\mbox{ or }a{=}n )\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$g(a){=}a \ \Rightarrow \ ( \mbox{element}(a,n)\mbox{ or }a{=}n )\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$\mbox{element}(a,g(a))\mbox{ or }\mbox{element}(a,n)\mbox{ or }a{=}n\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$\mbox{element}(g(a),a)\mbox{ or }\mbox{element}(a,n)\mbox{ or }a{=}n\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$( \mbox{element}(a,b)\mbox{ and }\mbox{element}(b,a)\mbox{ and }\mbox{element}(b,n) ) \ \Rightarrow \ ( a{=}b\mbox{ or }a{=}n\mbox{ or }b{=}n )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$a{=}m \ \Rightarrow \ ( \mbox{element}(a,k)\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$a{=}n \ \Rightarrow \ ( \mbox{element}(a,k)\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

$\mbox{element}(a,k) \ \Rightarrow \ ( a{=}n\mbox{ or }a{=}m\mbox{ or }a{=}k )\qquad\mbox{cnf}(c_{17},\mbox{negated\_conjecture})$

$\neg\,n{=}a\qquad\mbox{cnf}(c_{18},\mbox{negated\_conjecture})$

$\neg\,m{=}n\qquad\mbox{cnf}(c_{19},\mbox{negated\_conjecture})$

$n{=}k\qquad\mbox{cnf}(c_{20},\mbox{negated\_conjecture})$

$\neg\,m{=}a\qquad\mbox{cnf}(c_{21},\mbox{negated\_conjecture})$

$\neg\,k{=}a\qquad\mbox{cnf}(c_{22},\mbox{negated\_conjecture})$

$m{=}k\qquad\mbox{cnf}(c_{23},\mbox{negated\_conjecture})$

\medskip\textbf{SYN062+1.p} Pelletier Problem 32






$\forall x{:}\;( ( \mbox{big\_f}(x)\mbox{ and }( \mbox{big\_g}(x)\mbox{ or }\mbox{big\_h}(x) ) ) \ \Rightarrow \ \mbox{big\_i}(x) )\qquad\mbox{fof}(\mbox{pel32}_1,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_i}(x)\mbox{ and }\mbox{big\_h}(x) ) \ \Rightarrow \ \mbox{big\_j}(x) )\qquad\mbox{fof}(\mbox{pel32}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_k}(x) \ \Rightarrow \ \mbox{big\_h}(x) )\qquad\mbox{fof}(\mbox{pel32}_3,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_k}(x) ) \ \Rightarrow \ \mbox{big\_j}(x) )\qquad\mbox{fof}(\mbox{pel}_{32},\mbox{conjecture})$

\medskip\textbf{SYN041+1.p} Pelletier Problem 3

3: The hardest theorem proved by a breadth-first logic
 theorist.




$\neg\,p \ \Rightarrow \ q \ \Rightarrow \ ( q \ \Rightarrow \ p )\qquad\mbox{fof}(\mbox{pel}_3,\mbox{conjecture})$

\medskip\textbf{SYN952+1.p} Syntactic from Shults






$\forall x{:}\;p(x) \ \Rightarrow \ \exists y{:}\;p(y)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN070+1.p} Pelletier Problem 46






$\forall x,y{:}\;( ( \mbox{big\_f}(x)\mbox{ and }( ( \mbox{big\_f}(y)\mbox{ and }\mbox{big\_h}(y,x) ) \ \Rightarrow \ \mbox{big\_g}(y) ) ) \ \Rightarrow \ \mbox{big\_g}(x) )\qquad\mbox{fof}(\mbox{pel46}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }\neg\,\mbox{big\_g}(x) ) \ \Rightarrow \ \exists x_1{:}\;( \mbox{big\_f}(x_1)\mbox{ and }\neg\,\mbox{big\_g}(x_1)\mbox{ and }\forall y{:}\;( ( \mbox{big\_f}(y)\mbox{ and }\neg\,\mbox{big\_g}(y) ) \ \Rightarrow \ \mbox{big\_j}(x_1,y) ) )\qquad\mbox{fof}(\mbox{pel46}_2,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{big\_f}(x)\mbox{ and }\mbox{big\_f}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ \neg\,\mbox{big\_j}(y,x) )\qquad\mbox{fof}(\mbox{pel46}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(x) )\qquad\mbox{fof}(\mbox{pel}_{46},\mbox{conjecture})$

\medskip\textbf{SYN322-1.p} Church problem 46.4 (1)






$f(x,a) \ \Rightarrow \ f(a,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,a) \ \Rightarrow \ \neg\,f(a,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SYN551+1.p} Cute Little Problem






$( \exists x{:}\;x = f(g(x))\mbox{ and }\forall y,z{:}\;( ( y = f(g(y))\mbox{ and }z = f(g(z)) ) \ \Rightarrow \ y = z ) )\iff ( \exists x{:}\;x = g(f(x))\mbox{ and }\forall y,z{:}\;( ( y = g(f(y))\mbox{ and }z = g(f(z)) ) \ \Rightarrow \ y = z ) )\qquad\mbox{fof}(\mbox{prove\_this\_cute\_thing},\mbox{conjecture})$

\medskip\textbf{SYN058+1.p} Pelletier Problem 28






$\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \forall z{:}\;\mbox{big\_q}(z) )\qquad\mbox{fof}(\mbox{pel28}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_r}(x) ) \ \Rightarrow \ \exists x_1{:}\;( \mbox{big\_q}(x_1)\mbox{ and }\mbox{big\_s}(x_1) )\qquad\mbox{fof}(\mbox{pel28}_2,\mbox{axiom})$

$\exists x{:}\;\mbox{big\_s}(x) \ \Rightarrow \ \forall x_1{:}\;( \mbox{big\_f}(x_1) \ \Rightarrow \ \mbox{big\_g}(x_1) )\qquad\mbox{fof}(\mbox{pel28}_3,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_f}(x) ) \ \Rightarrow \ \mbox{big\_g}(x) )\qquad\mbox{fof}(\mbox{pel}_{28},\mbox{conjecture})$

\medskip\textbf{SYN386+1.p} Peter Andrews Problem X2138






$( \forall x{:}\;\exists y{:}\;\mbox{big\_f}(x,y)\mbox{ and }\exists x{:}\;\forall e{:}\;\exists n{:}\;\forall w{:}\;( \mbox{big\_s}(n,w) \ \Rightarrow \ \mbox{big\_d}(w,x,e) )\mbox{ and }\forall e{:}\;\exists d{:}\;\forall a,b{:}\;( \mbox{big\_d}(a,b,d) \ \Rightarrow \ \forall y,z{:}\;( ( \mbox{big\_f}(a,y)\mbox{ and }\mbox{big\_f}(b,z) ) \ \Rightarrow \ \mbox{big\_d}(y,z,e) ) ) ) \ \Rightarrow \ \exists y{:}\;\forall e{:}\;\exists m{:}\;\forall w{:}\;( \mbox{big\_s}(m,w) \ \Rightarrow \ \forall z{:}\;( \mbox{big\_f}(w,z) \ \Rightarrow \ \mbox{big\_d}(z,y,e) ) )\qquad\mbox{fof}(x_{2138},\mbox{conjecture})$

\medskip\textbf{SYN977$\wedge$4.p} Syntactic from Shults






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) ) )\qquad\mbox{thf}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN216-1.p} RPT63 synthetic problem 1 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_2(a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN978+1.p} Syntactic from Shults






$( a\mbox{ and }b ) \ \Rightarrow \ ( a\iff b )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN280-1.p} RPT63 synthetic problem 65 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(x,b,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN551+2.p} Cute Little Problem






$\exists x{:}\;\forall y{:}\;( y = f(g(y))\iff x = y )\iff \exists x{:}\;\forall y{:}\;( y = g(f(y))\iff x = y )\qquad\mbox{fof}(\mbox{prove\_this\_cute\_thing},\mbox{conjecture})$

\medskip\textbf{SYN403+1.p} Kalish and Montague Problem 234

Laws of alphabetic variance.




$\forall x{:}\;( ( ( f(x) \ \Rightarrow \ g(x) )\mbox{ and }( g(x) \ \Rightarrow \ h(x) ) ) \ \Rightarrow \ ( f(x) \ \Rightarrow \ h(x) ) )\qquad\mbox{fof}(\mbox{kalish}_{234},\mbox{conjecture})$

\medskip\textbf{SYN056-1.p} Pelletier Problem 26






$\mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_q}(a)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_q}(x) \ \Rightarrow \ \mbox{big\_p}(b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(y)\mbox{ and }\mbox{big\_r}(x) ) \ \Rightarrow \ \mbox{big\_s}(y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(y)\mbox{ and }\mbox{big\_s}(y) ) \ \Rightarrow \ \mbox{big\_r}(x)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_q}(x) ) \ \Rightarrow \ ( \mbox{big\_r}(x)\mbox{ or }\mbox{big\_s}(x) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_p}(c)\mbox{ or }\mbox{big\_r}(x) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_r}(c) ) \ \Rightarrow \ \mbox{big\_r}(x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$\mbox{big\_q}(x) \ \Rightarrow \ ( \mbox{big\_q}(d)\mbox{ or }\mbox{big\_s}(x) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{big\_q}(x)\mbox{ and }\mbox{big\_s}(d) ) \ \Rightarrow \ \mbox{big\_s}(x)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$\mbox{big\_p}(c)\mbox{ or }\mbox{big\_q}(d)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$\mbox{big\_r}(c) \ \Rightarrow \ \mbox{big\_q}(d)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$\mbox{big\_s}(d) \ \Rightarrow \ \mbox{big\_p}(c)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$\mbox{big\_r}(c) \ \Rightarrow \ \neg\,\mbox{big\_s}(d)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

\medskip\textbf{SYN040$\wedge$4.p} Pelletier Problem 1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iatom}@ q ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) ) ) )\qquad\mbox{thf}(\mbox{pel}_1,\mbox{conjecture})$

\medskip\textbf{SYN627-1.p} Harrison problem 4072






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_8(x_{70},x_{70})\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_6(x_{67},x_{67})\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_5(x_{64},x_{64})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_3(x_{40},x_{40})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_2(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{15}(c_{18},c_{19})\qquad\mbox{cnf}(\mbox{p15}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{23},c_{24})\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$p_3(c_{23},f_4(f_7(c_{17},c_{18}),c_{21}))\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$p_{16}(c_{17},c_{20},f_9(c_{19},c_{18}),c_{21},c_{22})\qquad\mbox{cnf}(\mbox{p16}_{10},\mbox{negated\_conjecture})$

$p_6(x_{55},x_{56}) \ \Rightarrow \ p_3(f_{14}(x_{55}),f_{14}(x_{56}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_{10}(x_7,x_8) \ \Rightarrow \ p_{10}(f_{12}(x_7),f_{12}(x_8))\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$p_6(x_{53},x_{54}) \ \Rightarrow \ p_3(f_{13}(x_{53}),f_{13}(x_{54}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_8(x_{70},x_{71})\mbox{ and }p_8(x_{70},x_{72}) ) \ \Rightarrow \ p_8(x_{71},x_{72})\qquad\mbox{cnf}(\mbox{p8}_{15},\mbox{negated\_conjecture})$

$( p_6(x_{67},x_{68})\mbox{ and }p_6(x_{67},x_{69}) ) \ \Rightarrow \ p_6(x_{68},x_{69})\qquad\mbox{cnf}(\mbox{p6}_{16},\mbox{negated\_conjecture})$

$( p_5(x_{64},x_{65})\mbox{ and }p_5(x_{64},x_{66}) ) \ \Rightarrow \ p_5(x_{65},x_{66})\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$( p_3(x_{40},x_{41})\mbox{ and }p_3(x_{40},x_{42}) ) \ \Rightarrow \ p_3(x_{41},x_{42})\qquad\mbox{cnf}(\mbox{p3}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{34})\mbox{ and }p_2(x_{33},x_{35}) ) \ \Rightarrow \ p_2(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$p_{15}(x_{32},c_{19}) \ \Rightarrow \ p_3(c_{24},f_4(f_7(c_{17},x_{32}),f_{14}(x_{32})))\qquad\mbox{cnf}(\mbox{p3}_{20},\mbox{negated\_conjecture})$

$p_{15}(x_{31},c_{19}) \ \Rightarrow \ p_3(c_{23},f_4(f_7(c_{17},x_{31}),f_{13}(x_{31})))\qquad\mbox{cnf}(\mbox{p3}_{21},\mbox{negated\_conjecture})$

$p_{15}(x_{31},c_{19}) \ \Rightarrow \ p_{16}(c_{17},c_{20},f_9(c_{19},x_{31}),f_{13}(x_{31}),c_{22})\qquad\mbox{cnf}(\mbox{p16}_{22},\mbox{negated\_conjecture})$

$p_{15}(x_{32},c_{19}) \ \Rightarrow \ p_{16}(c_{17},c_{20},f_9(c_{19},x_{32}),f_{14}(x_{32}),c_{25})\qquad\mbox{cnf}(\mbox{p16}_{23},\mbox{negated\_conjecture})$

$( p_6(x_{14},x_{12})\mbox{ and }p_8(x_{15},x_{13})\mbox{ and }p_{15}(x_{14},x_{15}) ) \ \Rightarrow \ p_{15}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p15}_{24},\mbox{negated\_conjecture})$

$( p_6(x_{74},x_{76})\mbox{ and }p_8(x_{73},x_{75}) ) \ \Rightarrow \ p_8(f_9(x_{73},x_{74}),f_9(x_{75},x_{76}))\qquad\mbox{cnf}(\mbox{p8}_{25},\mbox{negated\_conjecture})$

$( p_5(x_{36},x_{38})\mbox{ and }p_6(x_{37},x_{39}) ) \ \Rightarrow \ p_2(f_7(x_{36},x_{37}),f_7(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$( p_2(x_{57},x_{59})\mbox{ and }p_3(x_{58},x_{60}) ) \ \Rightarrow \ p_3(f_4(x_{57},x_{58}),f_4(x_{59},x_{60}))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$( p_{16}(c_{17},c_{20},x_4,x_6,x_3)\mbox{ and }p_{16}(c_{17},c_{20},x_4,x_5,c_{22}) ) \ \Rightarrow \ p_3(x_5,x_6)\qquad\mbox{cnf}(\mbox{p3}_{28},\mbox{negated\_conjecture})$

$( p_{16}(c_{17},c_{20},x_4,x_6,x_3)\mbox{ and }p_{16}(c_{17},c_{20},x_4,x_5,c_{22}) ) \ \Rightarrow \ p_{10}(c_{22},x_3)\qquad\mbox{cnf}(\mbox{p10}_{29},\mbox{negated\_conjecture})$

$( p_{16}(c_{17},c_{20},x_9,x_{11},c_{25})\mbox{ and }p_{16}(c_{17},c_{20},x_9,x_{10},f_{12}(c_{22})) ) \ \Rightarrow \ p_3(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p3}_{30},\mbox{negated\_conjecture})$

$( p_{16}(c_{17},c_{20},x_9,x_{11},c_{25})\mbox{ and }p_{16}(c_{17},c_{20},x_9,x_{10},f_{12}(c_{22})) ) \ \Rightarrow \ p_{10}(f_{12}(c_{22}),c_{25})\qquad\mbox{cnf}(\mbox{p10}_{31},\mbox{negated\_conjecture})$

$p_6(x_{61},x_{62})\mbox{ or }p_3(f_4(f_7(c_{17},x_{61}),f_4(f_7(c_{17},x_{62}),x_{63})),f_4(f_7(c_{17},x_{62}),f_4(f_7(c_{17},x_{61}),x_{63})))\qquad\mbox{cnf}(\mbox{p6}_{32},\mbox{negated\_conjecture})$

$( p_{15}(x_{28},x_{27})\mbox{ and }p_{16}(c_{17},x_{26},x_{27},x_{30},f_{12}(x_{29})) ) \ \Rightarrow \ p_3(x_{30},f_4(f_7(c_{17},x_{28}),f_{11}(x_{26},x_{29},x_{27},x_{30},x_{28})))\qquad\mbox{cnf}(\mbox{p3}_{33},\mbox{negated\_conjecture})$

$( p_{15}(x_{28},x_{27})\mbox{ and }p_{16}(c_{17},x_{26},x_{27},x_{30},f_{12}(x_{29})) ) \ \Rightarrow \ p_{16}(c_{17},x_{26},f_9(x_{27},x_{28}),f_{11}(x_{26},x_{29},x_{27},x_{30},x_{28}),x_{29})\qquad\mbox{cnf}(\mbox{p16}_{34},\mbox{negated\_conjecture})$

$( p_8(x_{24},x_{18})\mbox{ and }p_5(x_{22},x_{16})\mbox{ and }p_3(x_{25},x_{19})\mbox{ and }p_3(x_{23},x_{17})\mbox{ and }p_{10}(x_{21},x_{20})\mbox{ and }p_{16}(x_{22},x_{23},x_{24},x_{25},x_{21}) ) \ \Rightarrow \ p_{16}(x_{16},x_{17},x_{18},x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p16}_{35},\mbox{negated\_conjecture})$

$( p_6(x_{47},x_{52})\mbox{ and }p_8(x_{45},x_{50})\mbox{ and }p_{10}(x_{44},x_{49})\mbox{ and }p_3(x_{43},x_{48})\mbox{ and }p_3(x_{46},x_{51}) ) \ \Rightarrow \ p_3(f_{11}(x_{43},x_{44},x_{45},x_{46},x_{47}),f_{11}(x_{48},x_{49},x_{50},x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p3}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{SYN097-1.002.p} Plaisted problem sym(u(t(2,2)))






$\mbox{p\_1}_2 \ \Rightarrow \ u_1\qquad\mbox{cnf}(\mbox{u\_s2\_goal}_1,\mbox{negated\_conjecture})$

$\neg\,u_1\qquad\mbox{cnf}(\mbox{u\_1\_s2\_goal}_1,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ ( u_2\mbox{ or }\mbox{p\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type11}_1,\mbox{axiom})$

$u_2 \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{u\_2\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ ( u_3\mbox{ or }\mbox{p\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type11}_2,\mbox{axiom})$

$u_3 \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{u\_3\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{p\_2}_2\mbox{ and }\mbox{q\_1}_1 ) \ \Rightarrow \ ( u_4\mbox{ or }\mbox{q\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type12}_1,\mbox{axiom})$

$u_4 \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{u\_4\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{q\_2}_2\mbox{ and }\mbox{p\_1}_1 ) \ \Rightarrow \ ( u_5\mbox{ or }\mbox{q\_1}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type12}_2,\mbox{axiom})$

$u_5 \ \Rightarrow \ \mbox{q\_1}_2\qquad\mbox{cnf}(\mbox{u\_5\_s2\_type12}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ ( u_6\mbox{ or }\mbox{p\_1}_1 )\qquad\mbox{cnf}(\mbox{u\_s2\_type21}_1,\mbox{axiom})$

$u_6 \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_6\_s2\_type21}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ ( u_7\mbox{ or }\mbox{q\_1}_1 )\qquad\mbox{cnf}(\mbox{u\_s2\_type21}_2,\mbox{axiom})$

$u_7 \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_7\_s2\_type21}_2,\mbox{axiom})$

$\mbox{p\_1}_2 \ \Rightarrow \ ( u_8\mbox{ or }\mbox{p\_2}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type22}_1,\mbox{axiom})$

$u_8 \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_8\_s2\_type22}_1,\mbox{axiom})$

$\mbox{q\_1}_2 \ \Rightarrow \ ( u_9\mbox{ or }\mbox{q\_2}_2 )\qquad\mbox{cnf}(\mbox{u\_s2\_type22}_2,\mbox{axiom})$

$u_9 \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_9\_s2\_type22}_2,\mbox{axiom})$

$u_{10}\mbox{ or }\mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_t2}_1,\mbox{axiom})$

$u_{10} \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{u\_10\_t2}_1,\mbox{axiom})$

$u_{11}\mbox{ or }\mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_t2}_2,\mbox{axiom})$

$u_{11} \ \Rightarrow \ \mbox{q\_1}_1\qquad\mbox{cnf}(\mbox{u\_11\_t2}_2,\mbox{axiom})$

$u_{12}\mbox{ or }\mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_t2}_3,\mbox{axiom})$

$u_{12} \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{u\_12\_t2}_3,\mbox{axiom})$

$u_{13}\mbox{ or }\mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_t2}_4,\mbox{axiom})$

$u_{13} \ \Rightarrow \ \mbox{q\_2}_2\qquad\mbox{cnf}(\mbox{u\_13\_t2}_4,\mbox{axiom})$

$\mbox{sym\_u}_1 \ \Rightarrow \ \mbox{sym\_p\_1}_2\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_goal}_1,\mbox{axiom})$

$\mbox{sym\_u}_1\qquad\mbox{cnf}(\mbox{sym\_u\_1\_s2\_goal}_1,\mbox{axiom})$

$( \mbox{sym\_u}_2\mbox{ and }\mbox{sym\_p\_1}_2 ) \ \Rightarrow \ ( \mbox{sym\_p\_2}_2\mbox{ or }\mbox{sym\_p\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type11}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_2 \ \Rightarrow \ \mbox{sym\_u}_2\qquad\mbox{cnf}(\mbox{sym\_u\_2\_s2\_type11}_1,\mbox{axiom})$

$( \mbox{sym\_u}_3\mbox{ and }\mbox{sym\_p\_1}_2 ) \ \Rightarrow \ ( \mbox{sym\_q\_2}_2\mbox{ or }\mbox{sym\_q\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type11}_2,\mbox{axiom})$

$\mbox{sym\_p\_1}_2 \ \Rightarrow \ \mbox{sym\_u}_3\qquad\mbox{cnf}(\mbox{sym\_u\_3\_s2\_type11}_2,\mbox{axiom})$

$( \mbox{sym\_u}_4\mbox{ and }\mbox{sym\_q\_1}_2 ) \ \Rightarrow \ ( \mbox{sym\_p\_2}_2\mbox{ or }\mbox{sym\_q\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type12}_1,\mbox{axiom})$

$\mbox{sym\_q\_1}_2 \ \Rightarrow \ \mbox{sym\_u}_4\qquad\mbox{cnf}(\mbox{sym\_u\_4\_s2\_type12}_1,\mbox{axiom})$

$( \mbox{sym\_u}_5\mbox{ and }\mbox{sym\_q\_1}_2 ) \ \Rightarrow \ ( \mbox{sym\_q\_2}_2\mbox{ or }\mbox{sym\_p\_1}_1 )\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type12}_2,\mbox{axiom})$

$\mbox{sym\_q\_1}_2 \ \Rightarrow \ \mbox{sym\_u}_5\qquad\mbox{cnf}(\mbox{sym\_u\_5\_s2\_type12}_2,\mbox{axiom})$

$( \mbox{sym\_u}_6\mbox{ and }\mbox{sym\_p\_1}_1 ) \ \Rightarrow \ \mbox{sym\_p\_1}_2\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type21}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_1 \ \Rightarrow \ \mbox{sym\_u}_6\qquad\mbox{cnf}(\mbox{sym\_u\_6\_s2\_type21}_1,\mbox{axiom})$

$( \mbox{sym\_u}_7\mbox{ and }\mbox{sym\_q\_1}_1 ) \ \Rightarrow \ \mbox{sym\_q\_1}_2\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type21}_2,\mbox{axiom})$

$\mbox{sym\_q\_1}_1 \ \Rightarrow \ \mbox{sym\_u}_7\qquad\mbox{cnf}(\mbox{sym\_u\_7\_s2\_type21}_2,\mbox{axiom})$

$( \mbox{sym\_u}_8\mbox{ and }\mbox{sym\_p\_2}_2 ) \ \Rightarrow \ \mbox{sym\_p\_1}_2\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type22}_1,\mbox{axiom})$

$\mbox{sym\_p\_2}_2 \ \Rightarrow \ \mbox{sym\_u}_8\qquad\mbox{cnf}(\mbox{sym\_u\_8\_s2\_type22}_1,\mbox{axiom})$

$( \mbox{sym\_u}_9\mbox{ and }\mbox{sym\_q\_2}_2 ) \ \Rightarrow \ \mbox{sym\_q\_1}_2\qquad\mbox{cnf}(\mbox{sym\_u\_s2\_type22}_2,\mbox{axiom})$

$\mbox{sym\_q\_2}_2 \ \Rightarrow \ \mbox{sym\_u}_9\qquad\mbox{cnf}(\mbox{sym\_u\_9\_s2\_type22}_2,\mbox{axiom})$

$\mbox{sym\_u}_{10} \ \Rightarrow \ \neg\,\mbox{sym\_p\_1}_1\qquad\mbox{cnf}(\mbox{sym\_u\_t2}_1,\mbox{axiom})$

$\mbox{sym\_p\_1}_1 \ \Rightarrow \ \mbox{sym\_u}_{10}\qquad\mbox{cnf}(\mbox{sym\_u\_10\_t2}_1,\mbox{axiom})$

$\mbox{sym\_u}_{11} \ \Rightarrow \ \neg\,\mbox{sym\_q\_1}_1\qquad\mbox{cnf}(\mbox{sym\_u\_t2}_2,\mbox{axiom})$

$\mbox{sym\_q\_1}_1 \ \Rightarrow \ \mbox{sym\_u}_{11}\qquad\mbox{cnf}(\mbox{sym\_u\_11\_t2}_2,\mbox{axiom})$

$\mbox{sym\_u}_{12} \ \Rightarrow \ \neg\,\mbox{sym\_p\_2}_2\qquad\mbox{cnf}(\mbox{sym\_u\_t2}_3,\mbox{axiom})$

$\mbox{sym\_p\_2}_2 \ \Rightarrow \ \mbox{sym\_u}_{12}\qquad\mbox{cnf}(\mbox{sym\_u\_12\_t2}_3,\mbox{axiom})$

$\mbox{sym\_u}_{13} \ \Rightarrow \ \neg\,\mbox{sym\_q\_2}_2\qquad\mbox{cnf}(\mbox{sym\_u\_t2}_4,\mbox{axiom})$

$\mbox{sym\_q\_2}_2 \ \Rightarrow \ \mbox{sym\_u}_{13}\qquad\mbox{cnf}(\mbox{sym\_u\_13\_t2}_4,\mbox{axiom})$

\medskip\textbf{SYN351-1.p} Church problem 46.18 (3)






$f(a,y,a,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,b,x,y) \ \Rightarrow \ f(a,x,a,y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(a,x,a,y) \ \Rightarrow \ f(x,b,x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(a,x,a,y)\mbox{ or }f(a,z(x,y),x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(a,x,a,y)\mbox{ or }f(a,y,x,y)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$f(a,y,x,y)\mbox{ or }f(a,z(x,y),x,z(x,y))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( f(a,z(x,y),x,z(x,y))\mbox{ and }f(a,x,a,y) ) \ \Rightarrow \ \neg\,f(a,y,x,y)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN047-1.p} Pelletier Problem 17

A problem which appears to not be provable by [BBH72].
 For details of why not, see [Pel82] p.135f.




$p\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p \ \Rightarrow \ ( q\mbox{ or }s )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( p\mbox{ and }r ) \ \Rightarrow \ s\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,s\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ r\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN738-1.p} PSAT - K=4 C=20 V=8 D=1.8






$\mbox{ssRr}(u,\mbox{skf}_1(u))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssPv}_8(u) ) \ \Rightarrow \ ( \mbox{ssPv}_4(u)\mbox{ or }\mbox{ssPv}_5(u) )\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(u)\mbox{ and }\mbox{ssPv}_8(u) ) \ \Rightarrow \ ( \mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_2(u) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(u)\mbox{ or }\mbox{ssPv}_7(u) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_6(u)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w) ) \ \Rightarrow \ ( \mbox{ssPv}_5(w)\mbox{ or }\mbox{ssPv}_6(u)\mbox{ or }\mbox{ssPv}_8(u) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_5(v)\mbox{ or }\mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_5(u) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_8(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(w) ) \ \Rightarrow \ ( \mbox{ssPv}_5(u)\mbox{ or }\mbox{ssPv}_7(u) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_4(u)\mbox{ and }\mbox{ssPv}_5(u) ) \ \Rightarrow \ \mbox{ssPv}_6(w)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_8(u) ) \ \Rightarrow \ \mbox{ssPv}_8(w)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_8(u) ) \ \Rightarrow \ \mbox{ssPv}_7(w)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_6(u) ) \ \Rightarrow \ \mbox{ssPv}_7(w)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ \mbox{ssPv}_4(w)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(w)\mbox{ and }\mbox{ssPv}_1(u) ) \ \Rightarrow \ \mbox{ssPv}_8(u)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_2(w)\mbox{ and }\mbox{ssPv}_2(u) ) \ \Rightarrow \ \mbox{ssPv}_7(u)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(w)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_8(u)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_7(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_8(w)\mbox{ or }\mbox{ssPv}_3(x)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_8(w)\mbox{ or }\mbox{ssPv}_6(x) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_5(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_5(u) ) \ \Rightarrow \ ( \mbox{ssPv}_6(w)\mbox{ or }\mbox{ssPv}_4(x) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_6(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_3(w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_5(x)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(u,x)\mbox{ and }\mbox{ssPv}_7(u) ) \ \Rightarrow \ \mbox{ssPv}_7(x)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{SYN181-1.p} RPT63 synthetic problem 79 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_4(d,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN054+1.p} Pelletier Problem 24






$\neg\,\exists x{:}\;( \mbox{big\_s}(x)\mbox{ and }\mbox{big\_q}(x) )\qquad\mbox{fof}(\mbox{pel24}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_r}(x) ) )\qquad\mbox{fof}(\mbox{pel24}_2,\mbox{axiom})$

$\neg\,\exists x{:}\;\mbox{big\_p}(x) \ \Rightarrow \ \exists y{:}\;\mbox{big\_q}(y)\qquad\mbox{fof}(\mbox{pel24}_3,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_r}(x) ) \ \Rightarrow \ \mbox{big\_s}(x) )\qquad\mbox{fof}(\mbox{pel24}_4,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_p}(x)\mbox{ and }\mbox{big\_r}(x) )\qquad\mbox{fof}(\mbox{pel}_{24},\mbox{conjecture})$

\medskip\textbf{SYN350+1.p} Church problem 46.18 (2)






$\forall x{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( ( \mbox{big\_f}(x,z)\iff \mbox{big\_f}(z,x) ) \ \Rightarrow \ ( \mbox{big\_f}(x,z)\iff ( \mbox{big\_f}(y_2,z)\mbox{ and }( \mbox{big\_f}(y_1,z) \ \Rightarrow \ \mbox{big\_f}(y_1,y_2) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_18}_2,\mbox{conjecture})$

\medskip\textbf{SYN989$\wedge$3.p} All things are true or false






$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$p\mbox{ or }\neg\,p\qquad\mbox{thf}(\mbox{cTRIV}_0,\mbox{conjecture})$

\medskip\textbf{SYN340+1.p} Church problem 46.15 (5)






$\exists x{:}\;\forall y{:}\;\exists z_1,z_2{:}\;( \mbox{big\_f}(x,y,z_1,z_2,z_1) \ \Rightarrow \ \mbox{big\_f}(z_1,x,y,z_1,z_2) )\qquad\mbox{fof}(\mbox{church\_46\_15}_5,\mbox{conjecture})$

\medskip\textbf{SYN245-1.p} RPT63 synthetic problem 30 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(c,x,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN331+1.p} Church problem 46.14 (3)






$\exists x,y{:}\;\forall z{:}\;( \mbox{big\_f}(x,z) \ \Rightarrow \ ( \mbox{big\_f}(y,z) \ \Rightarrow \ ( ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(z,z) ) \ \Rightarrow \ ( ( \mbox{big\_f}(y,x)\mbox{ or }\mbox{big\_f}(z,z) ) \ \Rightarrow \ ( \mbox{big\_f}(z,x)\mbox{ or }\mbox{big\_f}(z,y) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_3,\mbox{conjecture})$

\medskip\textbf{SYN069+1.p} Pelletier Problem 45






$\forall x{:}\;( ( \mbox{big\_f}(x)\mbox{ and }\forall y{:}\;( ( \mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) ) \ \Rightarrow \ \mbox{big\_j}(x,y) ) ) \ \Rightarrow \ \forall y_1{:}\;( \mbox{big\_g}(y_1)\mbox{ and }\mbox{big\_h}(x,y_1)\mbox{ and }\mbox{big\_k}(y_1) ) )\qquad\mbox{fof}(\mbox{pel45}_1,\mbox{axiom})$

$\neg\,\exists y{:}\;( \mbox{big\_l}(y)\mbox{ and }\mbox{big\_k}(y) )\qquad\mbox{fof}(\mbox{pel45}_2,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }\forall y{:}\;( \mbox{big\_h}(x,y) \ \Rightarrow \ \mbox{big\_l}(y) )\mbox{ and }\forall y_1{:}\;( ( \mbox{big\_g}(y_1)\mbox{ and }\mbox{big\_h}(x,y_1) ) \ \Rightarrow \ \mbox{big\_j}(x,y_1) ) )\qquad\mbox{fof}(\mbox{pel45}_3,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }\neg\,\exists y{:}\;( \mbox{big\_g}(y)\mbox{ and }\mbox{big\_h}(x,y) ) )\qquad\mbox{fof}(\mbox{pel}_{45},\mbox{conjecture})$

\medskip\textbf{SYN236-1.p} RPT63 synthetic problem 21 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(e,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN957+1.p} Syntactic from Shults






$\neg\,\exists y{:}\;\forall x{:}\;( a(x,y)\iff \neg\,a(x,x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN915-1.p} FALSE






$\mbox{\$false}\qquad\mbox{cnf}(\mbox{false\_clause},\mbox{negated\_conjecture})$

\medskip\textbf{SYN267-1.p} RPT63 synthetic problem 52 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(x,y,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN079-1.p} Pelletier Problem 57






$\mbox{big\_f}(f(a,b),f(b,c))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_f}(f(b,c),f(a,c))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(y,z) ) \ \Rightarrow \ \mbox{big\_f}(x,z)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\neg\,\mbox{big\_f}(f(a,b),f(a,c))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN374+1.p} Peter Andrews Problem X2125






$\exists x{:}\;\forall y{:}\;( \mbox{big\_p}(x)\iff \mbox{big\_p}(y) )\iff ( \exists x{:}\;\mbox{big\_p}(x)\iff \forall y{:}\;\mbox{big\_p}(y) )\qquad\mbox{fof}(x_{2125},\mbox{conjecture})$

\medskip\textbf{SYN955+1.p} Syntactic from Shults






$\forall x{:}\;( a(x) \ \Rightarrow \ b(x) ) \ \Rightarrow \ ( \exists x{:}\;a(x) \ \Rightarrow \ \exists x{:}\;b(x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN327+1.p} Church problem 46.12 (2)






$\forall x{:}\;\exists y{:}\;\forall z{:}\;( \mbox{big\_f}(y,x) \ \Rightarrow \ ( ( \mbox{big\_f}(x,z) \ \Rightarrow \ \mbox{big\_f}(x,y) )\mbox{ and }( \mbox{big\_f}(x,y) \ \Rightarrow \ ( \neg\,\mbox{big\_f}(x,z) \ \Rightarrow \ ( \mbox{big\_f}(y,x)\mbox{ and }\mbox{big\_f}(z,y) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_12}_2,\mbox{conjecture})$

\medskip\textbf{SYN951+1.p} Syntactic from Shults






$\exists x{:}\;p(x) \ \Rightarrow \ ( \exists x{:}\;p(x)\mbox{ and }( a \ \Rightarrow \ ( ( b\mbox{ or }\neg\,b )\mbox{ and }( q \ \Rightarrow \ q ) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN971+1.p} Syntactic from Shults






$\exists y{:}\;( \exists x{:}\;p(x) \ \Rightarrow \ p(y) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN309-1.p} Problem for testing satisfiability






$s(h(x,y,z),a,h(x,y,z)) \ \Rightarrow \ p(f(x),a,g(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$r(f(a),a,b)\mbox{ or }s(h(a,y,z),b,h(a,y,z))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$p(k(x,y),k(x,y),b) \ \Rightarrow \ ( s(f(x),x,x)\mbox{ or }p(f(x),g(x),g(x)) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$s(f(x),x,x)\mbox{ or }p(f(y),y,f(y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$p(x,x,x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$p(a,b,c)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SYN328-1.p} Church problem 46.12 (3)






$( f(y(x))\mbox{ and }f(x) ) \ \Rightarrow \ g(y(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(y(x))\mbox{ or }f(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$g(y(x)) \ \Rightarrow \ f(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( f(y(x))\mbox{ and }g(x) ) \ \Rightarrow \ h(y(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(y(x))\mbox{ or }g(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$h(y(x)) \ \Rightarrow \ g(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$h(x) \ \Rightarrow \ ( f(y(x))\mbox{ or }h(y(x)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( g(y(x))\mbox{ and }h(x) ) \ \Rightarrow \ h(y(x))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$f(y(x)) \ \Rightarrow \ ( g(y(x))\mbox{ or }h(x) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$h(y(x)) \ \Rightarrow \ h(x)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( f(z(x))\mbox{ and }g(z(x)) ) \ \Rightarrow \ \neg\,h(z(x))\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{SYN982-1.p} Problem that's Horn but resists hyper-resolution






$\mbox{ssNonce}(\mbox{na})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{ssP}(t)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{ssBf}(\mbox{na})\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{ssP}(b)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{ssP}(a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{ssNonce}(\mbox{nb}(u))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{ssNonce}(\mbox{tb}(u))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$\neg\,\mbox{ssNonce}(\mbox{kt}(u))\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{ssTk}(\mbox{key}(\mbox{bt},b))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{ssTk}(\mbox{key}(\mbox{at},a))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{ssBk}(\mbox{key}(\mbox{bt},t))\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{ssSa}(\mbox{pair}(b,\mbox{na}))\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{ssAk}(\mbox{key}(\mbox{at},t))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{ssIm}(\mbox{pair}(u,v)) \ \Rightarrow \ \mbox{ssIm}(v)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$\mbox{ssIm}(\mbox{pair}(u,v)) \ \Rightarrow \ \mbox{ssIm}(u)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$\mbox{ssM}(\mbox{sent}(a,b,\mbox{pair}(a,\mbox{na})))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$\mbox{ssIm}(\mbox{triple}(u,v,w)) \ \Rightarrow \ \mbox{ssIm}(w)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$\mbox{ssIm}(\mbox{triple}(u,v,w)) \ \Rightarrow \ \mbox{ssIm}(v)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$\mbox{ssIm}(\mbox{triple}(u,v,w)) \ \Rightarrow \ \mbox{ssIm}(u)\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$\mbox{ssM}(\mbox{sent}(u,v,w)) \ \Rightarrow \ \mbox{ssIm}(w)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$\mbox{ssIm}(\mbox{quadr}(u,v,w,x)) \ \Rightarrow \ \mbox{ssIm}(x)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$\mbox{ssIm}(\mbox{quadr}(u,v,w,x)) \ \Rightarrow \ \mbox{ssIm}(w)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$\mbox{ssIm}(\mbox{quadr}(u,v,w,x)) \ \Rightarrow \ \mbox{ssIm}(v)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$\mbox{ssIm}(\mbox{quadr}(u,v,w,x)) \ \Rightarrow \ \mbox{ssIm}(u)\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssP}(v) ) \ \Rightarrow \ \mbox{ssIk}(\mbox{key}(u,v))\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssIm}(v) ) \ \Rightarrow \ \mbox{ssIm}(\mbox{pair}(u,v))\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssP}(v)\mbox{ and }\mbox{ssP}(w) ) \ \Rightarrow \ \mbox{ssM}(\mbox{sent}(v,w,u))\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssIm}(v)\mbox{ and }\mbox{ssIm}(w) ) \ \Rightarrow \ \mbox{ssIm}(\mbox{triple}(u,v,w))\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssIk}(\mbox{key}(v,w))\mbox{ and }\mbox{ssP}(w) ) \ \Rightarrow \ \mbox{ssIm}(\mbox{encr}(u,v))\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(u,b,\mbox{pair}(u,v)))\mbox{ and }\mbox{ssBf}(v) ) \ \Rightarrow \ \mbox{ssSb}(\mbox{pair}(u,v))\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{axiom})$

$( \mbox{ssIm}(u)\mbox{ and }\mbox{ssIm}(v)\mbox{ and }\mbox{ssIm}(w)\mbox{ and }\mbox{ssIm}(x) ) \ \Rightarrow \ \mbox{ssIm}(\mbox{quadr}(u,v,w,x))\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(t,a,\mbox{triple}(\mbox{encr}(\mbox{quadr}(u,v,w,x),\mbox{at}),y,z)))\mbox{ and }\mbox{ssSa}(\mbox{pair}(u,v)) ) \ \Rightarrow \ \mbox{ssAk}(\mbox{key}(w,u))\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(u,b,\mbox{pair}(u,v)))\mbox{ and }\mbox{ssBf}(v) ) \ \Rightarrow \ \mbox{ssM}(\mbox{sent}(b,t,\mbox{triple}(b,\mbox{nb}(v),\mbox{encr}(\mbox{triple}(u,v,\mbox{tb}(v)),\mbox{bt}))))\qquad\mbox{cnf}(\mbox{clause}_{33},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(u,b,\mbox{pair}(\mbox{encr}(\mbox{triple}(u,v,\mbox{tb}(w)),\mbox{bt}),\mbox{encr}(\mbox{nb}(w),v))))\mbox{ and }\mbox{ssSb}(\mbox{pair}(u,w)) ) \ \Rightarrow \ \mbox{ssBk}(\mbox{key}(v,u))\qquad\mbox{cnf}(\mbox{clause}_{34},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(t,a,\mbox{triple}(\mbox{encr}(\mbox{quadr}(u,v,w,x),\mbox{at}),y,z)))\mbox{ and }\mbox{ssSa}(\mbox{pair}(u,v)) ) \ \Rightarrow \ \mbox{ssM}(\mbox{sent}(a,u,\mbox{pair}(y,\mbox{encr}(z,w))))\qquad\mbox{cnf}(\mbox{clause}_{35},\mbox{axiom})$

$( \mbox{ssM}(\mbox{sent}(u,t,\mbox{triple}(u,v,\mbox{encr}(\mbox{triple}(w,x,y),z))))\mbox{ and }\mbox{ssTk}(\mbox{key}(z,u))\mbox{ and }\mbox{ssTk}(\mbox{key}(x_1,w))\mbox{ and }\mbox{ssNonce}(x) ) \ \Rightarrow \ \mbox{ssM}(\mbox{sent}(t,w,\mbox{triple}(\mbox{encr}(\mbox{quadr}(u,x,\mbox{kt}(x),y),x_1),\mbox{encr}(\mbox{triple}(w,\mbox{kt}(x),y),z),v)))\qquad\mbox{cnf}(\mbox{clause}_{36},\mbox{axiom})$

$\mbox{ssIk}(\mbox{key}(u,b)) \ \Rightarrow \ \neg\,\mbox{ssBk}(\mbox{key}(u,a))\qquad\mbox{cnf}(\mbox{clause}_{37},\mbox{negated\_conjecture})$

\medskip\textbf{SYN154-1.p} RPT63 synthetic problem 52 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_3(e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN613-1.p} Harrison problem 3862






$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{negated\_conjecture})$

$p_6(x_{48},x_{48})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$p_5(x_{41},x_{41})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{36},x_{36})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_{29},x_{29})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{16}(x_{14},x_{14})\qquad\mbox{cnf}(\mbox{p16}_6,\mbox{negated\_conjecture})$

$p_{11}(c_{23},f_{12}(f_9(c_{21})))\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_{11}(c_{24},f_{12}(f_9(c_{21})))\qquad\mbox{cnf}(\mbox{p11}_8,\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_9(x_{46}),f_9(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_9,\mbox{negated\_conjecture})$

$p_5(x_3,x_4) \ \Rightarrow \ p_{11}(f_{12}(x_3),f_{12}(x_4))\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_{11}(x_5,x_6) \ \Rightarrow \ p_{11}(f_{13}(x_5),f_{13}(x_6))\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_{11}(x_{11},x_{12}) \ \Rightarrow \ p_{11}(f_{15}(x_{11}),f_{15}(x_{12}))\qquad\mbox{cnf}(\mbox{p11}_{12},\mbox{negated\_conjecture})$

$p_2(x_{39},x_{40}) \ \Rightarrow \ p_3(f_4(x_{39}),f_4(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_2(x_{44},x_{45}) \ \Rightarrow \ p_5(f_{10}(x_{44}),f_{10}(x_{45}))\qquad\mbox{cnf}(\mbox{p5}_{14},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{15},\mbox{negated\_conjecture})$

$( p_6(x_{48},x_{49})\mbox{ and }p_6(x_{48},x_{50}) ) \ \Rightarrow \ p_6(x_{49},x_{50})\qquad\mbox{cnf}(\mbox{p6}_{16},\mbox{negated\_conjecture})$

$( p_5(x_{41},x_{42})\mbox{ and }p_5(x_{41},x_{43}) ) \ \Rightarrow \ p_5(x_{42},x_{43})\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$( p_3(x_{36},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(x_{37},x_{38})\qquad\mbox{cnf}(\mbox{p3}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{30})\mbox{ and }p_2(x_{29},x_{31}) ) \ \Rightarrow \ p_2(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_{16}(x_{14},x_{15})\mbox{ and }p_{16}(x_{14},x_{16}) ) \ \Rightarrow \ p_{16}(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p16}_{20},\mbox{negated\_conjecture})$

$p_{17}(c_{19},f_4(f_7(f_{10}(c_{20}),f_8(f_9(c_{21}),c_{22}))),c_{24})\qquad\mbox{cnf}(\mbox{p17}_{21},\mbox{negated\_conjecture})$

$\neg\,p_{17}(c_{19},f_4(f_7(f_{10}(c_{20}),f_8(f_9(c_{21}),c_{22}))),c_{23})\qquad\mbox{cnf}(\mbox{not\_p17}_{22},\mbox{negated\_conjecture})$

$( p_{11}(x_{26},x_{24})\mbox{ and }p_{18}(x_{25},x_{26})\mbox{ and }p_{11}(x_{25},x_{23}) ) \ \Rightarrow \ p_{18}(x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p18}_{23},\mbox{negated\_conjecture})$

$( p_{11}(x_7,x_9)\mbox{ and }p_{11}(x_8,x_{10}) ) \ \Rightarrow \ p_{11}(f_{14}(x_7,x_8),f_{14}(x_9,x_{10}))\qquad\mbox{cnf}(\mbox{p11}_{24},\mbox{negated\_conjecture})$

$( p_5(x_{51},x_{53})\mbox{ and }p_5(x_{52},x_{54}) ) \ \Rightarrow \ p_6(f_8(x_{51},x_{52}),f_8(x_{53},x_{54}))\qquad\mbox{cnf}(\mbox{p6}_{25},\mbox{negated\_conjecture})$

$( p_5(x_{32},x_{34})\mbox{ and }p_6(x_{33},x_{35}) ) \ \Rightarrow \ p_2(f_7(x_{32},x_{33}),f_7(x_{34},x_{35}))\qquad\mbox{cnf}(\mbox{p2}_{26},\mbox{negated\_conjecture})$

$p_{11}(x_{28},f_{12}(f_9(c_{21}))) \ \Rightarrow \ p_{18}(f_{13}(f_{14}(x_{28},c_{25})),f_{13}(f_{14}(c_{23},c_{25})))\qquad\mbox{cnf}(\mbox{p18}_{27},\mbox{negated\_conjecture})$

$p_{11}(x_{27},f_{12}(f_9(c_{21}))) \ \Rightarrow \ p_{18}(f_{13}(f_{14}(x_{27},c_{25})),f_{13}(f_{14}(c_{24},c_{25})))\qquad\mbox{cnf}(\mbox{p18}_{28},\mbox{negated\_conjecture})$

$( p_3(x_{22},x_{18})\mbox{ and }p_{16}(x_{21},x_{17})\mbox{ and }p_{11}(x_{20},x_{19})\mbox{ and }p_{17}(x_{21},x_{22},x_{20}) ) \ \Rightarrow \ p_{17}(x_{17},x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$( p_{11}(x_{13},f_{12}(f_9(c_{21})))\mbox{ and }p_{17}(c_{19},f_4(c_{20}),x_{13}) ) \ \Rightarrow \ ( p_{17}(c_{19},f_4(c_{20}),c_{23})\mbox{ or }p_{11}(f_{15}(x_{13}),f_{12}(f_9(c_{21}))) )\qquad\mbox{cnf}(\mbox{p17}_{30},\mbox{negated\_conjecture})$

$( p_{11}(x_{13},f_{12}(f_9(c_{21})))\mbox{ and }p_{17}(c_{19},f_4(c_{20}),x_{13})\mbox{ and }p_{18}(f_{13}(f_{14}(f_{15}(x_{13}),c_{25})),f_{13}(f_{14}(x_{13},c_{25}))) ) \ \Rightarrow \ p_{17}(c_{19},f_4(c_{20}),c_{23})\qquad\mbox{cnf}(\mbox{p17}_{31},\mbox{negated\_conjecture})$

\medskip\textbf{SYN065-1.p} Pelletier Problem 36






$\mbox{big\_f}(x,f(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_g}(x,g(x))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(y,z) ) \ \Rightarrow \ \mbox{big\_h}(x,z)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_g}(y,z) ) \ \Rightarrow \ \mbox{big\_h}(x,z)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( \mbox{big\_g}(x,y)\mbox{ and }\mbox{big\_f}(y,z) ) \ \Rightarrow \ \mbox{big\_h}(x,z)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$( \mbox{big\_g}(x,y)\mbox{ and }\mbox{big\_g}(y,z) ) \ \Rightarrow \ \mbox{big\_h}(x,z)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\neg\,\mbox{big\_h}(a,x)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN393+1.003.p} Pelletier 12






$( ( p_1\iff p_2 )\iff p_3 )\iff ( p_1\iff ( p_2\iff p_3 ) )\qquad\mbox{fof}(\mbox{pel}_{12},\mbox{conjecture})$

\medskip\textbf{SYN078+1.p} Pelletier Problem 56






$\forall x{:}\;( \exists y{:}\;( \mbox{big\_p}(y)\mbox{ and }x = f(y) ) \ \Rightarrow \ \mbox{big\_p}(x) )\iff \forall u{:}\;( \mbox{big\_p}(u) \ \Rightarrow \ \mbox{big\_p}(f(u)) )\qquad\mbox{fof}(\mbox{pel}_{56},\mbox{conjecture})$

\medskip\textbf{SYN081+1.p} Pelletier Problem 59






$\forall x{:}\;( \mbox{big\_f}(x)\iff \neg\,\mbox{big\_f}(f(x)) )\qquad\mbox{fof}(\mbox{pel59}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{big\_f}(x)\mbox{ and }\neg\,\mbox{big\_f}(f(x)) )\qquad\mbox{fof}(\mbox{pel}_{59},\mbox{conjecture})$

\medskip\textbf{SYN053-1.p} Pelletier Problem 23






$p\mbox{ or }\mbox{big\_f}(x)\mbox{ or }\mbox{big\_f}(y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p\mbox{ or }\mbox{big\_f}(x)\mbox{ or }\mbox{big\_f}(b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{big\_f}(a) \ \Rightarrow \ ( p\mbox{ or }\mbox{big\_f}(y) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(a) \ \Rightarrow \ \neg\,\mbox{big\_f}(b)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SYN390$\wedge$4.p} Pelletier 11






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ p ) )\qquad\mbox{thf}(\mbox{pel}_{11},\mbox{conjecture})$

\medskip\textbf{SYN580-1.p} Harrison problem 3332






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_4(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p4}_3,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{11}(c_{15},c_{16})\qquad\mbox{cnf}(\mbox{p11}_5,\mbox{negated\_conjecture})$

$p_9(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p9}_6,\mbox{negated\_conjecture})$

$p_2(c_{14},f_3(c_{15}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$\neg\,p_8(c_{12},c_{13},c_{14})\qquad\mbox{cnf}(\mbox{not\_p8}_8,\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_4(f_6(x_{17}),f_6(x_{18}))\qquad\mbox{cnf}(\mbox{p4}_9,\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_3(x_{10}),f_3(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_4(f_5(x_{15}),f_5(x_{16}))\qquad\mbox{cnf}(\mbox{p4}_{11},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{12},\mbox{negated\_conjecture})$

$( p_7(x_{19},x_{20})\mbox{ and }p_7(x_{19},x_{21}) ) \ \Rightarrow \ p_7(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$( p_4(x_{12},x_{13})\mbox{ and }p_4(x_{12},x_{14}) ) \ \Rightarrow \ p_4(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p4}_{14},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$( p_{11}(x_{29},c_{16})\mbox{ and }p_9(c_{12},x_{28}) ) \ \Rightarrow \ p_8(c_{12},x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p8}_{16},\mbox{negated\_conjecture})$

$p_9(c_{12},x_{30}) \ \Rightarrow \ ( p_9(c_{12},f_5(x_{31}))\mbox{ or }p_8(c_{12},x_{30},x_{31}) )\qquad\mbox{cnf}(\mbox{p9}_{17},\mbox{negated\_conjecture})$

$p_9(c_{12},x_{30}) \ \Rightarrow \ ( p_9(c_{12},f_6(x_{31}))\mbox{ or }p_8(c_{12},x_{30},f_3(x_{31})) )\qquad\mbox{cnf}(\mbox{p9}_{18},\mbox{negated\_conjecture})$

$( p_7(x_{35},x_{32})\mbox{ and }p_9(x_{35},x_{34})\mbox{ and }p_4(x_{34},x_{33}) ) \ \Rightarrow \ p_9(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p9}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{20},\mbox{negated\_conjecture})$

$( p_9(c_{12},x_{30})\mbox{ and }p_8(c_{12},f_5(x_{31}),f_3(x_{31})) ) \ \Rightarrow \ p_8(c_{12},x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p8}_{21},\mbox{negated\_conjecture})$

$( p_9(c_{12},x_{30})\mbox{ and }p_8(c_{12},f_6(x_{31}),x_{31}) ) \ \Rightarrow \ p_8(c_{12},x_{30},f_3(x_{31}))\qquad\mbox{cnf}(\mbox{p8}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{24})\mbox{ and }p_4(x_{26},x_{23})\mbox{ and }p_7(x_{27},x_{22})\mbox{ and }p_8(x_{27},x_{26},x_{25}) ) \ \Rightarrow \ p_8(x_{22},x_{23},x_{24})\qquad\mbox{cnf}(\mbox{p8}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{SYN332+1.p} Church problem 46.14 (4)






$\exists x,y{:}\;\forall z{:}\;( ( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(y,x) ) <~> \mbox{big\_f}(x,z) \ \Rightarrow \ ( ( \mbox{big\_f}(x,z)\iff \mbox{big\_f}(z,x) ) \ \Rightarrow \ ( ( \mbox{big\_f}(x,z)\iff \mbox{big\_f}(y,z) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(y,x) \ \Rightarrow \ \mbox{big\_f}(x,y) )\iff \mbox{big\_f}(z,z) ) \ \Rightarrow \ ( ( \mbox{big\_f}(x,y)\iff \mbox{big\_f}(y,x) )\iff \mbox{big\_f}(z,y) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_14}_4,\mbox{conjecture})$

\medskip\textbf{SYN358$\wedge$5.p} TPS problem X2109






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( p\mbox{ and }\mbox{cQ}@ \mbox{xx} )\iff ( p\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cX}_{2109},\mbox{conjecture})$

\medskip\textbf{SYN239-1.p} RPT63 synthetic problem 24 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(x,c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN733+1.p} Peter Andrews Problem Y2141






$\forall x{:}\;\exists y{:}\;( p(x)\mbox{ and }( q(y)\mbox{ or }q(x) ) ) \ \Rightarrow \ \exists z{:}\;( p(z)\mbox{ and }q(z) )\qquad\mbox{fof}(y_{2141},\mbox{conjecture})$

\medskip\textbf{SYN628-1.p} Harrison problem 4091






$p_2(x_4,x_4)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_9(x_{40},x_{40})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_6(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_5(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p5}_4,\mbox{negated\_conjecture})$

$p_3(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p3}_5,\mbox{negated\_conjecture})$

$p_9(c_{28},c_{22})\qquad\mbox{cnf}(\mbox{p9}_6,\mbox{negated\_conjecture})$

$p_{20}(c_{22},c_{28})\qquad\mbox{cnf}(\mbox{p20}_7,\mbox{negated\_conjecture})$

$p_{20}(c_{28},c_{26})\qquad\mbox{cnf}(\mbox{p20}_8,\mbox{negated\_conjecture})$

$p_{21}(c_{27},c_{22})\qquad\mbox{cnf}(\mbox{p21}_9,\mbox{negated\_conjecture})$

$p_{19}(c_{22},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{10},\mbox{negated\_conjecture})$

$p_{19}(c_{27},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{11},\mbox{negated\_conjecture})$

$p_{19}(c_{26},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{p19}_{12},\mbox{negated\_conjecture})$

$p_9(x_{53},x_{54}) \ \Rightarrow \ p_9(f_{16}(x_{53}),f_{16}(x_{54}))\qquad\mbox{cnf}(\mbox{p9}_{13},\mbox{negated\_conjecture})$

$p_2(x_{22},x_{23}) \ \Rightarrow \ p_3(f_4(x_{22}),f_4(x_{23}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_5(x_{27},x_{28}) \ \Rightarrow \ p_5(f_{12}(x_{27}),f_{12}(x_{28}))\qquad\mbox{cnf}(\mbox{p5}_{15},\mbox{negated\_conjecture})$

$p_5(x_{29},x_{30}) \ \Rightarrow \ p_5(f_{13}(x_{29}),f_{13}(x_{30}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$p_5(x_{31},x_{32}) \ \Rightarrow \ p_5(f_{14}(x_{31}),f_{14}(x_{32}))\qquad\mbox{cnf}(\mbox{p5}_{17},\mbox{negated\_conjecture})$

$p_5(x_{47},x_{48}) \ \Rightarrow \ p_9(f_{11}(x_{47}),f_{11}(x_{48}))\qquad\mbox{cnf}(\mbox{p9}_{18},\mbox{negated\_conjecture})$

$( p_2(x_4,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(x_5,x_6)\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_9(x_{40},x_{41})\mbox{ and }p_9(x_{40},x_{42}) ) \ \Rightarrow \ p_9(x_{41},x_{42})\qquad\mbox{cnf}(\mbox{p9}_{20},\mbox{negated\_conjecture})$

$( p_6(x_{33},x_{34})\mbox{ and }p_6(x_{33},x_{35}) ) \ \Rightarrow \ p_6(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p6}_{21},\mbox{negated\_conjecture})$

$( p_5(x_{24},x_{25})\mbox{ and }p_5(x_{24},x_{26}) ) \ \Rightarrow \ p_5(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p5}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{19},x_{20})\mbox{ and }p_3(x_{19},x_{21}) ) \ \Rightarrow \ p_3(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_3,x_1)\mbox{ and }p_9(x_2,x_0)\mbox{ and }p_{19}(x_2,x_3) ) \ \Rightarrow \ p_{19}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p19}_{24},\mbox{negated\_conjecture})$

$( p_9(x_{17},x_{15})\mbox{ and }p_9(x_{18},x_{16})\mbox{ and }p_{21}(x_{17},x_{18}) ) \ \Rightarrow \ p_{21}(x_{15},x_{16})\qquad\mbox{cnf}(\mbox{p21}_{25},\mbox{negated\_conjecture})$

$( p_9(x_{13},x_{11})\mbox{ and }p_9(x_{14},x_{12})\mbox{ and }p_{20}(x_{13},x_{14}) ) \ \Rightarrow \ p_{20}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p20}_{26},\mbox{negated\_conjecture})$

$( p_5(x_7,x_9)\mbox{ and }p_6(x_8,x_{10}) ) \ \Rightarrow \ p_2(f_7(x_7,x_8),f_7(x_9,x_{10}))\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_9(x_{59},x_{61})\mbox{ and }p_9(x_{60},x_{62}) ) \ \Rightarrow \ p_9(f_{18}(x_{59},x_{60}),f_{18}(x_{61},x_{62}))\qquad\mbox{cnf}(\mbox{p9}_{28},\mbox{negated\_conjecture})$

$( p_9(x_{55},x_{57})\mbox{ and }p_9(x_{56},x_{58}) ) \ \Rightarrow \ p_9(f_{17}(x_{55},x_{56}),f_{17}(x_{57},x_{58}))\qquad\mbox{cnf}(\mbox{p9}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{49},x_{51})\mbox{ and }p_9(x_{50},x_{52}) ) \ \Rightarrow \ p_9(f_{15}(x_{49},x_{50}),f_{15}(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p9}_{30},\mbox{negated\_conjecture})$

$( p_9(x_{43},x_{45})\mbox{ and }p_9(x_{44},x_{46}) ) \ \Rightarrow \ p_9(f_{10}(x_{43},x_{44}),f_{10}(x_{45},x_{46}))\qquad\mbox{cnf}(\mbox{p9}_{31},\mbox{negated\_conjecture})$

$( p_5(x_{36},x_{38})\mbox{ and }p_5(x_{37},x_{39}) ) \ \Rightarrow \ p_6(f_8(x_{36},x_{37}),f_8(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p6}_{32},\mbox{negated\_conjecture})$

$( p_{21}(x_{63},c_{26})\mbox{ and }p_{21}(c_{22},x_{63}) ) \ \Rightarrow \ \neg\,p_{19}(x_{63},f_4(f_7(c_{23},f_8(c_{24},c_{25}))))\qquad\mbox{cnf}(\mbox{not\_p21}_{33},\mbox{negated\_conjecture})$

$p_{20}(f_{16}(f_{17}(c_{28},c_{27})),f_{10}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{28}),f_{11}(f_{12}(f_{13}(f_{14}(c_{29}))))))\qquad\mbox{cnf}(\mbox{p20}_{34},\mbox{negated\_conjecture})$

$\neg\,p_{20}(f_{16}(f_{17}(c_{22},c_{27})),f_{10}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{22}),f_{11}(f_{12}(f_{13}(f_{14}(c_{29}))))))\qquad\mbox{cnf}(\mbox{not\_p20}_{35},\mbox{negated\_conjecture})$

$p_{20}(f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{28}),f_{18}(f_{11}(f_{12}(f_{13}(f_{14}(c_{29})))),f_{15}(f_7(c_{23},f_8(c_{24},c_{25})),c_{22})))\qquad\mbox{cnf}(\mbox{p20}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{SYN356$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cB}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$( \mbox{cR}@ \mbox{cA}@ \mbox{cB}\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ x@ y ) \ \Rightarrow \ ( \mbox{cR}@ y@ x\mbox{ and }\mbox{cQ}@ x@ y ) )\mbox{ and }\forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ u@ v ) \ \Rightarrow \ ( \mbox{cQ}@ u@ u ) ) ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{cA}@ \mbox{cA}\mbox{ and }\mbox{cQ}@ \mbox{cB}@ \mbox{cB} )\qquad\mbox{thf}(\mbox{cLX}_{2107},\mbox{conjecture})$

\medskip\textbf{SYN579-1.p} Harrison problem 3330






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_7(x_{21},x_{21})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_3(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_7,x_7)\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_9(c_{12},c_{13})\qquad\mbox{cnf}(\mbox{p9}_5,\mbox{negated\_conjecture})$

$p_{11}(c_{14},c_{15})\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_2(c_{16},f_5(c_{14}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_3(f_4(f_5(x_{17})),f_4(x_{17}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$p_3(f_6(f_5(x_{20})),f_6(x_{20}))\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$p_9(x_{32},f_6(c_{16})) \ \Rightarrow \ p_9(x_{32},c_{13})\qquad\mbox{cnf}(\mbox{p9}_{10},\mbox{negated\_conjecture})$

$p_9(x_{32},c_{13}) \ \Rightarrow \ p_9(x_{32},f_6(c_{16}))\qquad\mbox{cnf}(\mbox{p9}_{11},\mbox{negated\_conjecture})$

$p_2(x_{10},x_{11}) \ \Rightarrow \ p_2(f_5(x_{10}),f_5(x_{11}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{18},x_{19}) \ \Rightarrow \ p_3(f_6(x_{18}),f_6(x_{19}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_3(f_4(x_{15}),f_4(x_{16}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$( p_7(x_{21},x_{22})\mbox{ and }p_7(x_{21},x_{23}) ) \ \Rightarrow \ p_7(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{13})\mbox{ and }p_3(x_{12},x_{14}) ) \ \Rightarrow \ p_3(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$( p_2(x_7,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_2(x_8,x_9)\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_{11}(x_6,x_5)\mbox{ and }p_2(x_6,x_3)\mbox{ and }p_{10}(x_5,x_4) ) \ \Rightarrow \ p_{11}(x_3,x_4)\qquad\mbox{cnf}(\mbox{p11}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{31},x_{28})\mbox{ and }p_9(x_{31},x_{30})\mbox{ and }p_3(x_{30},x_{29}) ) \ \Rightarrow \ p_9(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p9}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{25},x_{27})\mbox{ and }p_3(x_{24},x_{26}) ) \ \Rightarrow \ p_7(f_8(x_{24},x_{25}),f_8(x_{26},x_{27}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_{11}(x_{34},c_{15})\mbox{ and }p_9(c_{12},x_{33}) ) \ \Rightarrow \ ( p_9(f_8(x_{33},x_{34}),x_{33})\mbox{ or }p_9(f_8(x_{33},x_{34}),f_6(x_{34})) )\qquad\mbox{cnf}(\mbox{p9}_{22},\mbox{negated\_conjecture})$

$( p_{11}(x_{34},c_{15})\mbox{ and }p_9(c_{12},x_{33})\mbox{ and }p_9(f_8(x_{33},x_{34}),x_{33}) ) \ \Rightarrow \ \neg\,p_9(f_8(x_{33},x_{34}),f_6(x_{34}))\qquad\mbox{cnf}(\mbox{not\_p11}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{SYN105-1.p} RPT63 synthetic problem 3 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_1(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN963+1.p} Syntactic from Shults






$\exists x{:}\;p(x)\iff \exists y{:}\;p(y)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN157-1.p} RPT63 synthetic problem 55 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_5(x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN290-1.p} RPT63 synthetic problem 75 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(c,x,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.005.p} Orevkov formula - size 5

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_5,z_4,z_3,z_2,z_1,z_0{:}\;( r(0,0,z_5)\mbox{ and }r(0,z_5,z_4)\mbox{ and }r(0,z_4,z_3)\mbox{ and }r(0,z_3,z_2)\mbox{ and }r(0,z_2,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN176-1.p} RPT63 synthetic problem 74 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_2(e,c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN214-1.p} RPT63 synthetic problem 112 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_5(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN179-1.p} RPT63 synthetic problem 77 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_3(d,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN221-1.p} RPT63 synthetic problem 6 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(x,c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN201-1.p} RPT63 synthetic problem 99 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_2(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN626-1.p} Harrison problem 4045






$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{negated\_conjecture})$

$p_7(x_{54},x_{54})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_6(x_{47},x_{47})\qquad\mbox{cnf}(\mbox{p6}_3,\mbox{negated\_conjecture})$

$p_3(x_{38},x_{38})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$p_2(x_{33},x_{33})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{11}(f_{13}(f_{12}(x_5)),x_5)\qquad\mbox{cnf}(\mbox{p11}_6,\mbox{negated\_conjecture})$

$p_{14}(f_4(f_5(c_{20})),f_4(c_{21}))\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_{15}(x_{14},x_{15}) \ \Rightarrow \ p_{16}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p16}_8,\mbox{negated\_conjecture})$

$p_{15}(c_{22},f_8(f_5(c_{20}),f_9(c_{23},c_{24})))\qquad\mbox{cnf}(\mbox{p15}_9,\mbox{negated\_conjecture})$

$p_7(f_{12}(f_{13}(x_{20})),x_{20}) \ \Rightarrow \ p_{16}(c_{22},x_{20})\qquad\mbox{cnf}(\mbox{p16}_{10},\mbox{negated\_conjecture})$

$p_7(x_3,x_4) \ \Rightarrow \ p_{11}(f_{13}(x_3),f_{13}(x_4))\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_{11}(x_{57},x_{58}) \ \Rightarrow \ p_7(f_{12}(x_{57}),f_{12}(x_{58}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_2(x_{45},x_{46}) \ \Rightarrow \ p_3(f_4(x_{45}),f_4(x_{46}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_2(x_{36},x_{37}) \ \Rightarrow \ p_2(f_5(x_{36}),f_5(x_{37}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_{16}(c_{22},x_{20}) \ \Rightarrow \ p_7(f_{12}(f_{13}(x_{20})),x_{20})\qquad\mbox{cnf}(\mbox{p7}_{15},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{16},\mbox{negated\_conjecture})$

$( p_7(x_{54},x_{55})\mbox{ and }p_7(x_{54},x_{56}) ) \ \Rightarrow \ p_7(x_{55},x_{56})\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$( p_6(x_{47},x_{48})\mbox{ and }p_6(x_{47},x_{49}) ) \ \Rightarrow \ p_6(x_{48},x_{49})\qquad\mbox{cnf}(\mbox{p6}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{38},x_{39})\mbox{ and }p_3(x_{38},x_{40}) ) \ \Rightarrow \ p_3(x_{39},x_{40})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{33},x_{34})\mbox{ and }p_2(x_{33},x_{35}) ) \ \Rightarrow \ p_2(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{16}(x_{14},x_{15})\mbox{ and }p_{19}(x_{14},x_{15}) ) \ \Rightarrow \ p_{15}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$( p_{16}(x_{14},x_{15})\mbox{ and }p_{18}(x_{14},x_{15}) ) \ \Rightarrow \ p_{15}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p15}_{22},\mbox{negated\_conjecture})$

$( p_{16}(x_{14},x_{15})\mbox{ and }p_{17}(x_{14},x_{15}) ) \ \Rightarrow \ p_{15}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p15}_{23},\mbox{negated\_conjecture})$

$\neg\,p_{15}(c_{22},f_{12}(f_{13}(f_8(f_5(c_{20}),f_9(c_{23},c_{24})))))\qquad\mbox{cnf}(\mbox{not\_p15}_{24},\mbox{negated\_conjecture})$

$p_{15}(x_{14},x_{15}) \ \Rightarrow \ ( p_{17}(x_{14},x_{15})\mbox{ or }p_{18}(x_{14},x_{15})\mbox{ or }p_{19}(x_{14},x_{15}) )\qquad\mbox{cnf}(\mbox{p17}_{25},\mbox{negated\_conjecture})$

$( p_3(x_8,x_6)\mbox{ and }p_3(x_9,x_7)\mbox{ and }p_{14}(x_8,x_9) ) \ \Rightarrow \ p_{14}(x_6,x_7)\qquad\mbox{cnf}(\mbox{p14}_{26},\mbox{negated\_conjecture})$

$( p_6(x_{31},x_{29})\mbox{ and }p_7(x_{32},x_{30})\mbox{ and }p_{19}(x_{31},x_{32}) ) \ \Rightarrow \ p_{19}(x_{29},x_{30})\qquad\mbox{cnf}(\mbox{p19}_{27},\mbox{negated\_conjecture})$

$( p_6(x_{27},x_{25})\mbox{ and }p_7(x_{28},x_{26})\mbox{ and }p_{18}(x_{27},x_{28}) ) \ \Rightarrow \ p_{18}(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p18}_{28},\mbox{negated\_conjecture})$

$( p_6(x_{23},x_{21})\mbox{ and }p_7(x_{24},x_{22})\mbox{ and }p_{17}(x_{23},x_{24}) ) \ \Rightarrow \ p_{17}(x_{21},x_{22})\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$( p_6(x_{18},x_{16})\mbox{ and }p_7(x_{19},x_{17})\mbox{ and }p_{16}(x_{18},x_{19}) ) \ \Rightarrow \ p_{16}(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p16}_{30},\mbox{negated\_conjecture})$

$( p_6(x_{12},x_{10})\mbox{ and }p_7(x_{13},x_{11})\mbox{ and }p_{15}(x_{12},x_{13}) ) \ \Rightarrow \ p_{15}(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p15}_{31},\mbox{negated\_conjecture})$

$p_3(f_{10}(c_{22},f_{12}(f_{13}(f_8(f_5(c_{20}),f_9(c_{23},c_{24}))))),f_4(c_{21}))\qquad\mbox{cnf}(\mbox{p3}_{32},\mbox{negated\_conjecture})$

$( p_2(x_{59},x_{61})\mbox{ and }p_6(x_{60},x_{62}) ) \ \Rightarrow \ p_7(f_8(x_{59},x_{60}),f_8(x_{61},x_{62}))\qquad\mbox{cnf}(\mbox{p7}_{33},\mbox{negated\_conjecture})$

$( p_6(x_{41},x_{43})\mbox{ and }p_7(x_{42},x_{44}) ) \ \Rightarrow \ p_3(f_{10}(x_{41},x_{42}),f_{10}(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p3}_{34},\mbox{negated\_conjecture})$

$( p_2(x_{51},x_{53})\mbox{ and }p_2(x_{50},x_{52}) ) \ \Rightarrow \ p_6(f_9(x_{50},x_{51}),f_9(x_{52},x_{53}))\qquad\mbox{cnf}(\mbox{p6}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{SYN116-1.p} RPT63 synthetic problem 14 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_5(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN947+1.p} Syntactic from Shults






$\forall x{:}\;\exists y{:}\;( p(x)\mbox{ and }q(y) ) \ \Rightarrow \ \exists z{:}\;\forall y{:}\;( p(y)\mbox{ or }r(z) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN077-1.p} Pelletier Problem 54

Montegue's paradox of grounded classes




$\mbox{big\_f}(x,f(y)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$x = y \ \Rightarrow \ \mbox{big\_f}(x,f(y))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( \mbox{big\_f}(x,a)\mbox{ and }\mbox{big\_f}(x,y) ) \ \Rightarrow \ \mbox{big\_f}(g(x,y),y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{big\_f}(x,a)\mbox{ and }\mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_f}(z,g(x,y)) ) \ \Rightarrow \ \neg\,\mbox{big\_f}(z,y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x,h(x))\mbox{ or }\mbox{big\_f}(x,a)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$\mbox{big\_f}(x,h(y)) \ \Rightarrow \ ( \mbox{big\_f}(i(y,x),x)\mbox{ or }\mbox{big\_f}(y,a) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\mbox{big\_f}(y,h(x)) \ \Rightarrow \ ( \mbox{big\_f}(x,a)\mbox{ or }\mbox{big\_f}(i(x,y),h(x)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN283-1.p} RPT63 synthetic problem 68 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(a,b,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN601-1.p} Harrison problem 3799






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_9(x_{52},x_{52})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_3(x_{37},x_{37})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{30},x_{30})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{12}(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{negated\_conjecture})$

$p_{17}(c_{22},c_{23})\qquad\mbox{cnf}(\mbox{p17}_6,\mbox{negated\_conjecture})$

$p_3(c_{22},c_{23})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$p_{18}(c_{22},f_8(c_{21}))\qquad\mbox{cnf}(\mbox{p18}_8,\mbox{negated\_conjecture})$

$p_{17}(f_5(f_6(f_7(c_{20}))),f_4(c_{21}))\qquad\mbox{cnf}(\mbox{p17}_9,\mbox{negated\_conjecture})$

$p_2(x_{50},x_{51}) \ \Rightarrow \ p_3(f_8(x_{50}),f_8(x_{51}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_3(x_9,x_{10}) \ \Rightarrow \ p_{10}(f_{16}(x_9),f_{16}(x_{10}))\qquad\mbox{cnf}(\mbox{p10}_{11},\mbox{negated\_conjecture})$

$p_2(x_{40},x_{41}) \ \Rightarrow \ p_3(f_{15}(x_{40}),f_{15}(x_{41}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_2(x_{42},x_{43}) \ \Rightarrow \ p_3(f_4(x_{42}),f_4(x_{43}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{44},x_{45}) \ \Rightarrow \ p_3(f_5(x_{44}),f_5(x_{45}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_3(x_{46},x_{47}) \ \Rightarrow \ p_3(f_6(x_{46}),f_6(x_{47}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_3(x_{48},x_{49}) \ \Rightarrow \ p_3(f_7(x_{48}),f_7(x_{49}))\qquad\mbox{cnf}(\mbox{p3}_{16},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{17},\mbox{negated\_conjecture})$

$( p_9(x_{52},x_{53})\mbox{ and }p_9(x_{52},x_{54}) ) \ \Rightarrow \ p_9(x_{53},x_{54})\qquad\mbox{cnf}(\mbox{p9}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{37},x_{38})\mbox{ and }p_3(x_{37},x_{39}) ) \ \Rightarrow \ p_3(x_{38},x_{39})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{30},x_{31})\mbox{ and }p_2(x_{30},x_{32}) ) \ \Rightarrow \ p_2(x_{31},x_{32})\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_{12}(x_{11},x_{12})\mbox{ and }p_{12}(x_{11},x_{13}) ) \ \Rightarrow \ p_{12}(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p12}_{21},\mbox{negated\_conjecture})$

$p_{19}(c_{21},f_{11}(f_{13}(f_{15}(c_{21}),f_{14}(f_4(c_{21}),c_{22})),c_{24},c_{25}))\qquad\mbox{cnf}(\mbox{p19}_{22},\mbox{negated\_conjecture})$

$\neg\,p_{19}(c_{21},f_{11}(f_{13}(f_{15}(c_{21}),f_{14}(f_4(c_{21}),c_{23})),c_{24},c_{25}))\qquad\mbox{cnf}(\mbox{not\_p19}_{23},\mbox{negated\_conjecture})$

$( p_{19}(x_{29},x_{28})\mbox{ and }p_2(x_{29},x_{26})\mbox{ and }p_{10}(x_{28},x_{27}) ) \ \Rightarrow \ p_{19}(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p19}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{20},x_{18})\mbox{ and }p_3(x_{21},x_{19})\mbox{ and }p_{17}(x_{20},x_{21}) ) \ \Rightarrow \ p_{17}(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p17}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{24},x_{22})\mbox{ and }p_3(x_{25},x_{23})\mbox{ and }p_{18}(x_{24},x_{25}) ) \ \Rightarrow \ p_{18}(x_{22},x_{23})\qquad\mbox{cnf}(\mbox{p18}_{26},\mbox{negated\_conjecture})$

$( p_3(x_{14},x_{16})\mbox{ and }p_3(x_{15},x_{17}) ) \ \Rightarrow \ p_{12}(f_{14}(x_{14},x_{15}),f_{14}(x_{16},x_{17}))\qquad\mbox{cnf}(\mbox{p12}_{27},\mbox{negated\_conjecture})$

$( p_{12}(x_{34},x_{36})\mbox{ and }p_3(x_{33},x_{35}) ) \ \Rightarrow \ p_2(f_{13}(x_{33},x_{34}),f_{13}(x_{35},x_{36}))\qquad\mbox{cnf}(\mbox{p2}_{28},\mbox{negated\_conjecture})$

$\neg\,p_{10}(f_{11}(f_{13}(f_{15}(c_{21}),f_{14}(f_4(c_{21}),c_{23})),c_{24},c_{25}),f_{16}(f_5(c_{20})))\qquad\mbox{cnf}(\mbox{not\_p10}_{29},\mbox{negated\_conjecture})$

$( p_2(x_3,x_6)\mbox{ and }p_9(x_4,x_7)\mbox{ and }p_{10}(x_5,x_8) ) \ \Rightarrow \ p_{10}(f_{11}(x_3,x_4,x_5),f_{11}(x_6,x_7,x_8))\qquad\mbox{cnf}(\mbox{p10}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN174-1.p} RPT63 synthetic problem 72 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(e,c,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN365$\wedge$5.p} TPS problem X2116






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx}@ ( g@ ( h@ \mbox{xy} ) )\mbox{ and }\mbox{cP}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cP}@ ( g@ \mbox{xw} )\mbox{ and }\mbox{cP}@ ( h@ \mbox{xw} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cP}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cX}_{2116},\mbox{conjecture})$

\medskip\textbf{SYN092-1.003.p} Plaisted problem sym(s(3,3))






$p_0 \ \Rightarrow \ \neg\,q_0\qquad\mbox{cnf}(\mbox{s3\_goal}_1,\mbox{negated\_conjecture})$

$( p_1\mbox{ and }p_2 ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{s3\_type11}_1,\mbox{axiom})$

$( q_1\mbox{ and }q_2 ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{s3\_type11}_2,\mbox{axiom})$

$( p_2\mbox{ and }p_3 ) \ \Rightarrow \ p_1\qquad\mbox{cnf}(\mbox{s3\_type11}_3,\mbox{axiom})$

$( q_2\mbox{ and }q_3 ) \ \Rightarrow \ p_1\qquad\mbox{cnf}(\mbox{s3\_type11}_4,\mbox{axiom})$

$( p_3\mbox{ and }p_4 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{s3\_type11}_5,\mbox{axiom})$

$( q_3\mbox{ and }q_4 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{s3\_type11}_6,\mbox{axiom})$

$( p_4\mbox{ and }p_5 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{s3\_type11}_7,\mbox{axiom})$

$( q_4\mbox{ and }q_5 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{s3\_type11}_8,\mbox{axiom})$

$( p_1\mbox{ and }q_2 ) \ \Rightarrow \ q_0\qquad\mbox{cnf}(\mbox{s3\_type12}_1,\mbox{axiom})$

$( q_1\mbox{ and }p_2 ) \ \Rightarrow \ q_0\qquad\mbox{cnf}(\mbox{s3\_type12}_2,\mbox{axiom})$

$( p_2\mbox{ and }q_3 ) \ \Rightarrow \ q_1\qquad\mbox{cnf}(\mbox{s3\_type12}_3,\mbox{axiom})$

$( q_2\mbox{ and }p_3 ) \ \Rightarrow \ q_1\qquad\mbox{cnf}(\mbox{s3\_type12}_4,\mbox{axiom})$

$( p_3\mbox{ and }q_4 ) \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{s3\_type12}_5,\mbox{axiom})$

$( q_3\mbox{ and }p_4 ) \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{s3\_type12}_6,\mbox{axiom})$

$( p_4\mbox{ and }q_5 ) \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{s3\_type12}_7,\mbox{axiom})$

$( q_4\mbox{ and }p_5 ) \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{s3\_type12}_8,\mbox{axiom})$

$p_2 \ \Rightarrow \ p_5\qquad\mbox{cnf}(\mbox{s3\_type2}_1,\mbox{axiom})$

$p_3 \ \Rightarrow \ p_6\qquad\mbox{cnf}(\mbox{s3\_type2}_2,\mbox{axiom})$

$q_2 \ \Rightarrow \ q_5\qquad\mbox{cnf}(\mbox{s3\_type2}_3,\mbox{axiom})$

$q_3 \ \Rightarrow \ q_6\qquad\mbox{cnf}(\mbox{s3\_type2}_4,\mbox{axiom})$

$\mbox{sym\_p}_0\mbox{ or }\mbox{sym\_q}_0\qquad\mbox{cnf}(\mbox{sym\_s3\_goal}_1,\mbox{axiom})$

$\mbox{sym\_p}_0 \ \Rightarrow \ ( \mbox{sym\_p}_1\mbox{ or }\mbox{sym\_p}_2 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_1,\mbox{axiom})$

$\mbox{sym\_p}_0 \ \Rightarrow \ ( \mbox{sym\_q}_1\mbox{ or }\mbox{sym\_q}_2 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_2,\mbox{axiom})$

$\mbox{sym\_p}_1 \ \Rightarrow \ ( \mbox{sym\_p}_2\mbox{ or }\mbox{sym\_p}_3 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_3,\mbox{axiom})$

$\mbox{sym\_p}_1 \ \Rightarrow \ ( \mbox{sym\_q}_2\mbox{ or }\mbox{sym\_q}_3 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_4,\mbox{axiom})$

$\mbox{sym\_p}_2 \ \Rightarrow \ ( \mbox{sym\_p}_3\mbox{ or }\mbox{sym\_p}_4 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_5,\mbox{axiom})$

$\mbox{sym\_p}_2 \ \Rightarrow \ ( \mbox{sym\_q}_3\mbox{ or }\mbox{sym\_q}_4 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_6,\mbox{axiom})$

$\mbox{sym\_p}_3 \ \Rightarrow \ ( \mbox{sym\_p}_4\mbox{ or }\mbox{sym\_p}_5 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_7,\mbox{axiom})$

$\mbox{sym\_p}_3 \ \Rightarrow \ ( \mbox{sym\_q}_4\mbox{ or }\mbox{sym\_q}_5 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type11}_8,\mbox{axiom})$

$\mbox{sym\_q}_0 \ \Rightarrow \ ( \mbox{sym\_p}_1\mbox{ or }\mbox{sym\_q}_2 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_1,\mbox{axiom})$

$\mbox{sym\_q}_0 \ \Rightarrow \ ( \mbox{sym\_q}_1\mbox{ or }\mbox{sym\_p}_2 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_2,\mbox{axiom})$

$\mbox{sym\_q}_1 \ \Rightarrow \ ( \mbox{sym\_p}_2\mbox{ or }\mbox{sym\_q}_3 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_3,\mbox{axiom})$

$\mbox{sym\_q}_1 \ \Rightarrow \ ( \mbox{sym\_q}_2\mbox{ or }\mbox{sym\_p}_3 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_4,\mbox{axiom})$

$\mbox{sym\_q}_2 \ \Rightarrow \ ( \mbox{sym\_p}_3\mbox{ or }\mbox{sym\_q}_4 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_5,\mbox{axiom})$

$\mbox{sym\_q}_2 \ \Rightarrow \ ( \mbox{sym\_q}_3\mbox{ or }\mbox{sym\_p}_4 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_6,\mbox{axiom})$

$\mbox{sym\_q}_3 \ \Rightarrow \ ( \mbox{sym\_p}_4\mbox{ or }\mbox{sym\_q}_5 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_7,\mbox{axiom})$

$\mbox{sym\_q}_3 \ \Rightarrow \ ( \mbox{sym\_q}_4\mbox{ or }\mbox{sym\_p}_5 )\qquad\mbox{cnf}(\mbox{sym\_s3\_type12}_8,\mbox{axiom})$

$\mbox{sym\_p}_5 \ \Rightarrow \ \mbox{sym\_p}_2\qquad\mbox{cnf}(\mbox{sym\_s3\_type2}_1,\mbox{axiom})$

$\mbox{sym\_p}_6 \ \Rightarrow \ \mbox{sym\_p}_3\qquad\mbox{cnf}(\mbox{sym\_s3\_type2}_2,\mbox{axiom})$

$\mbox{sym\_q}_5 \ \Rightarrow \ \mbox{sym\_q}_2\qquad\mbox{cnf}(\mbox{sym\_s3\_type2}_3,\mbox{axiom})$

$\mbox{sym\_q}_6 \ \Rightarrow \ \mbox{sym\_q}_3\qquad\mbox{cnf}(\mbox{sym\_s3\_type2}_4,\mbox{axiom})$

\medskip\textbf{SYN263-1.p} RPT63 synthetic problem 48 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_1(d,d,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN344+1.p} Church problem 46.16 (3)






$\forall x{:}\;\exists y_1,y_2{:}\;\forall z{:}\;( ( ( \mbox{big\_f}(x,z) \ \Rightarrow \ ( \mbox{big\_f}(y_1,z) \ \Rightarrow \ \mbox{big\_f}(y_2,x) ) ) \ \Rightarrow \ \mbox{big\_f}(x,x) ) \ \Rightarrow \ ( \mbox{big\_f}(x,x)\mbox{ and }( \mbox{big\_f}(y_1,y_2) \ \Rightarrow \ \mbox{big\_f}(z,z) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_16}_3,\mbox{conjecture})$

\medskip\textbf{SYN318-1.p} Church problem 46.2 (4)






$( f(x)\mbox{ and }f(b) ) \ \Rightarrow \ g(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$p(a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,g(b)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN732$\wedge$5.p} TPS problem X3411






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xu}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xv}@ \mbox{xz} ) \ \Rightarrow \ \exists \mbox{xw}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ \mbox{xw}\mbox{ or }\mbox{cQ}@ \mbox{xw}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cX}_{3411},\mbox{conjecture})$

\medskip\textbf{SYN124-1.p} RPT63 synthetic problem 22 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN049+1.p} Pelletier Problem 19






$\exists x{:}\;\forall y,z{:}\;( ( \mbox{big\_p}(y) \ \Rightarrow \ \mbox{big\_q}(z) ) \ \Rightarrow \ ( \mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_q}(x) ) )\qquad\mbox{fof}(\mbox{pel}_{19},\mbox{conjecture})$

\medskip\textbf{SYN317+1.p} Church problem 46.2 (3)






$\exists x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(x) )\iff \exists x,y{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_g}(y) )\qquad\mbox{fof}(\mbox{church\_46\_2}_3,\mbox{conjecture})$

\medskip\textbf{SYN015-1.p} A problem in quantification theory






$m\ne n\qquad\mbox{cnf}(c_1,\mbox{negated\_conjecture})$

$y = k \ \Rightarrow \ ( y = j\mbox{ or }\mbox{element}(y,j) )\qquad\mbox{cnf}(c_2,\mbox{negated\_conjecture})$

$\mbox{element}(y,j) \ \Rightarrow \ ( y = j\mbox{ or }y = k )\qquad\mbox{cnf}(c_3,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = m ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_4,\mbox{negated\_conjecture})$

$( \mbox{element}(y,m)\mbox{ and }f(y) = y ) \ \Rightarrow \ y = m\qquad\mbox{cnf}(c_5,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,f(y)) )\qquad\mbox{cnf}(c_6,\mbox{negated\_conjecture})$

$\mbox{element}(y,m) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(f(y),y) )\qquad\mbox{cnf}(c_7,\mbox{negated\_conjecture})$

$( \mbox{element}(y,v_1)\mbox{ and }\mbox{element}(v_1,y) ) \ \Rightarrow \ ( y = m\mbox{ or }\mbox{element}(y,m)\mbox{ or }v_1 = m\mbox{ or }v_1 = y )\qquad\mbox{cnf}(c_8,\mbox{negated\_conjecture})$

$g(y) = n \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_9,\mbox{negated\_conjecture})$

$g(y) = y \ \Rightarrow \ ( y = n\mbox{ or }\mbox{element}(y,n) )\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(y,g(y))\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$y = n\mbox{ or }\mbox{element}(y,n)\mbox{ or }\mbox{element}(g(y),y)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( \mbox{element}(y,n)\mbox{ and }\mbox{element}(y,v)\mbox{ and }\mbox{element}(v,y) ) \ \Rightarrow \ ( y = n\mbox{ or }v = n\mbox{ or }v = y )\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$y = m \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$y = n \ \Rightarrow \ ( y = k\mbox{ or }\mbox{element}(y,k) )\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$\mbox{element}(y,k) \ \Rightarrow \ ( y = k\mbox{ or }y = m\mbox{ or }y = n )\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{SYN056$\wedge$5.p} TPS problem PELL26






$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) )\mbox{ and }( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}\mbox{ and }\mbox{cQ}@ \mbox{xy} ) \ \Rightarrow \ ( ( ( \mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xy} ) )\mbox{ and }( ( \mbox{cS}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{26},\mbox{conjecture})$

\medskip\textbf{SYN581-1.p} Harrison problem 3466






$p_7(x_{34},x_{34})\qquad\mbox{cnf}(\mbox{p7}_1,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{31},x_{31})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_{12}(c_{13},c_{18})\qquad\mbox{cnf}(\mbox{p12}_4,\mbox{negated\_conjecture})$

$p_2(c_{18},c_{14})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(c_{18}),f_8(f_9(c_{17})))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_7,\mbox{negated\_conjecture})$

$p_{11}(f_4(f_6(c_{15},c_{14})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_8,\mbox{negated\_conjecture})$

$\neg\,p_{11}(f_4(f_6(c_{15},c_{18})),c_{16})\qquad\mbox{cnf}(\mbox{not\_p11}_9,\mbox{negated\_conjecture})$

$p_{12}(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{10},\mbox{negated\_conjecture})$

$p_2(x_{11},x_{12}) \ \Rightarrow \ p_{11}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p11}_{11},\mbox{negated\_conjecture})$

$p_7(x_{37},x_{38}) \ \Rightarrow \ p_7(f_9(x_{37}),f_9(x_{38}))\qquad\mbox{cnf}(\mbox{p7}_{12},\mbox{negated\_conjecture})$

$p_2(x_{21},x_{22}) \ \Rightarrow \ p_2(f_3(x_{21}),f_3(x_{22}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$p_2(x_{23},x_{24}) \ \Rightarrow \ p_2(f_4(x_{23}),f_4(x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_7(x_{29},x_{30}) \ \Rightarrow \ p_2(f_8(x_{29}),f_8(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{11}(x_{11},x_{12}) \ \Rightarrow \ ( p_{12}(x_{11},x_{12})\mbox{ or }p_2(x_{11},x_{12}) )\qquad\mbox{cnf}(\mbox{p12}_{16},\mbox{negated\_conjecture})$

$( p_7(x_{34},x_{35})\mbox{ and }p_7(x_{34},x_{36}) ) \ \Rightarrow \ p_7(x_{35},x_{36})\qquad\mbox{cnf}(\mbox{p7}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_5(x_{31},x_{32})\mbox{ and }p_5(x_{31},x_{33}) ) \ \Rightarrow \ p_5(x_{32},x_{33})\qquad\mbox{cnf}(\mbox{p5}_{19},\mbox{negated\_conjecture})$

$( p_{11}(x_6,c_{14})\mbox{ and }p_{11}(c_{13},x_6) ) \ \Rightarrow \ p_{10}(c_{15},f_3(x_6),x_6)\qquad\mbox{cnf}(\mbox{p10}_{20},\mbox{negated\_conjecture})$

$( p_2(x_9,x_7)\mbox{ and }p_2(x_{10},x_8)\mbox{ and }p_{11}(x_9,x_{10}) ) \ \Rightarrow \ p_{11}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p11}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{16},x_{14})\mbox{ and }p_2(x_{17},x_{15})\mbox{ and }p_{12}(x_{16},x_{17}) ) \ \Rightarrow \ p_{12}(x_{14},x_{15})\qquad\mbox{cnf}(\mbox{p12}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{26},x_{28})\mbox{ and }p_5(x_{25},x_{27}) ) \ \Rightarrow \ p_2(f_6(x_{25},x_{26}),f_6(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$( p_5(x_3,x_0)\mbox{ and }p_2(x_5,x_2)\mbox{ and }p_2(x_4,x_1)\mbox{ and }p_{10}(x_3,x_4,x_5) ) \ \Rightarrow \ p_{10}(x_0,x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{24},\mbox{negated\_conjecture})$

$( p_{12}(x_{13},c_{14})\mbox{ and }p_{12}(c_{13},x_{13})\mbox{ and }p_2(f_3(x_{13}),f_8(f_9(c_{17}))) ) \ \Rightarrow \ p_{11}(f_4(f_6(c_{15},x_{13})),c_{16})\qquad\mbox{cnf}(\mbox{p11}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.002.p} Orevkov formula - size 2

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_2,z_1,z_0{:}\;( r(0,0,z_2)\mbox{ and }r(0,z_2,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN392$\wedge$4.p} Pelletier 14






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iatom}@ p_1 ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{14},\mbox{conjecture})$

\medskip\textbf{SYN372+1.p} Peter Andrews Problem X2123






$\forall x{:}\;( \exists y{:}\;( \mbox{big\_p}(y) \ \Rightarrow \ \mbox{big\_q}(x) ) \ \Rightarrow \ \exists y{:}\;( \mbox{big\_p}(y) \ \Rightarrow \ \mbox{big\_q}(y) ) )\qquad\mbox{fof}(x_{2123},\mbox{conjecture})$

\medskip\textbf{SYN727+1.p} Peter Andrews Problem THM68






$( \forall x{:}\;\mbox{likes}(x,\mbox{bruce})\mbox{ and }\forall y{:}\;( \exists z{:}\;\mbox{likes}(y,z) \ \Rightarrow \ \mbox{likes}(\mbox{lyle},y) ) ) \ \Rightarrow \ \exists u{:}\;\forall v{:}\;\mbox{likes}(u,v)\qquad\mbox{fof}(\mbox{thm}_{68},\mbox{conjecture})$

\medskip\textbf{SYN353+1.p} Church problem 46.18 (5)






$\forall x{:}\;\exists y_1,y_2,y_3{:}\;\forall z{:}\;( ( \mbox{big\_f}(y_1,y_2,y_3) \ \Rightarrow \ ( \mbox{big\_f}(x,x,z) \ \Rightarrow \ ( \mbox{big\_f}(y_2,y_3,y_1)\mbox{ or }\mbox{big\_f}(y_3,y_1,y_2) ) ) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(y_3,y_1,y_2) \ \Rightarrow \ ( \mbox{big\_f}(y_1,y_2,y_3)\mbox{ and }\mbox{big\_f}(y_2,y_3,y_1) ) )\iff \mbox{big\_f}(y_2,y_1,z) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(y_2,y_3,y_1) \ \Rightarrow \ ( \mbox{big\_f}(y_1,y_2,y_3)\mbox{ and }\mbox{big\_f}(y_3,y_1,y_2) ) )\iff \mbox{big\_f}(y_1,z,y_2) ) \ \Rightarrow \ ( ( ( ( \mbox{big\_f}(y_3,y_1,y_2) \ \Rightarrow \ \neg\,\mbox{big\_f}(y_2,y_3,y_1) ) \ \Rightarrow \ \mbox{big\_f}(y_1,y_2,y_3) )\iff \mbox{big\_f}(z,y_2,y_1) ) \ \Rightarrow \ ( ( \mbox{big\_f}(y_1,y_2,y_3)\mbox{ and }\mbox{big\_f}(y_2,y_3,y_1)\mbox{ and }\mbox{big\_f}(y_3,y_1,y_2) )\iff \mbox{big\_f}(z,z,z) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_18}_5,\mbox{conjecture})$

\medskip\textbf{SYN609-1.p} Harrison problem 3829






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_7,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{27},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{28},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN072+1.p} Pelletier Problem 49






$\exists x,y{:}\;\forall z{:}\;( z = x\mbox{ or }z = y )\qquad\mbox{fof}(\mbox{pel49}_1,\mbox{axiom})$

$\mbox{big\_p}(a)\qquad\mbox{fof}(\mbox{pel49}_2,\mbox{axiom})$

$\mbox{big\_p}(b)\qquad\mbox{fof}(\mbox{pel49}_3,\mbox{axiom})$

$a\ne b\qquad\mbox{fof}(\mbox{pel49}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{big\_p}(x)\qquad\mbox{fof}(\mbox{pel}_{49},\mbox{conjecture})$

\medskip\textbf{SYN238-1.p} RPT63 synthetic problem 23 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(x,b,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN082+1.p} Pelletier Problem 60






$\forall x{:}\;( \mbox{big\_f}(x,f(x))\iff \exists y{:}\;( \forall z{:}\;( \mbox{big\_f}(z,y) \ \Rightarrow \ \mbox{big\_f}(z,f(x)) )\mbox{ and }\mbox{big\_f}(x,y) ) )\qquad\mbox{fof}(\mbox{pel}_{60},\mbox{conjecture})$

\medskip\textbf{SYN007$\wedge$4.014.p} Pelletier Problem 71






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_1\_type},\mbox{type})$

$p_{10}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_10\_type},\mbox{type})$

$p_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_11\_type},\mbox{type})$

$p_{12}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_12\_type},\mbox{type})$

$p_{13}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_13\_type},\mbox{type})$

$p_{14}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_14\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_5\_type},\mbox{type})$

$p_6{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_6\_type},\mbox{type})$

$p_7{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_7\_type},\mbox{type})$

$p_8{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_8\_type},\mbox{type})$

$p_9{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_9\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_7 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_8 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_9 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{10} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{11} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{12} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{13} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{14} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_7 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_8 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_9 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{10} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{11} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{12} )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_{13} )@ ( \mbox{iatom}@ p_{14} ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN919+1.p} From Smullyan






$( \forall x,y{:}\;( r(x,y) \ \Rightarrow \ r(y,x) )\mbox{ and }\forall x,y,z{:}\;( ( r(x,y)\mbox{ and }r(y,z) ) \ \Rightarrow \ r(x,z) ) ) \ \Rightarrow \ \forall x,y{:}\;( r(x,y) \ \Rightarrow \ r(x,x) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN247-1.p} RPT63 synthetic problem 32 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_1(d,x,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN003-1.006.p} Implications that form a contradiction

P1 \& Q1 $\to$ P2     P1 \& R1 $\to$ P2     Q $\to$ Q1     R $\to$ R1   :
 P2 \& Q2 $\to$ P3     P2 \& R2 $\to$ P3     Q $\to$ Q2     R $\to$ R2   :
 ......           ......        ....      ....           :
 Pk-1 \& Qk-1 $\to$Pk  Pk-1 \& Rk-1 $\to$ Pk Q $\to$ Qk-1   R $\to$ Rk-1 :
 P1                ~Pk               Q           R         :
 : The size is k, in the above.




$( p_1\mbox{ and }q_1 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{pqp}_1,\mbox{negated\_conjecture})$

$( p_2\mbox{ and }q_2 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{pqp}_2,\mbox{negated\_conjecture})$

$( p_3\mbox{ and }q_3 ) \ \Rightarrow \ p_4\qquad\mbox{cnf}(\mbox{pqp}_3,\mbox{negated\_conjecture})$

$( p_4\mbox{ and }q_4 ) \ \Rightarrow \ p_5\qquad\mbox{cnf}(\mbox{pqp}_4,\mbox{negated\_conjecture})$

$( p_5\mbox{ and }q_5 ) \ \Rightarrow \ p_6\qquad\mbox{cnf}(\mbox{pqp}_5,\mbox{negated\_conjecture})$

$( p_1\mbox{ and }r_1 ) \ \Rightarrow \ p_2\qquad\mbox{cnf}(\mbox{prp}_1,\mbox{negated\_conjecture})$

$( p_2\mbox{ and }r_2 ) \ \Rightarrow \ p_3\qquad\mbox{cnf}(\mbox{prp}_2,\mbox{negated\_conjecture})$

$( p_3\mbox{ and }r_3 ) \ \Rightarrow \ p_4\qquad\mbox{cnf}(\mbox{prp}_3,\mbox{negated\_conjecture})$

$( p_4\mbox{ and }r_4 ) \ \Rightarrow \ p_5\qquad\mbox{cnf}(\mbox{prp}_4,\mbox{negated\_conjecture})$

$( p_5\mbox{ and }r_5 ) \ \Rightarrow \ p_6\qquad\mbox{cnf}(\mbox{prp}_5,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ q_1\qquad\mbox{cnf}(\mbox{qq}_1,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ q_2\qquad\mbox{cnf}(\mbox{qq}_2,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ q_3\qquad\mbox{cnf}(\mbox{qq}_3,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ q_4\qquad\mbox{cnf}(\mbox{qq}_4,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ q_5\qquad\mbox{cnf}(\mbox{qq}_5,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ r_1\qquad\mbox{cnf}(\mbox{rr}_1,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ r_2\qquad\mbox{cnf}(\mbox{rr}_2,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ r_3\qquad\mbox{cnf}(\mbox{rr}_3,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ r_4\qquad\mbox{cnf}(\mbox{rr}_4,\mbox{negated\_conjecture})$

$r \ \Rightarrow \ r_5\qquad\mbox{cnf}(\mbox{rr}_5,\mbox{negated\_conjecture})$

$p_1\qquad\mbox{cnf}(\mbox{base}_1,\mbox{negated\_conjecture})$

$\neg\,p_6\qquad\mbox{cnf}(\mbox{base}_2,\mbox{negated\_conjecture})$

$q\qquad\mbox{cnf}(\mbox{base}_3,\mbox{negated\_conjecture})$

$r\qquad\mbox{cnf}(\mbox{base}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN057$\wedge$5.p} TPS problem PELL27






$\mbox{cI}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cI},\mbox{type})$

$\mbox{cJ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$\mbox{cH}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cH},\mbox{type})$

$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}\mbox{ and }\neg\,\mbox{cG}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cH}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cJ}@ \mbox{xx}\mbox{ and }\mbox{cI}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cF}@ \mbox{xx} ) )\mbox{ and }( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cH}@ \mbox{xx}\mbox{ and }\neg\,\mbox{cG}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cI}@ \mbox{xx} ) \ \Rightarrow \ \neg\,\mbox{cH}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cJ}@ \mbox{xx} ) \ \Rightarrow \ \neg\,\mbox{cI}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPELL}_{27},\mbox{conjecture})$

\medskip\textbf{SYN960+1.p} Syntactic from Shults






$\exists x,y{:}\;a(x,y)\iff \exists y,x{:}\;a(x,y)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN381+1.p} Peter Andrews Problem X2133






$( \forall x{:}\;( \exists y{:}\;\mbox{big\_q}(x,y) \ \Rightarrow \ \mbox{big\_p}(x) )\mbox{ and }\forall v{:}\;\exists u{:}\;\mbox{big\_q}(u,v)\mbox{ and }\forall w,z{:}\;( \mbox{big\_q}(w,z) \ \Rightarrow \ ( \mbox{big\_q}(z,w)\mbox{ or }\mbox{big\_q}(z,z) ) ) ) \ \Rightarrow \ \forall z{:}\;\mbox{big\_p}(z)\qquad\mbox{fof}(x_{2133},\mbox{conjecture})$

\medskip\textbf{SYN049-1.p} Pelletier Problem 19






$\mbox{big\_p}(f(x)) \ \Rightarrow \ \mbox{big\_q}(g(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{big\_p}(x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_q}(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SYN986+1.004.p} Orevkov formula - size 4

hyp(2,k,1) is defined. Each Ck has a non-normal natural deduction
 of size linear in k, and each normal deduction of Ck has at least
 hyp(2,k,1)=2\_k nodes.




$\mbox{include}(\mbox{'Axioms/SYN002+0.ax'})$

$\exists z_4,z_3,z_2,z_1,z_0{:}\;( r(0,0,z_4)\mbox{ and }r(0,z_4,z_3)\mbox{ and }r(0,z_3,z_2)\mbox{ and }r(0,z_2,z_1)\mbox{ and }r(0,z_1,z_0) )\qquad\mbox{fof}(\mbox{ck},\mbox{conjecture})$

\medskip\textbf{SYN002-1.007.008.p} Odd and Even Problem

Given by the clauses C1: p(X) $\vee$ p(f$\wedge$M(X)) and C2: ~p(X)
 $\vee$ ~p(f$\wedge$N(X)), where if M is odd N is even and vice versa,
 N $>$ M. The sizes are used for N and M.




$p(x)\mbox{ or }p(f(f(f(f(f(f(f(x))))))))\qquad\mbox{cnf}(\mbox{positive},\mbox{negated\_conjecture})$

$p(x) \ \Rightarrow \ \neg\,p(f(f(f(f(f(f(f(f(x)))))))))\qquad\mbox{cnf}(\mbox{negative},\mbox{negated\_conjecture})$

\medskip\textbf{SYN059+1.p} Pelletier Problem 29






$\exists x{:}\;\mbox{big\_f}(x)\qquad\mbox{fof}(\mbox{pel29}_1,\mbox{axiom})$

$\exists y{:}\;\mbox{big\_g}(y)\qquad\mbox{fof}(\mbox{pel29}_2,\mbox{axiom})$

$( \forall x{:}\;( \mbox{big\_f}(x) \ \Rightarrow \ \mbox{big\_h}(x) )\mbox{ and }\forall u{:}\;( \mbox{big\_g}(u) \ \Rightarrow \ \mbox{big\_j}(u) ) )\iff \forall w,y{:}\;( ( \mbox{big\_f}(w)\mbox{ and }\mbox{big\_g}(y) ) \ \Rightarrow \ ( \mbox{big\_h}(w)\mbox{ and }\mbox{big\_j}(y) ) )\qquad\mbox{fof}(\mbox{pel}_{29},\mbox{conjecture})$

\medskip\textbf{SYN768-1.p} PSAT inverse problem - K=4 C=20 V=4 D=1.6






$\mbox{ssRr}(\mbox{skf}_3(u),u)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{ssRr}(u,\mbox{skf}_2(u))\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ ( \mbox{ssPv}_1(v)\mbox{ or }\mbox{ssPv}_3(u) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssPv}_1(v) ) \ \Rightarrow \ ( \mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_3(v) )\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssPv}_4(u) ) \ \Rightarrow \ \mbox{ssPv}_1(u)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w) ) \ \Rightarrow \ ( \mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_1(v)\mbox{ or }\mbox{ssPv}_3(v) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_4(v) ) \ \Rightarrow \ ( \mbox{ssPv}_3(u)\mbox{ or }\mbox{ssPv}_1(w)\mbox{ or }\mbox{ssPv}_3(v) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_4(w) ) \ \Rightarrow \ ( \mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_4(v) )\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssRr}(v,w) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_2(v)\mbox{ or }\mbox{ssPv}_4(v) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssPv}_3(u) ) \ \Rightarrow \ ( \mbox{ssPv}_3(v)\mbox{ or }\mbox{ssPv}_2(w) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(u)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_1(v) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_4(v) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_4(v) ) \ \Rightarrow \ ( \mbox{ssPv}_3(w)\mbox{ or }\mbox{ssPv}_3(v) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_3(v) ) \ \Rightarrow \ ( \mbox{ssPv}_1(w)\mbox{ or }\mbox{ssPv}_2(v) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_2(w)\mbox{ and }\mbox{ssPv}_3(v) ) \ \Rightarrow \ \mbox{ssPv}_2(v)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_2(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssPv}_2(v) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_4(v)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_1(v)\mbox{ and }\mbox{ssRr}(u,w)\mbox{ and }\mbox{ssRr}(u,x) ) \ \Rightarrow \ ( \mbox{ssPv}_4(w)\mbox{ or }\mbox{ssPv}_1(x)\mbox{ or }\mbox{ssPv}_4(u) )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_4(w)\mbox{ and }\mbox{ssRr}(v,x)\mbox{ and }\mbox{ssPv}_1(x) ) \ \Rightarrow \ ( \mbox{ssPv}_2(u)\mbox{ or }\mbox{ssPv}_4(v) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssRr}(v,x)\mbox{ and }\mbox{ssPv}_2(v) ) \ \Rightarrow \ ( \mbox{ssPv}_2(w)\mbox{ or }\mbox{ssPv}_1(x) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_3(u)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(v,x) ) \ \Rightarrow \ ( \mbox{ssPv}_4(x)\mbox{ or }\mbox{ssPv}_2(v) )\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(u)\mbox{ and }\mbox{ssRr}(v,w)\mbox{ and }\mbox{ssPv}_3(w)\mbox{ and }\mbox{ssRr}(v,x)\mbox{ and }\mbox{ssPv}_1(x) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_1(v)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$( \mbox{ssRr}(u,v)\mbox{ and }\mbox{ssPv}_4(u)\mbox{ and }\mbox{ssRr}(w,v)\mbox{ and }\mbox{ssPv}_1(w)\mbox{ and }\mbox{ssRr}(v,x)\mbox{ and }\mbox{ssPv}_4(x) ) \ \Rightarrow \ \neg\,\mbox{ssPv}_4(v)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{SYN058-1.p} Pelletier Problem 28






$\mbox{big\_p}(x) \ \Rightarrow \ \mbox{big\_q}(y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{big\_q}(b) \ \Rightarrow \ \mbox{big\_q}(c)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{big\_q}(b) \ \Rightarrow \ \mbox{big\_s}(c)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{big\_r}(b) \ \Rightarrow \ \mbox{big\_q}(c)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{big\_r}(b) \ \Rightarrow \ \mbox{big\_s}(c)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$( \mbox{big\_s}(y)\mbox{ and }\mbox{big\_f}(x) ) \ \Rightarrow \ \mbox{big\_g}(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{big\_p}(d)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$\mbox{big\_f}(d)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$\neg\,\mbox{big\_g}(d)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

\medskip\textbf{SYN972+1.p} Syntactic from Shults






$\exists x{:}\;\forall y{:}\;p(x,y) \ \Rightarrow \ \forall y{:}\;\exists x{:}\;p(x,y)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN921+1.p} From Smullyan






$\exists x{:}\;\forall y{:}\;( ( p(y) \ \Rightarrow \ q(x) ) \ \Rightarrow \ ( p(x) \ \Rightarrow \ q(x) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN146-1.p} RPT63 synthetic problem 44 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_1(x,c,y)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN033-1.p} DM






$p(g(a,b),a,b)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$p(a,h(a,b),b)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( p(a,b,c)\mbox{ and }p(d,e,b)\mbox{ and }p(a,d,f) ) \ \Rightarrow \ p(f,e,c)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\neg\,p(k(a),a,k(a))\qquad\mbox{cnf}(\mbox{prove\_something},\mbox{negated\_conjecture})$

\medskip\textbf{SYN319+1.p} Church problem 46.2 (5)






$\exists x,y{:}\;\forall z_1,z_2{:}\;( ( ( \mbox{big\_f}(y) \ \Rightarrow \ \mbox{big\_g}(z_1) ) \ \Rightarrow \ ( \mbox{big\_g}(x)\mbox{ and }\neg\,\mbox{big\_f}(z_1) ) ) \ \Rightarrow \ ( ( ( \mbox{big\_f}(x)\mbox{ or }\mbox{big\_g}(x) ) \ \Rightarrow \ \mbox{big\_h}(x) ) \ \Rightarrow \ ( \mbox{big\_h}(z_2)\mbox{ and }( \mbox{big\_h}(y) \ \Rightarrow \ ( ( \mbox{big\_f}(z_2)\mbox{ or }\mbox{big\_g}(z_2) ) \ \Rightarrow \ \mbox{big\_h}(z_2) ) ) ) ) )\qquad\mbox{fof}(\mbox{church\_46\_2}_5,\mbox{conjecture})$

\medskip\textbf{SYN088-1.010.p} Plaisted problem s(4,10)






$\neg\,p(a,a,a,a,a,a,a,a,a,a)\qquad\mbox{cnf}(\mbox{s4\_goal}_1,\mbox{negated\_conjecture})$

$( q_1(x_1)\mbox{ and }q_2(x_2)\mbox{ and }q_3(x_3)\mbox{ and }q_4(x_4)\mbox{ and }q_5(x_5)\mbox{ and }q_6(x_6)\mbox{ and }q_7(x_7)\mbox{ and }q_8(x_8)\mbox{ and }q_9(x_9)\mbox{ and }q_{10}(x_{10}) ) \ \Rightarrow \ p(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10})\qquad\mbox{cnf}(\mbox{s4}_1,\mbox{axiom})$

$q_1(a)\qquad\mbox{cnf}(\mbox{s4}_2,\mbox{axiom})$

$q_1(b)\qquad\mbox{cnf}(\mbox{s4}_3,\mbox{axiom})$

$q_2(a)\qquad\mbox{cnf}(\mbox{s4}_4,\mbox{axiom})$

$q_2(b)\qquad\mbox{cnf}(\mbox{s4}_5,\mbox{axiom})$

$q_3(a)\qquad\mbox{cnf}(\mbox{s4}_6,\mbox{axiom})$

$q_3(b)\qquad\mbox{cnf}(\mbox{s4}_7,\mbox{axiom})$

$q_4(a)\qquad\mbox{cnf}(\mbox{s4}_8,\mbox{axiom})$

$q_4(b)\qquad\mbox{cnf}(\mbox{s4}_9,\mbox{axiom})$

$q_5(a)\qquad\mbox{cnf}(\mbox{s4}_{10},\mbox{axiom})$

$q_5(b)\qquad\mbox{cnf}(\mbox{s4}_{11},\mbox{axiom})$

$q_6(a)\qquad\mbox{cnf}(\mbox{s4}_{12},\mbox{axiom})$

$q_6(b)\qquad\mbox{cnf}(\mbox{s4}_{13},\mbox{axiom})$

$q_7(a)\qquad\mbox{cnf}(\mbox{s4}_{14},\mbox{axiom})$

$q_7(b)\qquad\mbox{cnf}(\mbox{s4}_{15},\mbox{axiom})$

$q_8(a)\qquad\mbox{cnf}(\mbox{s4}_{16},\mbox{axiom})$

$q_8(b)\qquad\mbox{cnf}(\mbox{s4}_{17},\mbox{axiom})$

$q_9(a)\qquad\mbox{cnf}(\mbox{s4}_{18},\mbox{axiom})$

$q_9(b)\qquad\mbox{cnf}(\mbox{s4}_{19},\mbox{axiom})$

$q_{10}(a)\qquad\mbox{cnf}(\mbox{s4}_{20},\mbox{axiom})$

$q_{10}(b)\qquad\mbox{cnf}(\mbox{s4}_{21},\mbox{axiom})$

\medskip\textbf{SYN103-1.p} RPT63 synthetic problem 1 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,k_0(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN001$\wedge$4.003.p} ILTP Problem SYJ212+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{iequiv}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ a_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_3 )@ ( \mbox{iequiv}@ ( \mbox{iatom}@ a_2 )@ ( \mbox{iatom}@ a_1 ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYN343+1.p} Church problem 46.16 (2)






$\exists x,y_1,y_2{:}\;\forall z{:}\;( ( ( \mbox{big\_f}(x,y_1) \ \Rightarrow \ \mbox{big\_f}(z,x) ) \ \Rightarrow \ \mbox{big\_f}(x,x) ) \ \Rightarrow \ ( \mbox{big\_f}(x,x)\mbox{ and }\mbox{big\_f}(y_1,y_2) ) )\qquad\mbox{fof}(\mbox{church\_46\_16}_2,\mbox{conjecture})$

\medskip\textbf{SYN726+1.p} Peter Andrews Problem THM400






$( ( \forall x,y,z{:}\;( ( p(x,y)\mbox{ and }p(y,z) ) \ \Rightarrow \ p(x,z) )\mbox{ and }\forall x,y,z{:}\;( ( q(x,y)\mbox{ and }q(y,z) ) \ \Rightarrow \ q(x,z) )\mbox{ and }\forall x,y{:}\;( q(x,y) \ \Rightarrow \ q(y,x) )\mbox{ and }\forall x,y{:}\;( p(x,y)\mbox{ or }q(x,y) ) ) \ \Rightarrow \ \forall x,y{:}\;p(x,y) )\mbox{ or }\forall x,y{:}\;q(x,y)\qquad\mbox{fof}(\mbox{thm}_{400},\mbox{conjecture})$

\medskip\textbf{SYN596-1.p} Harrison problem 3669






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_9(x_{50},x_{50})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_3(x_{35},x_{35})\qquad\mbox{cnf}(\mbox{p3}_3,\mbox{negated\_conjecture})$

$p_2(x_{28},x_{28})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_{12}(x_9,x_9)\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{negated\_conjecture})$

$p_{16}(c_{21},c_{22})\qquad\mbox{cnf}(\mbox{p16}_6,\mbox{negated\_conjecture})$

$p_{17}(c_{21},f_8(c_{20}))\qquad\mbox{cnf}(\mbox{p17}_7,\mbox{negated\_conjecture})$

$p_{16}(f_5(f_6(f_7(c_{19}))),f_4(c_{20}))\qquad\mbox{cnf}(\mbox{p16}_8,\mbox{negated\_conjecture})$

$p_2(x_{48},x_{49}) \ \Rightarrow \ p_3(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p3}_9,\mbox{negated\_conjecture})$

$p_2(x_{38},x_{39}) \ \Rightarrow \ p_3(f_{15}(x_{38}),f_{15}(x_{39}))\qquad\mbox{cnf}(\mbox{p3}_{10},\mbox{negated\_conjecture})$

$p_2(x_{40},x_{41}) \ \Rightarrow \ p_3(f_4(x_{40}),f_4(x_{41}))\qquad\mbox{cnf}(\mbox{p3}_{11},\mbox{negated\_conjecture})$

$p_3(x_{42},x_{43}) \ \Rightarrow \ p_3(f_5(x_{42}),f_5(x_{43}))\qquad\mbox{cnf}(\mbox{p3}_{12},\mbox{negated\_conjecture})$

$p_3(x_{44},x_{45}) \ \Rightarrow \ p_3(f_6(x_{44}),f_6(x_{45}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{46},x_{47}) \ \Rightarrow \ p_3(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{15},\mbox{negated\_conjecture})$

$( p_9(x_{50},x_{51})\mbox{ and }p_9(x_{50},x_{52}) ) \ \Rightarrow \ p_9(x_{51},x_{52})\qquad\mbox{cnf}(\mbox{p9}_{16},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{36})\mbox{ and }p_3(x_{35},x_{37}) ) \ \Rightarrow \ p_3(x_{36},x_{37})\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{28},x_{29})\mbox{ and }p_2(x_{28},x_{30}) ) \ \Rightarrow \ p_2(x_{29},x_{30})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_{12}(x_9,x_{10})\mbox{ and }p_{12}(x_9,x_{11}) ) \ \Rightarrow \ p_{12}(x_{10},x_{11})\qquad\mbox{cnf}(\mbox{p12}_{19},\mbox{negated\_conjecture})$

$p_{18}(c_{20},f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{21})),c_{23},c_{24}))\qquad\mbox{cnf}(\mbox{p18}_{20},\mbox{negated\_conjecture})$

$\neg\,p_{18}(c_{20},f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{22})),c_{23},c_{24}))\qquad\mbox{cnf}(\mbox{not\_p18}_{21},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{16})\mbox{ and }p_3(x_{19},x_{17})\mbox{ and }p_{16}(x_{18},x_{19}) ) \ \Rightarrow \ p_{16}(x_{16},x_{17})\qquad\mbox{cnf}(\mbox{p16}_{22},\mbox{negated\_conjecture})$

$( p_{18}(x_{27},x_{26})\mbox{ and }p_2(x_{27},x_{24})\mbox{ and }p_{10}(x_{26},x_{25}) ) \ \Rightarrow \ p_{18}(x_{24},x_{25})\qquad\mbox{cnf}(\mbox{p18}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{22},x_{20})\mbox{ and }p_3(x_{23},x_{21})\mbox{ and }p_{17}(x_{22},x_{23}) ) \ \Rightarrow \ p_{17}(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p17}_{24},\mbox{negated\_conjecture})$

$( p_{12}(x_{32},x_{34})\mbox{ and }p_3(x_{31},x_{33}) ) \ \Rightarrow \ p_2(f_{13}(x_{31},x_{32}),f_{13}(x_{33},x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{12},x_{14})\mbox{ and }p_3(x_{13},x_{15}) ) \ \Rightarrow \ p_{12}(f_{14}(x_{12},x_{13}),f_{14}(x_{14},x_{15}))\qquad\mbox{cnf}(\mbox{p12}_{26},\mbox{negated\_conjecture})$

$( p_2(x_3,x_6)\mbox{ and }p_9(x_4,x_7)\mbox{ and }p_{10}(x_5,x_8) ) \ \Rightarrow \ p_{10}(f_{11}(x_3,x_4,x_5),f_{11}(x_6,x_7,x_8))\qquad\mbox{cnf}(\mbox{p10}_{27},\mbox{negated\_conjecture})$

$p_{10}(f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{22})),c_{23},c_{24}),f_{11}(f_{13}(f_{15}(c_{20}),f_{14}(f_4(c_{20}),c_{21})),c_{23},c_{24}))\qquad\mbox{cnf}(\mbox{p10}_{28},\mbox{negated\_conjecture})$

\medskip\textbf{SYN379+1.p} Peter Andrews Problem X2131






$\forall x{:}\;\mbox{big\_p}(x) \ \Rightarrow \ \exists y{:}\;( \forall x,z{:}\;\mbox{big\_q}(x,y,z) \ \Rightarrow \ \neg\,\forall z{:}\;( \mbox{big\_p}(z)\mbox{ and }\neg\,\mbox{big\_q}(y,y,z) ) )\qquad\mbox{fof}(x_{2131},\mbox{conjecture})$

\medskip\textbf{SYN602-1.p} Harrison problem 3811






$p_{14}(c_{20})\qquad\mbox{cnf}(\mbox{c20\_is\_p14}_1,\mbox{negated\_conjecture})$

$p_2(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_{15}(c_{22},c_{21})\qquad\mbox{cnf}(\mbox{p15}_3,\mbox{negated\_conjecture})$

$p_2(f_3(c_{18}),c_{19})\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(c_{22},f_9(c_{20},c_{23}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(f_3(f_4(c_{16})),c_{17})\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$( p_{14}(x_{12})\mbox{ and }p_2(x_{12},x_{11}) ) \ \Rightarrow \ p_{14}(x_{11})\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_{15}(f_{10}(x_{17}),c_{21})\qquad\mbox{cnf}(\mbox{p15}_8,\mbox{negated\_conjecture})$

$p_2(x_{35},x_{36}) \ \Rightarrow \ p_2(f_5(x_{35}),f_5(x_{36}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{28}) \ \Rightarrow \ p_2(f_{10}(x_{27}),f_{10}(x_{28}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{29},x_{30}) \ \Rightarrow \ p_2(f_{11}(x_{29}),f_{11}(x_{30}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{31},x_{32}) \ \Rightarrow \ p_2(f_3(x_{31}),f_3(x_{32}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{33},x_{34}) \ \Rightarrow \ p_2(f_4(x_{33}),f_4(x_{34}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_2(x_{18},x_{19})\mbox{ and }p_2(x_{18},x_{20}) ) \ \Rightarrow \ p_2(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p2}_{14},\mbox{negated\_conjecture})$

$p_{13}(x_6,c_{18}) \ \Rightarrow \ p_{12}(x_6,f_3(x_6),f_3(f_5(x_6)))\qquad\mbox{cnf}(\mbox{p12}_{15},\mbox{negated\_conjecture})$

$p_2(f_7(f_8(c_{21},c_{22}),c_{20}),f_6(f_5(c_{18}),f_3(c_{18})))\qquad\mbox{cnf}(\mbox{p2}_{16},\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_2(f_{10}(x_{17}),f_9(c_{20},f_{11}(x_{17})))\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_2(x_9,x_7)\mbox{ and }p_2(x_{10},x_8)\mbox{ and }p_{13}(x_9,x_{10}) ) \ \Rightarrow \ p_{13}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p13}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{15},x_{13})\mbox{ and }p_2(x_{16},x_{14})\mbox{ and }p_{15}(x_{15},x_{16}) ) \ \Rightarrow \ p_{15}(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p15}_{19},\mbox{negated\_conjecture})$

$( p_2(x_{49},x_{51})\mbox{ and }p_2(x_{50},x_{52}) ) \ \Rightarrow \ p_2(f_9(x_{49},x_{50}),f_9(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p2}_{20},\mbox{negated\_conjecture})$

$( p_2(x_{37},x_{39})\mbox{ and }p_2(x_{38},x_{40}) ) \ \Rightarrow \ p_2(f_6(x_{37},x_{38}),f_6(x_{39},x_{40}))\qquad\mbox{cnf}(\mbox{p2}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{41},x_{43})\mbox{ and }p_2(x_{42},x_{44}) ) \ \Rightarrow \ p_2(f_7(x_{41},x_{42}),f_7(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p2}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{45},x_{47})\mbox{ and }p_2(x_{46},x_{48}) ) \ \Rightarrow \ p_2(f_8(x_{45},x_{46}),f_8(x_{47},x_{48}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_{13}(x_{17},f_5(c_{18})) \ \Rightarrow \ p_2(f_7(f_8(c_{21},f_{10}(x_{17})),c_{20}),f_6(f_5(x_{17}),f_3(x_{17})))\qquad\mbox{cnf}(\mbox{p2}_{24},\mbox{negated\_conjecture})$

$( p_2(x_5,x_2)\mbox{ and }p_2(x_4,x_1)\mbox{ and }p_2(x_3,x_0)\mbox{ and }p_{12}(x_3,x_4,x_5) ) \ \Rightarrow \ p_{12}(x_0,x_1,x_2)\qquad\mbox{cnf}(\mbox{p12}_{25},\mbox{negated\_conjecture})$

$( p_{15}(x_{53},c_{21})\mbox{ and }p_2(x_{53},f_9(c_{20},x_{54})) ) \ \Rightarrow \ \neg\,p_2(f_7(f_8(c_{21},x_{53}),c_{20}),f_6(f_5(c_{18}),c_{19}))\qquad\mbox{cnf}(\mbox{not\_p15}_{26},\mbox{negated\_conjecture})$

$( p_{15}(x_{21},c_{21})\mbox{ and }p_2(x_{21},f_9(c_{20},x_{22}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{21}),c_{20}),f_6(f_5(c_{18}),x_{23})) ) \ \Rightarrow \ p_2(x_{21},c_{22})\qquad\mbox{cnf}(\mbox{p2}_{27},\mbox{negated\_conjecture})$

$( p_{15}(x_{21},c_{21})\mbox{ and }p_2(x_{21},f_9(c_{20},x_{22}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{21}),c_{20}),f_6(f_5(c_{18}),x_{23})) ) \ \Rightarrow \ p_2(x_{23},f_3(c_{18}))\qquad\mbox{cnf}(\mbox{p2}_{28},\mbox{negated\_conjecture})$

$( p_{15}(x_{24},c_{21})\mbox{ and }p_{13}(x_{17},f_5(c_{18}))\mbox{ and }p_2(x_{24},f_9(c_{20},x_{25}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{24}),c_{20}),f_6(f_5(x_{17}),x_{26})) ) \ \Rightarrow \ p_2(x_{26},f_3(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{29},\mbox{negated\_conjecture})$

$( p_{15}(x_{24},c_{21})\mbox{ and }p_{13}(x_{17},f_5(c_{18}))\mbox{ and }p_2(x_{24},f_9(c_{20},x_{25}))\mbox{ and }p_2(f_7(f_8(c_{21},x_{24}),c_{20}),f_6(f_5(x_{17}),x_{26})) ) \ \Rightarrow \ p_2(x_{24},f_{10}(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN549+1.p} dia box(dia(p or dia q) $\leq$$>$ (dia p or dia q))






$\forall x{:}\;\mbox{reachable}(x,x)\qquad\mbox{fof}(\mbox{reflexivity\_of\_reachable},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{reachable}(x,y)\mbox{ and }\mbox{reachable}(y,z) ) \ \Rightarrow \ \mbox{reachable}(x,z) )\qquad\mbox{fof}(\mbox{transitivity\_of\_reachable},\mbox{axiom})$

$\exists x{:}\;( \mbox{reachable}(\mbox{initial\_world},x)\mbox{ and }\forall y{:}\;( \mbox{reachable}(x,y) \ \Rightarrow \ ( \exists z{:}\;( \mbox{reachable}(y,z)\mbox{ and }( p(z)\mbox{ or }\exists v{:}\;( \mbox{reachable}(z,v)\mbox{ and }q(v) ) ) )\iff ( \exists w{:}\;( \mbox{reachable}(y,w)\mbox{ and }p(w) )\mbox{ or }\exists w{:}\;( \mbox{reachable}(y,w)\mbox{ and }q(w) ) ) ) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN608-1.p} Harrison problem 3828






$\neg\,p_{16}(c_{21})\qquad\mbox{cnf}(\mbox{c21\_is\_not\_p16}_1,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{11})\qquad\mbox{cnf}(\mbox{p2}_2,\mbox{negated\_conjecture})$

$p_5(x_{43},x_{43})\qquad\mbox{cnf}(\mbox{p5}_3,\mbox{negated\_conjecture})$

$p_3(x_{18},x_{18})\qquad\mbox{cnf}(\mbox{p3}_4,\mbox{negated\_conjecture})$

$\neg\,p_5(c_{21},f_7(c_{20}))\qquad\mbox{cnf}(\mbox{not\_p5}_5,\mbox{negated\_conjecture})$

$p_3(f_{12}(c_{22},c_{21}),f_{15}(c_{22}))\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(c_{22}),c_{21}),c_{22})\qquad\mbox{cnf}(\mbox{p3}_7,\mbox{negated\_conjecture})$

$\neg\,p_3(c_{19},f_4(f_6(f_7(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p3}_8,\mbox{negated\_conjecture})$

$( p_{16}(x_1)\mbox{ and }p_5(x_1,x_0) ) \ \Rightarrow \ p_{16}(x_0)\qquad\mbox{cnf}(\mbox{p16}_9,\mbox{negated\_conjecture})$

$p_5(x_{48},x_{49}) \ \Rightarrow \ p_5(f_8(x_{48}),f_8(x_{49}))\qquad\mbox{cnf}(\mbox{p5}_{10},\mbox{negated\_conjecture})$

$p_5(x_{14},x_{15}) \ \Rightarrow \ p_2(f_6(x_{14}),f_6(x_{15}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_3(x_{16},x_{17}) \ \Rightarrow \ p_2(f_9(x_{16}),f_9(x_{17}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_5(x_{33},x_{34}) \ \Rightarrow \ p_3(f_{13}(x_{33}),f_{13}(x_{34}))\qquad\mbox{cnf}(\mbox{p3}_{13},\mbox{negated\_conjecture})$

$p_3(x_{39},x_{40}) \ \Rightarrow \ p_3(f_{15}(x_{39}),f_{15}(x_{40}))\qquad\mbox{cnf}(\mbox{p3}_{14},\mbox{negated\_conjecture})$

$p_2(x_{41},x_{42}) \ \Rightarrow \ p_3(f_4(x_{41}),f_4(x_{42}))\qquad\mbox{cnf}(\mbox{p3}_{15},\mbox{negated\_conjecture})$

$p_5(x_{46},x_{47}) \ \Rightarrow \ p_5(f_7(x_{46}),f_7(x_{47}))\qquad\mbox{cnf}(\mbox{p5}_{16},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{12})\mbox{ and }p_2(x_{11},x_{13}) ) \ \Rightarrow \ p_2(x_{12},x_{13})\qquad\mbox{cnf}(\mbox{p2}_{17},\mbox{negated\_conjecture})$

$( p_5(x_{43},x_{44})\mbox{ and }p_5(x_{43},x_{45}) ) \ \Rightarrow \ p_5(x_{44},x_{45})\qquad\mbox{cnf}(\mbox{p5}_{18},\mbox{negated\_conjecture})$

$( p_3(x_{18},x_{19})\mbox{ and }p_3(x_{18},x_{20}) ) \ \Rightarrow \ p_3(x_{19},x_{20})\qquad\mbox{cnf}(\mbox{p3}_{19},\mbox{negated\_conjecture})$

$( p_2(x_4,x_2)\mbox{ and }p_2(x_5,x_3)\mbox{ and }p_{17}(x_4,x_5) ) \ \Rightarrow \ p_{17}(x_2,x_3)\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_5(x_9,x_7)\mbox{ and }p_5(x_{10},x_8)\mbox{ and }p_{18}(x_9,x_{10}) ) \ \Rightarrow \ p_{18}(x_7,x_8)\qquad\mbox{cnf}(\mbox{p18}_{21},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_4(f_6(f_7(f_8(c_{20})))),c_{21}),f_4(f_6(f_7(f_8(c_{20})))))\qquad\mbox{cnf}(\mbox{p3}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{35},x_{37})\mbox{ and }p_3(x_{36},x_{38}) ) \ \Rightarrow \ p_3(f_{14}(x_{35},x_{36}),f_{14}(x_{37},x_{38}))\qquad\mbox{cnf}(\mbox{p3}_{23},\mbox{negated\_conjecture})$

$( p_3(x_{21},x_{23})\mbox{ and }p_3(x_{22},x_{24}) ) \ \Rightarrow \ p_3(f_{10}(x_{21},x_{22}),f_{10}(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p3}_{24},\mbox{negated\_conjecture})$

$( p_3(x_{25},x_{27})\mbox{ and }p_3(x_{26},x_{28}) ) \ \Rightarrow \ p_3(f_{11}(x_{25},x_{26}),f_{11}(x_{27},x_{28}))\qquad\mbox{cnf}(\mbox{p3}_{25},\mbox{negated\_conjecture})$

$( p_3(x_{29},x_{31})\mbox{ and }p_5(x_{30},x_{32}) ) \ \Rightarrow \ p_3(f_{12}(x_{29},x_{30}),f_{12}(x_{31},x_{32}))\qquad\mbox{cnf}(\mbox{p3}_{26},\mbox{negated\_conjecture})$

$p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{15}(c_{22}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{p17}_{27},\mbox{negated\_conjecture})$

$p_3(f_{12}(f_{15}(f_4(f_6(f_7(f_8(c_{20}))))),c_{21}),f_{15}(f_4(f_6(f_7(f_8(c_{20}))))))\qquad\mbox{cnf}(\mbox{p3}_{28},\mbox{negated\_conjecture})$

$\neg\,p_{17}(f_9(f_{10}(f_{14}(f_4(f_9(c_{19})),c_{19}),f_{12}(x_{50},c_{21}))),f_6(f_7(f_8(c_{20}))))\qquad\mbox{cnf}(\mbox{not\_p17}_{29},\mbox{negated\_conjecture})$

$p_{18}(x_6,c_{21}) \ \Rightarrow \ ( p_5(x_6,f_7(c_{20}))\mbox{ or }p_3(c_{19},f_4(f_6(f_7(c_{20}))))\mbox{ or }p_{17}(f_9(f_{10}(f_4(f_6(f_7(f_8(c_{20})))),f_{11}(c_{19},f_{12}(f_{13}(x_6),x_6)))),f_6(f_7(f_8(c_{20})))) )\qquad\mbox{cnf}(\mbox{p5}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{SYN398+1.p} Kalish and Montague Problem 215

Laws of confinement.




$\forall x{:}\;( p\mbox{ and }f(x) )\iff ( p\mbox{ and }\forall y{:}\;f(y) )\qquad\mbox{fof}(\mbox{kalish}_{215},\mbox{conjecture})$

\medskip\textbf{SYN141-1.p} RPT63 synthetic problem 39 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,m_5(c,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN974+1.p} Syntactic from Shults






$\exists x{:}\;p(x) \ \Rightarrow \ \exists y{:}\;p(y)\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN385+1.p} Peter Andrews Problem X2137






$\exists x{:}\;\forall y{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_q}(x)\mbox{ or }\mbox{big\_p}(y) ) )\qquad\mbox{fof}(x_{2137},\mbox{conjecture})$

\medskip\textbf{SYN621-1.p} Harrison problem 3998






$p_{10}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p10}_1,\mbox{negated\_conjecture})$

$p_8(x_{78},x_{78})\qquad\mbox{cnf}(\mbox{p8}_2,\mbox{negated\_conjecture})$

$p_7(x_{71},x_{71})\qquad\mbox{cnf}(\mbox{p7}_3,\mbox{negated\_conjecture})$

$p_4(x_{38},x_{38})\qquad\mbox{cnf}(\mbox{p4}_4,\mbox{negated\_conjecture})$

$p_2(x_{12},x_{12})\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_{13}(x_7,x_7)\qquad\mbox{cnf}(\mbox{p13}_6,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{23},f_3(c_{24}))\qquad\mbox{cnf}(\mbox{not\_p2}_7,\mbox{negated\_conjecture})$

$p_7(x_{10},x_{11}) \ \Rightarrow \ p_{13}(f_{14}(x_{10}),f_{14}(x_{11}))\qquad\mbox{cnf}(\mbox{p13}_8,\mbox{negated\_conjecture})$

$p_4(x_{59},x_{60}) \ \Rightarrow \ p_4(f_5(x_{59}),f_5(x_{60}))\qquad\mbox{cnf}(\mbox{p4}_9,\mbox{negated\_conjecture})$

$p_2(x_{45},x_{46}) \ \Rightarrow \ p_4(f_{16}(x_{45}),f_{16}(x_{46}))\qquad\mbox{cnf}(\mbox{p4}_{10},\mbox{negated\_conjecture})$

$p_2(x_{24},x_{25}) \ \Rightarrow \ p_2(f_3(x_{24}),f_3(x_{25}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{22},x_{23}) \ \Rightarrow \ p_2(f_{18}(x_{22}),f_{18}(x_{23}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{20},x_{21}) \ \Rightarrow \ p_2(f_{17}(x_{20}),f_{17}(x_{21}))\qquad\mbox{cnf}(\mbox{p2}_{13},\mbox{negated\_conjecture})$

$( p_{10}(x_0,x_1)\mbox{ and }p_{10}(x_0,x_2) ) \ \Rightarrow \ p_{10}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$( p_8(x_{78},x_{79})\mbox{ and }p_8(x_{78},x_{80}) ) \ \Rightarrow \ p_8(x_{79},x_{80})\qquad\mbox{cnf}(\mbox{p8}_{15},\mbox{negated\_conjecture})$

$( p_7(x_{71},x_{72})\mbox{ and }p_7(x_{71},x_{73}) ) \ \Rightarrow \ p_7(x_{72},x_{73})\qquad\mbox{cnf}(\mbox{p7}_{16},\mbox{negated\_conjecture})$

$( p_4(x_{38},x_{39})\mbox{ and }p_4(x_{38},x_{40}) ) \ \Rightarrow \ p_4(x_{39},x_{40})\qquad\mbox{cnf}(\mbox{p4}_{17},\mbox{negated\_conjecture})$

$( p_{21}(x_{30},x_{32})\mbox{ and }p_{21}(x_{32},x_{31}) ) \ \Rightarrow \ p_{21}(x_{30},x_{31})\qquad\mbox{cnf}(\mbox{p21}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{12},x_{13})\mbox{ and }p_2(x_{12},x_{14}) ) \ \Rightarrow \ p_2(x_{13},x_{14})\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_{13}(x_7,x_8)\mbox{ and }p_{13}(x_7,x_9) ) \ \Rightarrow \ p_{13}(x_8,x_9)\qquad\mbox{cnf}(\mbox{p13}_{20},\mbox{negated\_conjecture})$

$( p_4(x_{28},x_{26})\mbox{ and }p_4(x_{29},x_{27})\mbox{ and }p_{21}(x_{28},x_{29}) ) \ \Rightarrow \ p_{21}(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p21}_{21},\mbox{negated\_conjecture})$

$( p_{22}(x_{37},x_{36})\mbox{ and }p_4(x_{37},x_{34})\mbox{ and }p_{13}(x_{36},x_{35}) ) \ \Rightarrow \ p_{22}(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p22}_{22},\mbox{negated\_conjecture})$

$( p_{10}(x_{75},x_{77})\mbox{ and }p_2(x_{74},x_{76}) ) \ \Rightarrow \ p_7(f_{11}(x_{74},x_{75}),f_{11}(x_{76},x_{77}))\qquad\mbox{cnf}(\mbox{p7}_{23},\mbox{negated\_conjecture})$

$( p_2(x_3,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_{10}(f_{12}(x_3,x_4),f_{12}(x_5,x_6))\qquad\mbox{cnf}(\mbox{p10}_{24},\mbox{negated\_conjecture})$

$( p_4(x_{41},x_{43})\mbox{ and }p_4(x_{42},x_{44}) ) \ \Rightarrow \ p_4(f_{15}(x_{41},x_{42}),f_{15}(x_{43},x_{44}))\qquad\mbox{cnf}(\mbox{p4}_{25},\mbox{negated\_conjecture})$

$( p_4(x_{48},x_{50})\mbox{ and }p_7(x_{47},x_{49}) ) \ \Rightarrow \ p_4(f_{19}(x_{47},x_{48}),f_{19}(x_{49},x_{50}))\qquad\mbox{cnf}(\mbox{p4}_{26},\mbox{negated\_conjecture})$

$( p_4(x_{61},x_{63})\mbox{ and }p_4(x_{62},x_{64}) ) \ \Rightarrow \ p_4(f_6(x_{61},x_{62}),f_6(x_{63},x_{64}))\qquad\mbox{cnf}(\mbox{p4}_{27},\mbox{negated\_conjecture})$

$p_2(x_{15},f_3(c_{24}))\mbox{ or }p_{22}(f_{20}(x_{17},x_{15},x_{18},x_{16}),f_{14}(f_{11}(x_{17},f_{12}(x_{15},x_{18}))))\qquad\mbox{cnf}(\mbox{p2}_{28},\mbox{negated\_conjecture})$

$( p_7(x_{65},x_{68})\mbox{ and }p_8(x_{66},x_{69})\mbox{ and }p_4(x_{67},x_{70}) ) \ \Rightarrow \ p_4(f_9(x_{65},x_{66},x_{67}),f_9(x_{68},x_{69},x_{70}))\qquad\mbox{cnf}(\mbox{p4}_{29},\mbox{negated\_conjecture})$

$( p_2(x_{53},x_{57})\mbox{ and }p_4(x_{54},x_{58})\mbox{ and }p_2(x_{51},x_{55})\mbox{ and }p_2(x_{52},x_{56}) ) \ \Rightarrow \ p_4(f_{20}(x_{51},x_{52},x_{53},x_{54}),f_{20}(x_{55},x_{56},x_{57},x_{58}))\qquad\mbox{cnf}(\mbox{p4}_{30},\mbox{negated\_conjecture})$

$p_{22}(x_{33},f_{14}(f_{11}(c_{25},f_{12}(c_{23},c_{26})))) \ \Rightarrow \ p_{21}(f_5(f_6(c_{27},f_9(f_{11}(c_{25},f_{12}(c_{23},c_{26})),c_{28},c_{27}))),f_5(f_6(c_{27},x_{33})))\qquad\mbox{cnf}(\mbox{p21}_{31},\mbox{negated\_conjecture})$

$\neg\,p_{21}(f_5(f_6(c_{27},f_9(f_{11}(c_{25},f_{12}(c_{23},c_{26})),c_{28},c_{27}))),f_{15}(f_{19}(f_{11}(c_{25},f_{12}(c_{23},c_{26})),c_{27}),f_{16}(f_3(f_{17}(f_{18}(c_{24}))))))\qquad\mbox{cnf}(\mbox{not\_p21}_{32},\mbox{negated\_conjecture})$

$p_{22}(x_{19},f_{14}(f_{11}(x_{17},f_{12}(x_{15},x_{18})))) \ \Rightarrow \ ( p_2(x_{15},f_3(c_{24}))\mbox{ or }p_{21}(f_5(f_6(x_{16},f_{20}(x_{17},x_{15},x_{18},x_{16}))),f_5(f_6(x_{16},x_{19}))) )\qquad\mbox{cnf}(\mbox{p2}_{33},\mbox{negated\_conjecture})$

$p_2(x_{15},f_3(c_{24}))\mbox{ or }p_{21}(f_5(f_6(x_{16},f_{20}(x_{17},x_{15},x_{18},x_{16}))),f_{15}(f_{19}(f_{11}(x_{17},f_{12}(x_{15},x_{18})),x_{16}),f_{16}(f_3(f_{17}(f_{18}(c_{24}))))))\qquad\mbox{cnf}(\mbox{p2}_{34},\mbox{negated\_conjecture})$

\medskip\textbf{SYN071+1.p} Pelletier Problem 48






$a = b\mbox{ or }c = d\qquad\mbox{fof}(\mbox{pel48}_1,\mbox{axiom})$

$a = c\mbox{ or }b = d\qquad\mbox{fof}(\mbox{pel48}_2,\mbox{axiom})$

$a = d\mbox{ or }b = c\qquad\mbox{fof}(\mbox{pel}_{48},\mbox{conjecture})$

\medskip\textbf{SYN325+1.p} Church problem 46.9 (2)






$\exists x{:}\;\forall y{:}\;( ( ( \mbox{big\_f}(x,x) \ \Rightarrow \ \mbox{big\_f}(y,y) ) \ \Rightarrow \ ( \mbox{big\_f}(x,y)\mbox{ and }\mbox{big\_g}(x) ) ) \ \Rightarrow \ \mbox{big\_g}(y) )\qquad\mbox{fof}(\mbox{church\_46\_9}_2,\mbox{conjecture})$

\medskip\textbf{SYN010-1.005.005.p} Example for Proposition 5.2 in [LMG94]

Example to show that connection tableaux with factorization
 cannot polynomially simulate simulate connection tableaux with
 folding up.




$\neg\,p_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( \mbox{p\_1}_1\mbox{ and }\mbox{p\_1}_2\mbox{ and }\mbox{p\_1}_3\mbox{ and }\mbox{p\_1}_4\mbox{ and }\mbox{p\_1}_5 ) \ \Rightarrow \ p_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_1\mbox{ and }\mbox{p\_2}_2\mbox{ and }\mbox{p\_2}_3\mbox{ and }\mbox{p\_2}_4\mbox{ and }\mbox{p\_2}_5 ) \ \Rightarrow \ \mbox{p\_1}_1\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_1\mbox{ and }\mbox{p\_2}_2\mbox{ and }\mbox{p\_2}_3\mbox{ and }\mbox{p\_2}_4\mbox{ and }\mbox{p\_2}_5 ) \ \Rightarrow \ \mbox{p\_1}_2\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_1\mbox{ and }\mbox{p\_2}_2\mbox{ and }\mbox{p\_2}_3\mbox{ and }\mbox{p\_2}_4\mbox{ and }\mbox{p\_2}_5 ) \ \Rightarrow \ \mbox{p\_1}_3\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_1\mbox{ and }\mbox{p\_2}_2\mbox{ and }\mbox{p\_2}_3\mbox{ and }\mbox{p\_2}_4\mbox{ and }\mbox{p\_2}_5 ) \ \Rightarrow \ \mbox{p\_1}_4\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$( \mbox{p\_2}_1\mbox{ and }\mbox{p\_2}_2\mbox{ and }\mbox{p\_2}_3\mbox{ and }\mbox{p\_2}_4\mbox{ and }\mbox{p\_2}_5 ) \ \Rightarrow \ \mbox{p\_1}_5\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

$( \mbox{p\_3}_1\mbox{ and }\mbox{p\_3}_2\mbox{ and }\mbox{p\_3}_3\mbox{ and }\mbox{p\_3}_4\mbox{ and }\mbox{p\_3}_5 ) \ \Rightarrow \ \mbox{p\_2}_1\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{negated\_conjecture})$

$( \mbox{p\_3}_1\mbox{ and }\mbox{p\_3}_2\mbox{ and }\mbox{p\_3}_3\mbox{ and }\mbox{p\_3}_4\mbox{ and }\mbox{p\_3}_5 ) \ \Rightarrow \ \mbox{p\_2}_2\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{negated\_conjecture})$

$( \mbox{p\_3}_1\mbox{ and }\mbox{p\_3}_2\mbox{ and }\mbox{p\_3}_3\mbox{ and }\mbox{p\_3}_4\mbox{ and }\mbox{p\_3}_5 ) \ \Rightarrow \ \mbox{p\_2}_3\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{negated\_conjecture})$

$( \mbox{p\_3}_1\mbox{ and }\mbox{p\_3}_2\mbox{ and }\mbox{p\_3}_3\mbox{ and }\mbox{p\_3}_4\mbox{ and }\mbox{p\_3}_5 ) \ \Rightarrow \ \mbox{p\_2}_4\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{negated\_conjecture})$

$( \mbox{p\_3}_1\mbox{ and }\mbox{p\_3}_2\mbox{ and }\mbox{p\_3}_3\mbox{ and }\mbox{p\_3}_4\mbox{ and }\mbox{p\_3}_5 ) \ \Rightarrow \ \mbox{p\_2}_5\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{negated\_conjecture})$

$( \mbox{p\_4}_1\mbox{ and }\mbox{p\_4}_2\mbox{ and }\mbox{p\_4}_3\mbox{ and }\mbox{p\_4}_4\mbox{ and }\mbox{p\_4}_5 ) \ \Rightarrow \ \mbox{p\_3}_1\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{negated\_conjecture})$

$( \mbox{p\_4}_1\mbox{ and }\mbox{p\_4}_2\mbox{ and }\mbox{p\_4}_3\mbox{ and }\mbox{p\_4}_4\mbox{ and }\mbox{p\_4}_5 ) \ \Rightarrow \ \mbox{p\_3}_2\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{negated\_conjecture})$

$( \mbox{p\_4}_1\mbox{ and }\mbox{p\_4}_2\mbox{ and }\mbox{p\_4}_3\mbox{ and }\mbox{p\_4}_4\mbox{ and }\mbox{p\_4}_5 ) \ \Rightarrow \ \mbox{p\_3}_3\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{negated\_conjecture})$

$( \mbox{p\_4}_1\mbox{ and }\mbox{p\_4}_2\mbox{ and }\mbox{p\_4}_3\mbox{ and }\mbox{p\_4}_4\mbox{ and }\mbox{p\_4}_5 ) \ \Rightarrow \ \mbox{p\_3}_4\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{negated\_conjecture})$

$( \mbox{p\_4}_1\mbox{ and }\mbox{p\_4}_2\mbox{ and }\mbox{p\_4}_3\mbox{ and }\mbox{p\_4}_4\mbox{ and }\mbox{p\_4}_5 ) \ \Rightarrow \ \mbox{p\_3}_5\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{negated\_conjecture})$

$( \mbox{p\_5}_1\mbox{ and }\mbox{p\_5}_2\mbox{ and }\mbox{p\_5}_3\mbox{ and }\mbox{p\_5}_4\mbox{ and }\mbox{p\_5}_5 ) \ \Rightarrow \ \mbox{p\_4}_1\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{negated\_conjecture})$

$( \mbox{p\_5}_1\mbox{ and }\mbox{p\_5}_2\mbox{ and }\mbox{p\_5}_3\mbox{ and }\mbox{p\_5}_4\mbox{ and }\mbox{p\_5}_5 ) \ \Rightarrow \ \mbox{p\_4}_2\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{negated\_conjecture})$

$( \mbox{p\_5}_1\mbox{ and }\mbox{p\_5}_2\mbox{ and }\mbox{p\_5}_3\mbox{ and }\mbox{p\_5}_4\mbox{ and }\mbox{p\_5}_5 ) \ \Rightarrow \ \mbox{p\_4}_3\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{negated\_conjecture})$

$( \mbox{p\_5}_1\mbox{ and }\mbox{p\_5}_2\mbox{ and }\mbox{p\_5}_3\mbox{ and }\mbox{p\_5}_4\mbox{ and }\mbox{p\_5}_5 ) \ \Rightarrow \ \mbox{p\_4}_4\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{negated\_conjecture})$

$( \mbox{p\_5}_1\mbox{ and }\mbox{p\_5}_2\mbox{ and }\mbox{p\_5}_3\mbox{ and }\mbox{p\_5}_4\mbox{ and }\mbox{p\_5}_5 ) \ \Rightarrow \ \mbox{p\_4}_5\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{negated\_conjecture})$

$\mbox{p\_5}_1\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{negated\_conjecture})$

$\mbox{p\_5}_2\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{negated\_conjecture})$

$\mbox{p\_5}_3\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{negated\_conjecture})$

$\mbox{p\_5}_4\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{negated\_conjecture})$

$\mbox{p\_5}_5\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{SYN371+1.p} Peter Andrews Problem X2122






$( \exists x{:}\;\mbox{big\_r}(x,x) \ \Rightarrow \ \forall y{:}\;\mbox{big\_r}(y,y) ) \ \Rightarrow \ \exists u{:}\;\forall v{:}\;( \mbox{big\_r}(u,u) \ \Rightarrow \ \mbox{big\_r}(v,v) )\qquad\mbox{fof}(x_{2122},\mbox{conjecture})$

\medskip\textbf{SYN324-1.p} Church problem 46.9 (1)






$f(x,x) \ \Rightarrow \ f(x,y(x))\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(x,y(x)) \ \Rightarrow \ f(x,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$f(x,y(x))\mbox{ or }f(y(x),y(x))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$f(x,y(x)) \ \Rightarrow \ \neg\,f(y(x),y(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SYN256-1.p} RPT63 synthetic problem 41 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,n_2(c)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN589-1.p} Harrison problem 3578






$p_2(x_8,x_8)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_7(x_{25},x_{25})\qquad\mbox{cnf}(\mbox{p7}_2,\mbox{negated\_conjecture})$

$p_2(c_{21},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{p2}_3,\mbox{negated\_conjecture})$

$\neg\,p_2(c_{16},f_3(c_{17}))\qquad\mbox{cnf}(\mbox{not\_p2}_4,\mbox{negated\_conjecture})$

$p_{14}(c_{18},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_5,\mbox{negated\_conjecture})$

$p_{14}(c_{23},f_3(f_4(f_5(c_{17}))))\qquad\mbox{cnf}(\mbox{p14}_6,\mbox{negated\_conjecture})$

$p_{14}(c_{19},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_7,\mbox{negated\_conjecture})$

$p_{14}(c_{24},f_6(f_3(f_4(f_5(c_{17}))),c_{16}))\qquad\mbox{cnf}(\mbox{p14}_8,\mbox{negated\_conjecture})$

$p_7(x_{34},x_{35}) \ \Rightarrow \ p_7(f_{12}(x_{34}),f_{12}(x_{35}))\qquad\mbox{cnf}(\mbox{p7}_9,\mbox{negated\_conjecture})$

$p_2(x_{15},x_{16}) \ \Rightarrow \ p_2(f_3(x_{15}),f_3(x_{16}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_{17},x_{18}) \ \Rightarrow \ p_2(f_4(x_{17}),f_4(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$p_2(x_{19},x_{20}) \ \Rightarrow \ p_2(f_5(x_{19}),f_5(x_{20}))\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$p_2(x_{32},x_{33}) \ \Rightarrow \ p_7(f_{11}(x_{32}),f_{11}(x_{33}))\qquad\mbox{cnf}(\mbox{p7}_{13},\mbox{negated\_conjecture})$

$( p_7(x_{25},x_{26})\mbox{ and }p_7(x_{25},x_{27}) ) \ \Rightarrow \ p_7(x_{26},x_{27})\qquad\mbox{cnf}(\mbox{p7}_{14},\mbox{negated\_conjecture})$

$( p_2(x_8,x_9)\mbox{ and }p_2(x_8,x_{10}) ) \ \Rightarrow \ p_2(x_9,x_{10})\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{15}(f_6(f_3(f_4(f_5(c_{17}))),f_{13}(c_{16},f_3(f_5(c_{17})))),c_{24})\qquad\mbox{cnf}(\mbox{p15}_{16},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{14}(x_2,x_3) ) \ \Rightarrow \ p_{14}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p14}_{17},\mbox{negated\_conjecture})$

$( p_2(x_6,x_4)\mbox{ and }p_2(x_7,x_5)\mbox{ and }p_{15}(x_6,x_7) ) \ \Rightarrow \ p_{15}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p15}_{18},\mbox{negated\_conjecture})$

$( p_2(x_{11},x_{13})\mbox{ and }p_2(x_{12},x_{14}) ) \ \Rightarrow \ p_2(f_{13}(x_{11},x_{12}),f_{13}(x_{13},x_{14}))\qquad\mbox{cnf}(\mbox{p2}_{19},\mbox{negated\_conjecture})$

$( p_7(x_{40},x_{42})\mbox{ and }p_7(x_{41},x_{43}) ) \ \Rightarrow \ p_7(f_9(x_{40},x_{41}),f_9(x_{42},x_{43}))\qquad\mbox{cnf}(\mbox{p7}_{20},\mbox{negated\_conjecture})$

$( p_7(x_{36},x_{38})\mbox{ and }p_7(x_{37},x_{39}) ) \ \Rightarrow \ p_7(f_8(x_{36},x_{37}),f_8(x_{38},x_{39}))\qquad\mbox{cnf}(\mbox{p7}_{21},\mbox{negated\_conjecture})$

$( p_2(x_{29},x_{31})\mbox{ and }p_7(x_{28},x_{30}) ) \ \Rightarrow \ p_7(f_{10}(x_{28},x_{29}),f_{10}(x_{30},x_{31}))\qquad\mbox{cnf}(\mbox{p7}_{22},\mbox{negated\_conjecture})$

$( p_2(x_{21},x_{23})\mbox{ and }p_2(x_{22},x_{24}) ) \ \Rightarrow \ p_2(f_6(x_{21},x_{22}),f_6(x_{23},x_{24}))\qquad\mbox{cnf}(\mbox{p2}_{23},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{25}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{24},\mbox{negated\_conjecture})$

$p_7(c_{20},f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{p7}_{25},\mbox{negated\_conjecture})$

$\neg\,p_7(f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{18}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{21}),f_9(f_{11}(c_{19}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))),f_8(f_{10}(f_{12}(f_{11}(f_3(f_5(c_{17})))),c_{23}),f_8(f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{25}),f_9(f_{11}(c_{24}),f_{10}(f_{11}(f_3(f_4(f_5(c_{17})))),c_{22})))))\qquad\mbox{cnf}(\mbox{not\_p7}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{SYN220-1.p} RPT63 synthetic problem 5 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(x,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN269-1.p} RPT63 synthetic problem 54 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,p_4(b,a,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN365+1.p} Peter Andrews Problem X2116






$( \forall x{:}\;\exists y{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ ( \mbox{big\_r}(x,g(h(y)))\mbox{ and }\mbox{big\_p}(y) ) )\mbox{ and }\forall w{:}\;( \mbox{big\_p}(w) \ \Rightarrow \ ( \mbox{big\_p}(g(w))\mbox{ and }\mbox{big\_p}(h(w)) ) ) ) \ \Rightarrow \ \forall x{:}\;( \mbox{big\_p}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{big\_r}(x,y)\mbox{ and }\mbox{big\_p}(y) ) )\qquad\mbox{fof}(x_{2116},\mbox{conjecture})$

\medskip\textbf{SYN276-1.p} RPT63 synthetic problem 61 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_0(d,b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN946+1.p} Syntactic from Shults






$( \forall x{:}\;p(x)\mbox{ and }\exists y{:}\;q(y) ) \ \Rightarrow \ \exists z{:}\;\forall y{:}\;( p(y)\mbox{ or }r(z) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN409+1.p} Kalish and Montague Problem 246

Laws of alphabetic variance.




$\forall x{:}\;f(x)\iff \forall y,z{:}\;( f(y)\mbox{ and }f(z) )\qquad\mbox{fof}(\mbox{kalish}_{246},\mbox{conjecture})$

\medskip\textbf{SYN225-1.p} RPT63 synthetic problem 10 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(b,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN408+1.p} Kalish and Montague Problem 244

Laws of alphabetic variance.




$\neg\,\exists x{:}\;f(x) \ \Rightarrow \ \forall y{:}\;( f(y) \ \Rightarrow \ g(y) )\qquad\mbox{fof}(\mbox{kalish}_{244},\mbox{conjecture})$

\medskip\textbf{SYN319-1.p} Church problem 46.2 (5)






$f(y)\mbox{ or }g(x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(z_1(x,y)) \ \Rightarrow \ f(y)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$g(z_1(x,y)) \ \Rightarrow \ g(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$g(z_1(x,y)) \ \Rightarrow \ \neg\,f(z_1(x,y))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$f(x) \ \Rightarrow \ h(x)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$g(x) \ \Rightarrow \ h(x)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{negated\_conjecture})$

$\neg\,h(z_2(x,y))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SYN126-1.p} RPT63 synthetic problem 24 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_4(b)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN223-1.p} RPT63 synthetic problem 8 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(x,e)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN298-1.p} RPT63 synthetic problem 83 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,s_3(b,x)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN989$\wedge$2.p} All things are true or false






$\forall x{:}\;\mbox{\$o}{:}\;( x\mbox{ or }\neg\,x )\qquad\mbox{thf}(\mbox{thm},\mbox{axiom})$

\medskip\textbf{SYN997$\wedge$1.p} Test for validity of axiom of choice






$\exists e{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists y{:}\;\mbox{\$i}{:}\;( p@ y ) \ \Rightarrow \ ( p@ ( e@ p ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYN295-1.p} RPT63 synthetic problem 80 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,q_1(d,d,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN988$\wedge$2.p} All things are false






$\forall x{:}\;\mbox{\$o}{:}\;\neg\,x\qquad\mbox{thf}(\mbox{thm},\mbox{axiom})$

\medskip\textbf{SYN234-1.p} RPT63 synthetic problem 19 (skewed distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,l_3(e,a)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN562-1.p} Harrison problem 2559






$p_2(x_4,x_4)\qquad\mbox{cnf}(\mbox{p2}_1,\mbox{negated\_conjecture})$

$p_6(x_{19},x_{19})\qquad\mbox{cnf}(\mbox{p6}_2,\mbox{negated\_conjecture})$

$\neg\,p_{10}(x_{22},x_{22})\qquad\mbox{cnf}(\mbox{not\_p10}_3,\mbox{negated\_conjecture})$

$p_2(f_9(c_{11}),f_3(c_{12}))\qquad\mbox{cnf}(\mbox{p2}_4,\mbox{negated\_conjecture})$

$p_2(f_8(c_{11}),f_3(c_{12}))\qquad\mbox{cnf}(\mbox{p2}_5,\mbox{negated\_conjecture})$

$p_2(x_7,x_8) \ \Rightarrow \ p_2(f_3(x_7),f_3(x_8))\qquad\mbox{cnf}(\mbox{p2}_6,\mbox{negated\_conjecture})$

$p_6(x_{17},x_{18}) \ \Rightarrow \ p_2(f_9(x_{17}),f_9(x_{18}))\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_6(x_{15},x_{16}) \ \Rightarrow \ p_2(f_8(x_{15}),f_8(x_{16}))\qquad\mbox{cnf}(\mbox{p2}_8,\mbox{negated\_conjecture})$

$p_6(x_{13},x_{14}) \ \Rightarrow \ p_2(f_7(x_{13}),f_7(x_{14}))\qquad\mbox{cnf}(\mbox{p2}_9,\mbox{negated\_conjecture})$

$p_2(x_{11},x_{12}) \ \Rightarrow \ p_2(f_5(x_{11}),f_5(x_{12}))\qquad\mbox{cnf}(\mbox{p2}_{10},\mbox{negated\_conjecture})$

$p_2(x_9,x_{10}) \ \Rightarrow \ p_2(f_4(x_9),f_4(x_{10}))\qquad\mbox{cnf}(\mbox{p2}_{11},\mbox{negated\_conjecture})$

$( p_2(x_4,x_5)\mbox{ and }p_2(x_4,x_6) ) \ \Rightarrow \ p_2(x_5,x_6)\qquad\mbox{cnf}(\mbox{p2}_{12},\mbox{negated\_conjecture})$

$( p_6(x_{19},x_{20})\mbox{ and }p_6(x_{19},x_{21}) ) \ \Rightarrow \ p_6(x_{20},x_{21})\qquad\mbox{cnf}(\mbox{p6}_{13},\mbox{negated\_conjecture})$

$( p_2(x_2,x_0)\mbox{ and }p_2(x_3,x_1)\mbox{ and }p_{10}(x_2,x_3) ) \ \Rightarrow \ p_{10}(x_0,x_1)\qquad\mbox{cnf}(\mbox{p10}_{14},\mbox{negated\_conjecture})$

$p_2(f_7(c_{11}),f_3(f_4(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(c_{12})))))))))))))))))))\qquad\mbox{cnf}(\mbox{p2}_{15},\mbox{negated\_conjecture})$

$p_{10}(f_7(c_{11}),f_3(f_4(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(f_5(c_{12})))))))))))))))))))\qquad\mbox{cnf}(\mbox{p10}_{16},\mbox{negated\_conjecture})$

\medskip\textbf{SYN632-1.p} Harrison problem 4144






$p_{11}(x_0,x_0)\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{negated\_conjecture})$

$p_9(x_{70},x_{70})\qquad\mbox{cnf}(\mbox{p9}_2,\mbox{negated\_conjecture})$

$p_8(x_{67},x_{67})\qquad\mbox{cnf}(\mbox{p8}_3,\mbox{negated\_conjecture})$

$p_6(x_{60},x_{60})\qquad\mbox{cnf}(\mbox{p6}_4,\mbox{negated\_conjecture})$

$p_4(x_{53},x_{53})\qquad\mbox{cnf}(\mbox{p4}_5,\mbox{negated\_conjecture})$

$p_3(x_{46},x_{46})\qquad\mbox{cnf}(\mbox{p3}_6,\mbox{negated\_conjecture})$

$p_2(x_{27},x_{27})\qquad\mbox{cnf}(\mbox{p2}_7,\mbox{negated\_conjecture})$

$p_{19}(x_{24},x_{24})\qquad\mbox{cnf}(\mbox{p19}_8,\mbox{negated\_conjecture})$

$p_{17}(x_{17},x_{17})\qquad\mbox{cnf}(\mbox{p17}_9,\mbox{negated\_conjecture})$

$p_{15}(x_{10},x_{10})\qquad\mbox{cnf}(\mbox{p15}_{10},\mbox{negated\_conjecture})$

$p_{13}(x_3,x_3)\qquad\mbox{cnf}(\mbox{p13}_{11},\mbox{negated\_conjecture})$

$( p_{11}(x_0,x_1)\mbox{ and }p_{11}(x_0,x_2) ) \ \Rightarrow \ p_{11}(x_1,x_2)\qquad\mbox{cnf}(\mbox{p11}_{12},\mbox{negated\_conjecture})$

$( p_9(x_{70},x_{71})\mbox{ and }p_9(x_{70},x_{72}) ) \ \Rightarrow \ p_9(x_{71},x_{72})\qquad\mbox{cnf}(\mbox{p9}_{13},\mbox{negated\_conjecture})$

$( p_8(x_{67},x_{68})\mbox{ and }p_8(x_{67},x_{69}) ) \ \Rightarrow \ p_8(x_{68},x_{69})\qquad\mbox{cnf}(\mbox{p8}_{14},\mbox{negated\_conjecture})$

$( p_6(x_{60},x_{61})\mbox{ and }p_6(x_{60},x_{62}) ) \ \Rightarrow \ p_6(x_{61},x_{62})\qquad\mbox{cnf}(\mbox{p6}_{15},\mbox{negated\_conjecture})$

$( p_4(x_{53},x_{54})\mbox{ and }p_4(x_{53},x_{55}) ) \ \Rightarrow \ p_4(x_{54},x_{55})\qquad\mbox{cnf}(\mbox{p4}_{16},\mbox{negated\_conjecture})$

$( p_3(x_{46},x_{47})\mbox{ and }p_3(x_{46},x_{48}) ) \ \Rightarrow \ p_3(x_{47},x_{48})\qquad\mbox{cnf}(\mbox{p3}_{17},\mbox{negated\_conjecture})$

$( p_2(x_{27},x_{28})\mbox{ and }p_2(x_{27},x_{29}) ) \ \Rightarrow \ p_2(x_{28},x_{29})\qquad\mbox{cnf}(\mbox{p2}_{18},\mbox{negated\_conjecture})$

$( p_{19}(x_{24},x_{25})\mbox{ and }p_{19}(x_{24},x_{26}) ) \ \Rightarrow \ p_{19}(x_{25},x_{26})\qquad\mbox{cnf}(\mbox{p19}_{19},\mbox{negated\_conjecture})$

$( p_{17}(x_{17},x_{18})\mbox{ and }p_{17}(x_{17},x_{19}) ) \ \Rightarrow \ p_{17}(x_{18},x_{19})\qquad\mbox{cnf}(\mbox{p17}_{20},\mbox{negated\_conjecture})$

$( p_{15}(x_{10},x_{11})\mbox{ and }p_{15}(x_{10},x_{12}) ) \ \Rightarrow \ p_{15}(x_{11},x_{12})\qquad\mbox{cnf}(\mbox{p15}_{21},\mbox{negated\_conjecture})$

$( p_{13}(x_3,x_4)\mbox{ and }p_{13}(x_3,x_5) ) \ \Rightarrow \ p_{13}(x_4,x_5)\qquad\mbox{cnf}(\mbox{p13}_{22},\mbox{negated\_conjecture})$

$( p_3(x_{37},x_{35})\mbox{ and }p_4(x_{36},x_{34})\mbox{ and }p_{21}(x_{36},x_{37}) ) \ \Rightarrow \ p_{21}(x_{34},x_{35})\qquad\mbox{cnf}(\mbox{p21}_{23},\mbox{negated\_conjecture})$

$p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{28},f_{12}(c_{25},c_{26}))),c_{22}),c_{23}),c_{27})\qquad\mbox{cnf}(\mbox{p21}_{24},\mbox{negated\_conjecture})$

$( p_{11}(x_{73},x_{75})\mbox{ and }p_9(x_{74},x_{76}) ) \ \Rightarrow \ p_9(f_{12}(x_{73},x_{74}),f_{12}(x_{75},x_{76}))\qquad\mbox{cnf}(\mbox{p9}_{25},\mbox{negated\_conjecture})$

$( p_{15}(x_6,x_8)\mbox{ and }p_2(x_7,x_9) ) \ \Rightarrow \ p_{13}(f_{16}(x_6,x_7),f_{16}(x_8,x_9))\qquad\mbox{cnf}(\mbox{p13}_{26},\mbox{negated\_conjecture})$

$( p_{17}(x_{13},x_{15})\mbox{ and }p_3(x_{14},x_{16}) ) \ \Rightarrow \ p_{15}(f_{18}(x_{13},x_{14}),f_{18}(x_{15},x_{16}))\qquad\mbox{cnf}(\mbox{p15}_{27},\mbox{negated\_conjecture})$

$( p_{19}(x_{20},x_{22})\mbox{ and }p_9(x_{21},x_{23}) ) \ \Rightarrow \ p_{17}(f_{20}(x_{20},x_{21}),f_{20}(x_{22},x_{23}))\qquad\mbox{cnf}(\mbox{p17}_{28},\mbox{negated\_conjecture})$

$( p_2(x_{31},x_{33})\mbox{ and }p_6(x_{30},x_{32}) ) \ \Rightarrow \ p_2(f_7(x_{30},x_{31}),f_7(x_{32},x_{33}))\qquad\mbox{cnf}(\mbox{p2}_{29},\mbox{negated\_conjecture})$

$( p_{13}(x_{49},x_{51})\mbox{ and }p_3(x_{50},x_{52}) ) \ \Rightarrow \ p_3(f_{14}(x_{49},x_{50}),f_{14}(x_{51},x_{52}))\qquad\mbox{cnf}(\mbox{p3}_{30},\mbox{negated\_conjecture})$

$( p_2(x_{56},x_{58})\mbox{ and }p_3(x_{57},x_{59}) ) \ \Rightarrow \ p_4(f_5(x_{56},x_{57}),f_5(x_{58},x_{59}))\qquad\mbox{cnf}(\mbox{p4}_{31},\mbox{negated\_conjecture})$

$( p_8(x_{63},x_{65})\mbox{ and }p_9(x_{64},x_{66}) ) \ \Rightarrow \ p_6(f_{10}(x_{63},x_{64}),f_{10}(x_{65},x_{66}))\qquad\mbox{cnf}(\mbox{p6}_{32},\mbox{negated\_conjecture})$

$p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{25},c_{26})),x_{42}),x_{43}),x_{44}) \ \Rightarrow \ p_3(x_{43},x_{44})\qquad\mbox{cnf}(\mbox{p3}_{33},\mbox{negated\_conjecture})$

$p_3(x_{43},x_{44}) \ \Rightarrow \ p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{25},c_{26})),x_{42}),x_{43}),x_{44})\qquad\mbox{cnf}(\mbox{p21}_{34},\mbox{negated\_conjecture})$

$p_{21}(f_5(c_{22},c_{23}),x_{77}) \ \Rightarrow \ \neg\,p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{25},c_{26})),c_{22}),x_{77}),c_{27})\qquad\mbox{cnf}(\mbox{not\_p21}_{35},\mbox{negated\_conjecture})$

$p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{28},x_{39})),x_{38}),x_{40}),x_{41}) \ \Rightarrow \ p_{21}(f_5(x_{38},f_{14}(f_{16}(f_{18}(f_{20}(c_{29},x_{39}),x_{40}),x_{38}),x_{41})),x_{41})\qquad\mbox{cnf}(\mbox{p21}_{36},\mbox{negated\_conjecture})$

$( p_{21}(f_5(x_{38},x_{45}),x_{41})\mbox{ and }p_{21}(f_5(f_7(f_{10}(c_{24},x_{39}),x_{38}),x_{40}),x_{45}) ) \ \Rightarrow \ p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{28},x_{39})),x_{38}),x_{40}),x_{41})\qquad\mbox{cnf}(\mbox{p21}_{37},\mbox{negated\_conjecture})$

$p_{21}(f_5(f_7(f_{10}(c_{24},f_{12}(c_{28},x_{39})),x_{38}),x_{40}),x_{41}) \ \Rightarrow \ p_{21}(f_5(f_7(f_{10}(c_{24},x_{39}),x_{38}),x_{40}),f_{14}(f_{16}(f_{18}(f_{20}(c_{29},x_{39}),x_{40}),x_{38}),x_{41}))\qquad\mbox{cnf}(\mbox{p21}_{38},\mbox{negated\_conjecture})$

\medskip\textbf{SYN190-1.p} RPT63 synthetic problem 88 (quasi-uniform distribution)






$\mbox{include}(\mbox{'Axioms/SYN001-0.ax'})$

$\neg\,r_3(a,x,d)\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{SYN404+1.p} Kalish and Montague Problem 238

Laws of alphabetic variance.




$\forall x{:}\;f(x) \ \Rightarrow \ \exists y{:}\;f(y)\qquad\mbox{fof}(\mbox{kalish}_{238},\mbox{conjecture})$

\medskip\textbf{SYN918+1.p} From Smullyan






$( \forall x{:}\;( ( ( f(x)\mbox{ and }g(x) ) \ \Rightarrow \ h(x) ) \ \Rightarrow \ \exists y{:}\;( f(y)\mbox{ and }\neg\,g(y) ) )\mbox{ and }( \forall w{:}\;( f(w) \ \Rightarrow \ g(w) )\mbox{ or }\forall z{:}\;( f(z) \ \Rightarrow \ h(z) ) ) ) \ \Rightarrow \ ( \forall r{:}\;( ( f(r)\mbox{ and }h(r) ) \ \Rightarrow \ g(r) ) \ \Rightarrow \ \exists v{:}\;( f(v)\mbox{ and }g(v)\mbox{ and }\neg\,h(v) ) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SYN305-1.p} Problem for testing satisfiability






$f(g_1(x)) = x\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(g_2(x)) = x\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$g_1(x)\ne g_2(x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip
\end{document}