\documentclass{amsart}
    \advance\textheight by 1.75in
    \advance\textwidth by 2in
    \advance\topmargin by -1in
    \advance\oddsidemargin by -1in
    \advance\evensidemargin by -1in
    \usepackage{amssymb}
    \parindent0pt
    \begin{document}
    \textbf{ARI691=1.p} Rewrite modulo commutativity of multiplication






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

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

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

$\mbox{\$product}(a,b) = f(\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$f(f(\mbox{\$product}(a,b))) = \mbox{\$product}(b,a)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI132=1.p} Integer: - - 2 is 2






$\mbox{\$uminus}(\mbox{\$uminus}(2)) = 2\qquad\mbox{tff}(\mbox{uminus\_uminus}_2,\mbox{conjecture})$

\medskip\textbf{ARI355=1.p} Real: Something less than 0.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(x,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{real\_less\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI696=1.p} Expand and rewrite polynomial






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(a,a),\mbox{\$product}(\mbox{\$product}(\mbox{\$sum}(a,b),\mbox{\$sum}(\mbox{\$difference}(c,d),1)),\mbox{\$difference}(a,2))),b),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),d)),\mbox{\$sum}(2,c)))) = 0\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(d,b),a),\mbox{\$product}(c,b)),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(2,\mbox{\$product}(a,a)),b),\mbox{\$product}(b,a)),\mbox{\$product}(d,b)),\mbox{\$product}(2,c)),\mbox{\$product}(-1,\mbox{\$product}(c,a))),\mbox{\$product}(c,c)),\mbox{\$product}(\mbox{\$product}(c,a),a)),d),\mbox{\$product}(2,\mbox{\$product}(d,a))),\mbox{\$product}(d,b)),\mbox{\$product}(\mbox{\$product}(c,b),a)),\mbox{\$product}(d,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(d,a),a))),d))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI672=1.p} Prove that a*b = 1 and a = b \& (a = 1 | a = -1) are equivalent






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

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

$\mbox{\$product}(a,b) = 1\iff ( ( a = -1\mbox{ or }a = 1 )\mbox{ and }a = b )\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI459=1.p} Real: -3.25 is - 3.25






$-\mbox{3.}_{25} = \mbox{\$uminus}(\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI017=1.p} Integer: Something lesseq to 13






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(x,13)\qquad\mbox{tff}(\mbox{something\_lesseq}_{13},\mbox{conjecture})$

\medskip\textbf{ARI660=1.p} Prove that a*a*a $>$= 11 and a $>$= 3 are equivalent






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

$\mbox{\$lesseq}(11,\mbox{\$product}(\mbox{\$product}(a,a),a))\iff \mbox{\$lesseq}(3,a)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI071=1.p} Integer: Sum 2 and -5 is something






$\exists y{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(2,-5) = y\qquad\mbox{tff}(\mbox{sum\_2\_n5\_what},\mbox{conjecture})$

\medskip\textbf{ARI225=1.p} Rational: -33/4 not greater than 33/4






$\neg\,\mbox{\$greater}(-33/4,33/4)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI244=1.p} Rational: 2/5 not equal to 1/16






$2/5\ne 1/16\qquad\mbox{tff}(\mbox{rat\_not\_equal\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI383=1.p} Real: -32500.0 greater than something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(-\mbox{32500.}_0,x)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI126=1.p} Integer: Product of something and itself is not 0






$\exists x{:}\;\mbox{\$int}{:}\;( x\ne 0\mbox{ and }\mbox{\$product}(x,x) = 0 )\qquad\mbox{tff}(\mbox{anti\_product\_what\_what}_0,\mbox{conjecture})$

\medskip\textbf{ARI039=1.p} Integer: Something greater than -5






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(x,-5)\qquad\mbox{tff}(\mbox{something\_greater\_n}_5,\mbox{conjecture})$

\medskip\textbf{ARI617=1.p} Two different definitions of absolute value agree






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

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

$( \forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,f(x))\mbox{ and }\mbox{\$lesseq}(\mbox{\$uminus}(x),f(x))\mbox{ and }( \mbox{\$lesseq}(f(x),x)\mbox{ or }\mbox{\$lesseq}(f(x),\mbox{\$uminus}(x)) ) )\mbox{ and }\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(0,g(x))\mbox{ and }( g(x) = x\mbox{ or }g(x) = \mbox{\$uminus}(x) ) ) ) \ \Rightarrow \ \forall x{:}\;\mbox{\$int}{:}\;f(x) = g(x)\qquad\mbox{tff}(\mbox{absolute\_value\_defs},\mbox{conjecture})$

\medskip\textbf{ARI057=1.p} Integer: Sum 2 and 3 is 5






$\mbox{\$sum}(2,3) = 5\qquad\mbox{tff}(\mbox{sum\_2\_3}_5,\mbox{conjecture})$

\medskip\textbf{ARI349=1.p} Real: 7.0 less than something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(\mbox{7.}_0,x)\qquad\mbox{tff}(\mbox{real\_less\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI303=1.p} Rational: Product -15/16 and -4/5 is 3/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(-15/16,-4/5) = 3/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{20},\mbox{conjecture})$

\medskip\textbf{ARI176=1.p} Integer: Formula equals 22






$\exists u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(\mbox{\$product}(3,u),\mbox{\$product}(5,v)) = 22\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI203=1.p} Rational: 5/12 lesseq to 5/12






$\mbox{\$lesseq}(5/12,5/12)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI432=1.p} Real: Difference 16.05 and only 4.0 is 12.05






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$difference}(\mbox{16.}_{05},x) = \mbox{12.}_{05} \ \Rightarrow \ x = \mbox{4.}_0 )\qquad\mbox{tff}(\mbox{real\_difference\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI529=1.p} Mixed: Mad mixture 2






$\mbox{\$less}(\mbox{\$to\_int}(\mbox{\$difference}(\mbox{10.}_0,\mbox{0.}_{0001})),\mbox{\$product}(\mbox{\$to\_int}(11/2),2))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{34},\mbox{conjecture})$

\medskip\textbf{ARI520=1.p} Mixed: Sum 2 and 3 is integer






$\mbox{\$is\_int}(\mbox{\$sum}(2,3))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{25},\mbox{conjecture})$

\medskip\textbf{ARI218=1.p} Rational: 3/4 not greater 7/8






$\neg\,\mbox{\$greater}(3/4,7/8)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI290=1.p} Rational: Product -3/40 and -12/1 is 9/10






$\mbox{\$product}(-3/40,-12/1) = 9/10\qquad\mbox{tff}(\mbox{rat\_product\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI605=1.p} If f(X) $>$ X, then a + b $<$ f(a) + b $<$ f(a) + f(b)






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

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

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

$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(x),x) \ \Rightarrow \ \exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(\mbox{\$sum}(a,b),y)\mbox{ and }\mbox{\$less}(y,\mbox{\$sum}(f(a),f(b))) )\qquad\mbox{tff}(\mbox{fX\_gt\_X\_implies\_f\_a\_b},\mbox{conjecture})$

\medskip\textbf{ARI298=1.p} Rational: Product 5/8 and only 2/5 is 1/4






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$product}(5/8,x) = 1/4 \ \Rightarrow \ x = 2/5 )\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI099=1.p} Integer: Product of 11 and 11 is 121






$\mbox{\$product}(11,11) = 121\qquad\mbox{tff}(\mbox{product\_11\_11}_{121},\mbox{conjecture})$

\medskip\textbf{ARI395=1.p} Real: -8.69 not greatereq to -3.25






$\neg\,\mbox{\$greatereq}(-\mbox{8.}_{69},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI028=1.p} Integer: 4 greater than 3






$\mbox{\$greater}(4,3)\qquad\mbox{tff}(\mbox{prove\_4\_greater}_3,\mbox{conjecture})$

\medskip\textbf{ARI707=1.p} Simple rewriting: d*d + c = 2*d*d implies c*d*d = c*c






$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$product}(d,d),c) = \mbox{\$product}(\mbox{\$product}(2,d),d)\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(c,d),d) = \mbox{\$product}(c,c)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI256=1.p} Rational: Sum of 7/2 and 81/20 is not 11/2






$\mbox{\$sum}(7/2,81/20)\ne 11/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI411=1.p} Real: Sum 4.25 and only 5.75 is 10.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(\mbox{4.}_{25},x) = \mbox{10.}_0 \ \Rightarrow \ x = \mbox{5.}_{75} )\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI586=1.p} For positive X, there is a Y between X and 3X (e.g., Y = 2X)






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,x) \ \Rightarrow \ \exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,y)\mbox{ and }\mbox{\$less}(y,\mbox{\$product}(x,3)) ) )\qquad\mbox{tff}(\mbox{exists\_Y\_between\_X\_and\_3X},\mbox{conjecture})$

\medskip\textbf{ARI044=1.p} Integer: Something greatereq to 15






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(x,15)\qquad\mbox{tff}(\mbox{something\_greatereq}_{15},\mbox{conjecture})$

\medskip\textbf{ARI447=1.p} Real: Product 7.25 and 4.0 is only 29.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(\mbox{7.}_{25},\mbox{4.}_0) = x \ \Rightarrow \ x = \mbox{29.}_0 )\qquad\mbox{tff}(\mbox{real\_product\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI732=1.p} If X is an integer and 5*X+Y is an integer, then Y is an integer






$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( ( \mbox{\$is\_int}(x)\mbox{ and }\mbox{\$is\_int}(\mbox{\$sum}(\mbox{\$product}(\mbox{5.}_0,x),y)) ) \ \Rightarrow \ \mbox{\$is\_int}(y) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI300=1.p} Rational: Product 1/4 and 18/25 is -9/50






$\mbox{\$product}(1/4,-18/25) = -9/50\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI648=1.p} 2*a $>$ 0 | -4*b + 2*a $<$ 8 implies a $>$= 1 | a-2*b$\leq$3 | 20*a-30*b=7






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

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

$\mbox{\$greater}(8,\mbox{\$difference}(\mbox{\$product}(2,a),\mbox{\$product}(4,b)))\mbox{ or }\mbox{\$less}(0,\mbox{\$product}(2,a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(20,a),\mbox{\$product}(-1,\mbox{\$product}(30,b))) = 7\mbox{ or }\mbox{\$greatereq}(3,\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(2,b))))\mbox{ or }\mbox{\$lesseq}(1,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI062=1.p} Integer: Sum 2 and 3 is not 6






$\mbox{\$sum}(2,3)\ne 6\qquad\mbox{tff}(\mbox{sum\_2\_3\_not}_6,\mbox{conjecture})$

\medskip\textbf{ARI002=1.p} Integer: 3 not less than 2






$\neg\,\mbox{\$less}(3,2)\qquad\mbox{tff}(\mbox{prove\_3\_not\_less}_2,\mbox{conjecture})$

\medskip\textbf{ARI461=1.p} Real: Sum 11.38 and - 11.38 is 0.0






$\mbox{\$sum}(\mbox{11.}_{38},\mbox{\$uminus}(\mbox{11.}_{38})) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI621=2.p} 12 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$rat}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{pow}_2(x)\iff x = 1/( 1\mbox{ or }( \mbox{\$lesseq}(2/1,x)\mbox{ and }\exists y{:}\;\mbox{\$rat}{:}\;( \mbox{\$product}(2/1,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(12/1)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_{10},\mbox{conjecture})$

\medskip\textbf{ARI727=1.p} Every integer is greater than 3.8 or less than 3.2






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$is\_int}(x) \ \Rightarrow \ ( \mbox{\$greater}(x,\mbox{3.}_8)\mbox{ or }\mbox{\$less}(x,\mbox{3.}_2) ) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI396=1.p} Real: Something greatereq to 0.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(x,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI585=1.p} Interval (X+5,X+8) is covered by (Y,Y+4), e.g. for Y = X + 5






$\forall x{:}\;\mbox{\$int}{:}\;\exists y{:}\;\mbox{\$int}{:}\;\forall z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(\mbox{\$sum}(x,5),z)\mbox{ and }\mbox{\$less}(z,\mbox{\$sum}(x,8)) ) \ \Rightarrow \ ( \mbox{\$less}(y,z)\mbox{ and }\mbox{\$less}(z,\mbox{\$sum}(y,4)) ) )\qquad\mbox{tff}(\mbox{interv\_4\_can\_cover\_interv}_3,\mbox{conjecture})$

\medskip\textbf{ARI053=1.p} Integer: Something greatereq to -5






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(x,-5)\qquad\mbox{tff}(\mbox{something\_greatereq\_n}_5,\mbox{conjecture})$

\medskip\textbf{ARI348=1.p} Real: Something less than 12.8






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(x,\mbox{12.}_8)\qquad\mbox{tff}(\mbox{real\_less\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI162=1.p} Integer: Sum is 8 and difference is 0






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8\mbox{ and }\mbox{\$difference}(x,y) = 0 )\qquad\mbox{tff}(\mbox{sum\_and\_difference},\mbox{conjecture})$

\medskip\textbf{ARI600=1.p} Inequations imply a+1 = b-1, hence f(a+1,b-1) $\leq$ f(b-1,a+1) + 1






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

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

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

$( \mbox{\$lesseq}(a,\mbox{\$sum}(b,2))\mbox{ and }\mbox{\$lesseq}(b,\mbox{\$difference}(a,2)) ) \ \Rightarrow \ \mbox{\$lesseq}(f(\mbox{\$sum}(a,1),\mbox{\$difference}(b,1)),\mbox{\$sum}(1,f(\mbox{\$difference}(b,1),\mbox{\$sum}(a,1))))\qquad\mbox{tff}(\mbox{ineq\_imply\_f\_leq\_fplus}_1,\mbox{conjecture})$

\medskip\textbf{ARI709=1.p} Simple rewriting: a * 1 = 3 implies a = 3






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

$\mbox{\$product}(1,a) = 3\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$a = 3\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI251=1.p} Rational: Sum of 17/4 and 2/1 is 25/4






$\mbox{\$sum}(17/4,2/1) = 25/4\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI735=1.p} Verification example






$\forall i{:}\;\mbox{\$int}{:}\;( \mbox{\$greatereq}(i,0) \ \Rightarrow \ ( ( i = 0 \ \Rightarrow \ \mbox{\$true} )\mbox{ and }( i\ne 0 \ \Rightarrow \ \exists i_1{:}\;\mbox{\$int}{:}\;( i_1 = \mbox{\$difference}(i,1)\mbox{ and }\mbox{\$greatereq}(i_1,0) ) ) ) )\qquad\mbox{tff}(\mbox{formula},\mbox{conjecture})$

\medskip\textbf{ARI174=1.p} Integer: Formula equals 24






$\exists u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(\mbox{\$product}(3,u),\mbox{\$product}(5,v)) = 24\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI128=1.p} Integer: -2 equal - 2






$-2 = \mbox{\$uminus}(2)\qquad\mbox{tff}(\mbox{n2\_equal\_uminus}_2,\mbox{conjecture})$

\medskip\textbf{ARI626=1.p} Overflow checking on the integers

A simple test that should go over 2$\wedge$64 (more than machine 
 integers) and therefore detect whether the prover uses arbitrary 
 precision arithmetic.




$\exists x{:}\;\mbox{\$int}{:}\;( x = \mbox{\$sum}(18446744073709551616,18446744073709551616)\mbox{ and }\mbox{\$greater}(x,0) )\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI214=1.p} Rational: Something lesseq to 0/1






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(x,0/1)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI163=1.p} Integer: Between -1 and 1 must be 0






$\forall x{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(-1,x)\mbox{ and }\mbox{\$less}(x,1) ) \ \Rightarrow \ \mbox{\$sum}(21,x) = 21 )\qquad\mbox{tff}(\mbox{sum\_something\_0\_samething},\mbox{conjecture})$

\medskip\textbf{ARI023=1.p} Integer: 2 not lesseq to -2






$\neg\,\mbox{\$lesseq}(2,-2)\qquad\mbox{tff}(\mbox{prove\_2\_not\_lesseq\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI633=1.p} Example 7






$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$w{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{w\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(f(x),\mbox{1.}_0)\qquad\mbox{tff}(\mbox{f\_less\_equal}_1,\mbox{axiom})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(w,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(u,\mbox{\$product}(w,f(x))),\mbox{\$sum}(v,w))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI700=1.p} Solve a simple system of nonlinear equations






$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

$\mbox{\$product}(\mbox{\$product}(98164184,y),y) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(6472,x),y) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$product}(5235848,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(123,x),x),a)) = \mbox{\$product}(12270523,\mbox{\$product}(\mbox{\$product}(123,b),b))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI277=1.p} Rational: Difference 4/7 and -3/7 is 1/1






$\mbox{\$difference}(4/7,-3/7) = 1/1\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI519=1.p} Mixed: Something is greatereq to 9 coerced to real






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(x,\mbox{\$to\_real}(9))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{24},\mbox{conjecture})$

\medskip\textbf{ARI048=1.p} Integer: 4 greatereq to -6






$\mbox{\$greatereq}(4,-6)\qquad\mbox{tff}(\mbox{prove\_4\_greatereq\_n}_6,\mbox{conjecture})$

\medskip\textbf{ARI112=1.p} Integer: Product of 2 and -5 is something






$\exists y{:}\;\mbox{\$int}{:}\;\mbox{\$product}(2,-5) = y\qquad\mbox{tff}(\mbox{product\_2\_n5\_what},\mbox{conjecture})$

\medskip\textbf{ARI010=1.p} Integer: -2 not less than -4






$\neg\,\mbox{\$less}(-2,-4)\qquad\mbox{tff}(\mbox{n2\_not\_less\_n}_4,\mbox{conjecture})$

\medskip\textbf{ARI546=1.p} Integer: Difference -4 and sum 0 and -3 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(-4,\mbox{\$sum}(0,-3)) = x\qquad\mbox{tff}(\mbox{int\_combined\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI027=1.p} Integer: -2 lesseq to something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(-2,x)\qquad\mbox{tff}(\mbox{n2\_lesseq\_something},\mbox{conjecture})$

\medskip\textbf{ARI194=1.p} Rational: 13/24 less than something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(13/24,x)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI377=1.p} Real: Something greater than something else






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(x,y)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI444=1.p} Real: Product something and 5.5 is 30.25






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(x,\mbox{5.}_5) = \mbox{30.}_{25}\qquad\mbox{tff}(\mbox{real\_product\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI016=1.p} Integer: 3 not lesseq to 2






$\neg\,\mbox{\$lesseq}(3,2)\qquad\mbox{tff}(\mbox{prove\_3\_not\_lesseq}_2,\mbox{conjecture})$

\medskip\textbf{ARI703=1.p} Sum-of-squares decomposition






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$t{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{t\_type},\mbox{type})$

$u{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$w{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{w\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(x,x),\mbox{\$product}(y,y)),\mbox{\$product}(z,z)),\mbox{\$product}(t,t)),\mbox{\$product}(u,u)),\mbox{\$product}(v,v)),\mbox{\$product}(w,w)) = 1234567890987654321\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

\medskip\textbf{ARI254=1.p} Rational: Sum of something and 3/16 is 1/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,3/16) = 1/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI728=1.p} If 2*X is an integer, then X is an integer or X+0.5 is an integer






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$is\_int}(\mbox{\$product}(\mbox{2.}_0,x)) \ \Rightarrow \ ( \mbox{\$is\_int}(x)\mbox{ or }\mbox{\$is\_int}(\mbox{\$sum}(x,\mbox{0.}_5)) ) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI122=1.p} Integer: Associative product






$\forall z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( ( \mbox{\$product}(2,3) = z_1\mbox{ and }\mbox{\$product}(z_1,6) = z_2\mbox{ and }\mbox{\$product}(3,6) = z_3\mbox{ and }\mbox{\$product}(2,z_3) = z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{tff}(\mbox{associative\_product},\mbox{conjecture})$

\medskip\textbf{ARI688=1.p} Verify gcd computation






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

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

$\mbox{\$product}(98164184,x) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(6472,x) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(8,x),\mbox{\$product}(-1,\mbox{\$difference}(\mbox{\$product}(4353078,b),\mbox{\$product}(287,a)))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI494=1.p} Real: (Something * 3.2) + -0.75 is less than -12.8






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(x,\mbox{3.}_2),-\mbox{0.}_{75}),-\mbox{12.}_8)\qquad\mbox{tff}(\mbox{real\_combined\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI167=1.p} Integer: -4 * (127 - 99) $<$ (- 3) + 27






$\mbox{\$less}(\mbox{\$product}(-4,\mbox{\$difference}(127,99)),\mbox{\$sum}(\mbox{\$uminus}(3),27))\qquad\mbox{tff}(\mbox{complex}_2,\mbox{conjecture})$

\medskip\textbf{ARI621=1.p} 12 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$int}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{pow}_2(x)\iff ( x = 1\mbox{ or }( \mbox{\$lesseq}(2,x)\mbox{ and }\exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(12)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_{10},\mbox{conjecture})$

\medskip\textbf{ARI015=1.p} Integer: 2 lesseq to 3






$\mbox{\$lesseq}(2,3)\qquad\mbox{tff}(\mbox{prove\_2\_lesseq}_3,\mbox{conjecture})$

\medskip\textbf{ARI340=1.p} Rational: 15/2 is less than sum of 29/10 and 24/5






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(29/10,24/5) = x \ \Rightarrow \ \mbox{\$less}(15/2,x) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI424=1.p} Real: Sum something and 0.0 is itself






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,\mbox{0.}_0) = x\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{24},\mbox{conjecture})$

\medskip\textbf{ARI645=1.p} Prove that d $>$= 0, b $>$= a, a $>$ 0 imply d / b $\leq$ d / a






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

$b{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{a\_type}_{001},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{a\_type}_{002},\mbox{type})$

$( \mbox{\$greatereq}(d,0)\mbox{ and }\mbox{\$greatereq}(b,a)\mbox{ and }\mbox{\$greater}(a,0) ) \ \Rightarrow \ \mbox{\$lesseq}(\mbox{\$quotient\_e}(d,b),\mbox{\$quotient\_e}(d,a))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI456=1.p} Real: Product 0.08 and -14.25 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(\mbox{0.}_{08},-\mbox{14.}_{25}) = x\qquad\mbox{tff}(\mbox{real\_product\_problem}_{21},\mbox{conjecture})$

\medskip\textbf{ARI670=1.p} Prove that a*a $>$= a






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$lesseq}(a,\mbox{\$product}(a,a))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI291=1.p} Rational: Product 11/2 and 11/2 is 121/4






$\mbox{\$product}(11/2,11/2) = 121/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI653=1.p} Prove that 5*a $>$= 1, 7*a $\leq$ 6 are unsat






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

$\mbox{\$greatereq}(6,\mbox{\$product}(7,a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(1,\mbox{\$product}(5,a))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

\medskip\textbf{ARI051=1.p} Integer: -5 not greatereq to -3






$\neg\,\mbox{\$greatereq}(-5,-3)\qquad\mbox{tff}(\mbox{n5\_not\_greatereq\_n}_3,\mbox{conjecture})$

\medskip\textbf{ARI576=1.p} Inequation system is solvable (e.g., X = 10)






$\exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(\mbox{\$product}(2,x),21)\mbox{ and }\mbox{\$lesseq}(29,\mbox{\$product}(3,x)) )\qquad\mbox{tff}(\mbox{ineq\_sys\_solvable}_1,\mbox{conjecture})$

\medskip\textbf{ARI079=1.p} Integer: Sum something and another thing is the first thing






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,y) = x\qquad\mbox{tff}(\mbox{sum\_something\_anotherthing\_firstthing},\mbox{conjecture})$

\medskip\textbf{ARI104=1.p} Integer: Product of 2 and 3 is only 6






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,3) = x \ \Rightarrow \ x = 6 )\qquad\mbox{tff}(\mbox{product\_2\_3\_only}_6,\mbox{conjecture})$

\medskip\textbf{ARI656=1.p} a $>$= 0, b $>$= c imply a*b $>$= a*c






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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

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

$\mbox{\$lesseq}(c,b)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(0,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(\mbox{\$product}(a,c),\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI684=1.p} Expand polynomial a * (a + b) * c






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$product}(\mbox{\$product}(a,\mbox{\$sum}(a,b)),c) = \mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(a,a),c),\mbox{\$product}(\mbox{\$product}(a,b),c))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI102=1.p} Integer: Product of 11 and something is 121






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(11,x) = 121\qquad\mbox{tff}(\mbox{product\_11\_something}_{121},\mbox{conjecture})$

\medskip\textbf{ARI685=1.p} Expand and rewrite polynomial






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(a,a),\mbox{\$product}(\mbox{\$product}(\mbox{\$sum}(a,b),\mbox{\$sum}(\mbox{\$sum}(c,d),1)),\mbox{\$sum}(a,2))),b),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),d),\mbox{\$sum}(2,c)))) = 0\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(d,b),a),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$product}(-2,\mbox{\$product}(a,a)),\mbox{\$product}(-1,\mbox{\$product}(c,a))),b),\mbox{\$product}(-1,\mbox{\$product}(b,a))),\mbox{\$product}(2,c)),\mbox{\$product}(-1,\mbox{\$product}(c,b))),\mbox{\$product}(c,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(c,a),a))),\mbox{\$product}(-1,\mbox{\$product}(2,\mbox{\$product}(d,a)))),\mbox{\$product}(-1,\mbox{\$product}(2,\mbox{\$product}(d,b)))),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(c,b),a))),\mbox{\$product}(d,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(d,a),a))),\mbox{\$product}(2,d)))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI196=1.p} Rational: -1/4 less than 1/4






$\mbox{\$less}(-1/4,1/4)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI372=1.p} Real: 3.0 greater than 2.5






$\mbox{\$greater}(\mbox{3.}_0,\mbox{2.}_5)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI052=1.p} Integer: Something greatereq to 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(x,0)\qquad\mbox{tff}(\mbox{something\_greatereq}_0,\mbox{conjecture})$

\medskip\textbf{ARI359=1.p} Real: 3.25 lesseq to 7.8






$\mbox{\$lesseq}(\mbox{3.}_{25},\mbox{7.}_8)\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI547=1.p} Integer: Product something and 3 is 27 means 8 less than that






$\exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,3) = 27\mbox{ and }\mbox{\$less}(8,x) )\qquad\mbox{tff}(\mbox{int\_combined\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI601=1.p} If f(X) $>$ X, then 3 $<$ a implies 4 $<$ a+1 $<$ f(a+1)






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

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

$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(x),x) \ \Rightarrow \ ( \mbox{\$less}(3,a) \ \Rightarrow \ \mbox{\$less}(4,f(\mbox{\$sum}(a,1))) )\qquad\mbox{tff}(\mbox{fX\_gt\_X\_implies\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI223=1.p} Rational: 13/121 greater than -13/121






$\mbox{\$greater}(13/121,-13/121)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI042=1.p} Integer: 4 greatereq to 3






$\mbox{\$greatereq}(4,3)\qquad\mbox{tff}(\mbox{prove\_4\_greatereq}_3,\mbox{conjecture})$

\medskip\textbf{ARI252=1.p} Rational: Sum of 7/2 and 1/2 is 4/1






$\mbox{\$sum}(7/2,1/2) = 4/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI730=1.p} If X =$<$ 7.2 and X is an integer, then X =$<$ 7






$\forall x{:}\;\mbox{\$real}{:}\;( ( \mbox{\$is\_int}(x)\mbox{ and }\mbox{\$lesseq}(x,\mbox{7.}_2) ) \ \Rightarrow \ \mbox{\$lesseq}(x,\mbox{7.}_0) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI296=1.p} Rational: Product 5/8 and 2/5 is only 1/4






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$product}(5/8,2/5) = x \ \Rightarrow \ x = 1/4 )\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI454=1.p} Real: Product -14.25 and -0.08 is 1.14






$\mbox{\$product}(-\mbox{14.}_{25},-\mbox{0.}_{08}) = \mbox{1.}_{14}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI552=1.p} Rational: -35/4 less than difference 0/1 and -53/12






$\mbox{\$less}(-35/4,\mbox{\$difference}(0/1,-53/12))\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI632=1.p} Example 6






$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$greatereq}(x,y) \ \Rightarrow \ \mbox{\$greatereq}(f(x),f(y)) )\qquad\mbox{tff}(\mbox{f\_non\_decreasing},\mbox{axiom})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(u,f(x)),\mbox{\$sum}(v,f(y)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI559=1.p} Rational: Product of sum -1/2 and -1/3, and 0/1 is 0/1






$\mbox{\$product}(\mbox{\$sum}(-1/2,-1/3),0/1) = 0/1\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI233=1.p} Rational: 19/25 greatereq to something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(19/25,x)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI242=1.p} Rational: -3/5 greatereq to something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(-3/5,x)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI616=1.p} If intervals intersect, then sum\_of\_radii $>$= distance\_of\_centers






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

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(\mbox{\$sum}(y,\mbox{\$uminus}(z)),x)\mbox{ and }\mbox{\$lesseq}(x,\mbox{\$sum}(y,z)) )\iff p(x,y,z) ) \ \Rightarrow \ \forall y_1{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},y_2{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( \exists x{:}\;\mbox{\$int}{:}\;( p(x,y_1,z_1)\mbox{ and }p(x,y_2,z_2) ) \ \Rightarrow \ \mbox{\$lesseq}(\mbox{\$sum}(y_1,\mbox{\$uminus}(y_2)),\mbox{\$sum}(z_1,z_2)) )\qquad\mbox{tff}(\mbox{sum\_of\_radii\_gt\_distance\_of\_centers},\mbox{conjecture})$

\medskip\textbf{ARI690=1.p} Solve simple system of linear equations






$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

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

$\mbox{\$product}(2,d) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$difference}(c,d) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$a = \mbox{\$product}(2,c)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI631=1.p} Example 3






$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$w{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{w\_type},\mbox{type})$

$z{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$\mbox{\$greater}(u,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(z,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(\mbox{1.}_0,z),w)\qquad\mbox{tff}(\mbox{hypothesis}_{03},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$product}(\mbox{\$product}(\mbox{\$sum}(\mbox{\$sum}(u,v),z),\mbox{\$sum}(\mbox{\$sum}(u,v),z)),\mbox{\$sum}(\mbox{\$sum}(u,v),z)),\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(u,v),w),\mbox{1.}_0),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(u,v),w),\mbox{1.}_0)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(u,v),w),\mbox{1.}_0)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(u,v),w),\mbox{1.}_0)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(u,v),w),\mbox{1.}_0)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI429=1.p} Real: Difference 7.48 and 0.65 is 6.83






$\mbox{\$difference}(\mbox{7.}_{48},\mbox{0.}_{65}) = \mbox{6.}_{83}\qquad\mbox{tff}(\mbox{real\_difference\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI297=1.p} Rational: Product only 5/8 and 2/5 is 1/4






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$product}(x,2/5) = 1/4 \ \Rightarrow \ x = 5/8 )\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI370=1.p} Real: Something lesseq to -3.25






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(x,-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI664=1.p} 5*a + 11*b = 1 implies a*b $\leq$ -36 | a*b $>$= -2






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

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

$\mbox{\$sum}(\mbox{\$product}(5,a),\mbox{\$product}(11,b)) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(-2,\mbox{\$product}(a,b))\mbox{ or }\mbox{\$greatereq}(-36,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI495=1.p} Real: (3.2 * something) + (6.8 * something else) is 10.92






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{\$product}(\mbox{3.}_2,x),\mbox{\$product}(\mbox{6.}_8,y)) = \mbox{10.}_{92}\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI360=1.p} Real: 7.8 not lesseq to 3.25






$\neg\,\mbox{\$lesseq}(\mbox{7.}_8,\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI634=1.p} Example 8






$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$w{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{w\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(f(x),\mbox{2.}_0)\qquad\mbox{tff}(\mbox{f\_less\_equal}_2,\mbox{axiom})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(w,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(u,\mbox{\$product}(w,\mbox{\$sum}(-\mbox{1.}_0,f(x)))),\mbox{\$sum}(v,w))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI619=2.p} 5 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$rat}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{pow}_2(x)\iff x = 1/( 1\mbox{ or }( \mbox{\$lesseq}(2/1,x)\mbox{ and }\exists y{:}\;\mbox{\$rat}{:}\;( \mbox{\$product}(2/1,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(5/1)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_5,\mbox{conjecture})$

\medskip\textbf{ARI625=2.p} There is no enumeration of the reals






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

$\neg\,\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( x = y\mbox{ or }\mbox{\$less}(f(x),f(y))\mbox{ or }\mbox{\$greatereq}(f(x),\mbox{\$sum}(f(y),1/1)) )\qquad\mbox{tff}(\mbox{not\_ex\_enum\_of\_rats},\mbox{conjecture})$

\medskip\textbf{ARI717=1.p} floor(X+0.5) can be greater than X






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_5)),x)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI622=1.p} There exist two powers of 2 whose sum equals 10






$\mbox{pow}_2{:}\;\mbox{\$int}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{pow}_2(x)\iff ( x = 1\mbox{ or }( \mbox{\$lesseq}(2,x)\mbox{ and }\exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{pow}_2(x)\mbox{ and }\mbox{pow}_2(y)\mbox{ and }\mbox{\$sum}(x,y) = 10 )\qquad\mbox{tff}(\mbox{sum\_of\_pows\_of\_2\_eq}_{10},\mbox{conjecture})$

\medskip\textbf{ARI405=1.p} Real: Sum 4.25 and 2.0 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{4.}_{25},\mbox{2.}_0) = x\qquad\mbox{tff}(\mbox{real\_sum\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI597=1.p} Either a or b or their sum is even






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

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

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

$( p(a)\mbox{ and }p(b)\mbox{ and }p(\mbox{\$sum}(a,b)) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(2,x))\qquad\mbox{tff}(\mbox{a\_or\_b\_or\_aplusb\_even},\mbox{conjecture})$

\medskip\textbf{ARI659=1.p} Prove that a*a*a $\leq$ 3 and a $\leq$ 1 are equivalent






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

$\mbox{\$greatereq}(3,\mbox{\$product}(\mbox{\$product}(a,a),a))\iff \mbox{\$lesseq}(a,1)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI189=1.p} Integer: Product of 2 and something is 10 in a predicate






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

$\forall u{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(2,u)) \ \Rightarrow \ p(10)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI439=1.p} Real: Product 2.4 and 7.0 is 16.8






$\mbox{\$product}(\mbox{2.}_4,\mbox{7.}_0) = \mbox{16.}_8\qquad\mbox{tff}(\mbox{real\_product\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI235=1.p} Rational: Something greatereq something else






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(x,y)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI036=1.p} Integer: -3 not greater than 3






$\neg\,\mbox{\$greater}(-3,3)\qquad\mbox{tff}(\mbox{prove\_n3\_not\_greater}_3,\mbox{conjecture})$

\medskip\textbf{ARI643=1.p} Prove that a != 0 implies 0 / a = 0






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

$a\ne 0 \ \Rightarrow \ \mbox{\$quotient\_e}(0,a) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI030=1.p} Integer: 17 greater than 8






$\mbox{\$greater}(17,8)\qquad\mbox{tff}(\mbox{prove\_17\_greater}_8,\mbox{conjecture})$

\medskip\textbf{ARI563=1.p} Real: Product 14.375 and 7.5 lesseq to 123.8






$\mbox{\$lesseq}(\mbox{\$product}(\mbox{14.}_{375},\mbox{7.}_5),\mbox{123.}_8)\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI539=1.p} Integer: Sum 9 and 3 greater than -21






$\mbox{\$greater}(\mbox{\$sum}(9,3),-21)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI381=1.p} Real: -8.69 not greater than -3.25






$\neg\,\mbox{\$greater}(-\mbox{8.}_{69},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greater\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI593=1.p} There is a number in {5,6,7} that is divisible by 3






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

$( p(5)\mbox{ and }p(6)\mbox{ and }p(7) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(3,x))\qquad\mbox{tff}(\mbox{exists\_X\_in\_5\_6\_7\_div}_3,\mbox{conjecture})$

\medskip\textbf{ARI081=1.p} Integer: Communative sum of 6 and 7






$\forall z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(6,7) = z_1\mbox{ and }\mbox{\$sum}(7,6) = z_2 ) \ \Rightarrow \ z_1 = z_2 )\qquad\mbox{tff}(\mbox{communative\_sum\_6}_7,\mbox{conjecture})$

\medskip\textbf{ARI125=1.p} Integer: Product of 2 and 2 is not 5






$\mbox{\$product}(2,2) = 5\qquad\mbox{tff}(\mbox{anti\_product\_2\_2}_5,\mbox{conjecture})$

\medskip\textbf{ARI706=1.p} Simple rewriting: d*d = 2*d*d implies d*d = d*3*d






$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$product}(d,d) = \mbox{\$product}(\mbox{\$product}(2,d),d)\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$product}(d,d) = \mbox{\$product}(\mbox{\$product}(d,3),d)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI202=1.p} Rational: -13/4 less than something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(-13/4,x)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI500=1.p} Mixed: 3/4 is a rational






$\mbox{\$is\_rat}(3/4)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI312=1.p} Rational: - - 3/4 is 3/4






$\mbox{\$uminus}(\mbox{\$uminus}(3/4)) = 3/4\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI357=1.p} Real: -32500.0 less than something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(-\mbox{32500.}_0,x)\qquad\mbox{tff}(\mbox{real\_less\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI050=1.p} Integer: -3 not greatereq to 3






$\neg\,\mbox{\$greatereq}(-3,3)\qquad\mbox{tff}(\mbox{prove\_n3\_not\_greatereq}_3,\mbox{conjecture})$

\medskip\textbf{ARI489=1.p} Real: Something less than sum something and 1






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(x,\mbox{1.}_0) = y\mbox{ and }\mbox{\$less}(x,y) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI018=1.p} Integer: 12 lesseq to something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(12,x)\qquad\mbox{tff}(\mbox{prove\_12\_lesseq\_something},\mbox{conjecture})$

\medskip\textbf{ARI628=1.p} Example 0






$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$\mbox{\$greater}(u,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{00},\mbox{hypothesis})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$less}(v,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$greatereq}(x,\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{03},\mbox{hypothesis})$

$\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{hypothesis}_{04},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$product}(\mbox{\$product}(\mbox{2.}_0,\mbox{\$product}(u,u)),x),\mbox{\$product}(\mbox{\$product}(y,y),v))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI352=1.p} Real: -8.68 less than 3.25






$\mbox{\$less}(-\mbox{8.}_{69},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_less\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI179=1.p} Integer: It must be 2






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int},w{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(u,5)\mbox{ and }\mbox{\$less}(v,3)\mbox{ and }\mbox{\$greater}(\mbox{\$sum}(u,\mbox{\$product}(2,v)),7) ) \ \Rightarrow \ v = 2 )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI222=1.p} Rational: Something greater than something else






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(x,y)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI129=1.p} Integer: 2 equal - -2






$2 = \mbox{\$uminus}(-2)\qquad\mbox{tff}(\mbox{n2\_equal\_uminus\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI386=1.p} Real: 7.8 greatereq to 3.25






$\mbox{\$greatereq}(\mbox{7.}_8,\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI450=1.p} Real: Product -0.05 and -70.4 is 3.52






$\mbox{\$product}(-\mbox{0.}_{05},-\mbox{70.}_4) = \mbox{3.}_{52}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI358=1.p} Real: 3.25 lesseq to 3.25






$\mbox{\$lesseq}(\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI658=1.p} Prove that a*a $\leq$ 3 and a $>$= -1 \& a $\leq$ 1 are equivalent






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

$\mbox{\$greatereq}(3,\mbox{\$product}(a,a))\iff ( \mbox{\$lesseq}(a,1)\mbox{ and }\mbox{\$greatereq}(a,-1) )\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI205=1.p} Rational: 5/12 not lesseq to 1/4






$\neg\,\mbox{\$lesseq}(5/12,1/4)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI003=1.p} Integer: 2 less than 13






$\mbox{\$less}(2,13)\qquad\mbox{tff}(\mbox{prove\_2\_less}_{13},\mbox{conjecture})$

\medskip\textbf{ARI599=1.p} Inequations imply a = b, hence f(a,b) = f(b,a)






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

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

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

$( \mbox{\$lesseq}(\mbox{\$product}(2,a),\mbox{\$product}(2,b))\mbox{ and }\mbox{\$lesseq}(\mbox{\$product}(3,b),\mbox{\$product}(3,a)) ) \ \Rightarrow \ f(a,b) = f(b,a)\qquad\mbox{tff}(\mbox{ineq\_imply\_f\_eq\_f},\mbox{conjecture})$

\medskip\textbf{ARI165=1.p} Integer: Sum of 2 and 3 is less than 6






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,3) = x \ \Rightarrow \ \mbox{\$less}(x,6) )\qquad\mbox{tff}(\mbox{sum\_2\_3\_less}_6,\mbox{conjecture})$

\medskip\textbf{ARI584=1.p} Interval (Y,Y+3) cannot cover interval (X,X+5)






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\exists z{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,z)\mbox{ and }\mbox{\$less}(z,\mbox{\$sum}(x,5))\mbox{ and }\neg\,\mbox{\$less}(y,z)\mbox{ and }\mbox{\$less}(z,\mbox{\$sum}(y,3)) )\qquad\mbox{tff}(\mbox{interv\_3\_cannot\_cover\_interv}_5,\mbox{conjecture})$

\medskip\textbf{ARI438=1.p} Real: Product 2.38 and 1.5 is 3.57






$\mbox{\$product}(\mbox{2.}_{38},\mbox{1.}_5) = \mbox{3.}_{57}\qquad\mbox{tff}(\mbox{real\_product\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI100=1.p} Integer: Product of 11 and 11 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(11,11) = x\qquad\mbox{tff}(\mbox{product\_11\_11\_something},\mbox{conjecture})$

\medskip\textbf{ARI066=1.p} Integer: Sum -2 and -5 is -7






$\mbox{\$sum}(-2,-5) = -7\qquad\mbox{tff}(\mbox{sum\_n2\_n5\_n}_7,\mbox{conjecture})$

\medskip\textbf{ARI580=1.p} Inequation system is solvable (choose, e.g., Y = X + 1)






$\forall x{:}\;\mbox{\$int}{:}\;\exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,y)\mbox{ and }\mbox{\$less}(y,\mbox{\$sum}(x,3)) )\qquad\mbox{tff}(\mbox{mix\_quant\_ineq\_sys\_solvable}_1,\mbox{conjecture})$

\medskip\textbf{ARI341=1.p} Rational: -2/5 * (145/2 - 569/5) less than (- 76/25) + 271/10






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

$\mbox{\$less}(\mbox{\$product}(-2/5,\mbox{\$difference}(145/2,569/5)),\mbox{\$sum}(\mbox{\$uminus}(76/25),271/10))\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI579=3.p} Inequation system is solvable over \$real (e.g., X = Y = 1/2)






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$less}(\mbox{0.}_0,x)\mbox{ and }\mbox{\$less}(\mbox{0.}_0,y)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(\mbox{3.}_0,x),\mbox{\$product}(\mbox{4.}_0,y)),\mbox{6.}_0) )\qquad\mbox{tff}(\mbox{ineq\_sys\_rat\_solvable},\mbox{conjecture})$

\medskip\textbf{ARI673=1.p} Prove that a*a = 1 and a = 1 | a = -1 are equivalent






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

$\mbox{\$product}(a,a) = 1\iff ( a = -1\mbox{ or }a = 1 )\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI356=1.p} Real: Something less than -32500.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(x,-\mbox{32500.}_0)\qquad\mbox{tff}(\mbox{real\_less\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI012=1.p} Integer: Something less than -2






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(x,-2)\qquad\mbox{tff}(\mbox{something\_less\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI667=1.p} 11*a + 7*b = 1 implies a*b $\leq$ -40 | a*b $>$= -6






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

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

$\mbox{\$sum}(\mbox{\$product}(11,a),\mbox{\$product}(7,b)) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(-6,\mbox{\$product}(a,b))\mbox{ or }\mbox{\$greatereq}(-40,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI578=1.p} Inequation system is solvable (e.g., X = 3, Y = 6)






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(2,x)\mbox{ and }\mbox{\$lesseq}(2,y)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(x,y),9)\mbox{ and }\mbox{\$lesseq}(12,\mbox{\$sum}(\mbox{\$product}(2,x),y)) )\qquad\mbox{tff}(\mbox{ineq\_sys\_solvable}_3,\mbox{conjecture})$

\medskip\textbf{ARI292=1.p} Rational: Product 11/2 and 11/2 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(11/2,11/2) = x\qquad\mbox{tff}(\mbox{rat\_product\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI219=1.p} Rational: 4/15 greather than 1/5






$\mbox{\$greater}(4/15,1/5)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI644=1.p} Prove that a != 0 implies b / a * a $\leq$ b






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

$b{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{a\_type}_{001},\mbox{type})$

$a\ne 0 \ \Rightarrow \ \mbox{\$lesseq}(\mbox{\$quotient\_e}(b,\mbox{\$product}(a,a)),b)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI095=1.p} Integer: Lower boundary for bytes






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(-128,1) = x\qquad\mbox{tff}(\mbox{lower\_boundary},\mbox{conjecture})$

\medskip\textbf{ARI058=1.p} Integer: Sum 23 and 34 is 57






$\mbox{\$sum}(23,34) = 57\qquad\mbox{tff}(\mbox{sum\_23\_34}_{57},\mbox{conjecture})$

\medskip\textbf{ARI729=1.p} If floor(X)=floor(Y), the X-Y $<$ 1






$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$floor}(x) = \mbox{\$floor}(y) \ \Rightarrow \ \mbox{\$less}(\mbox{\$difference}(x,y),\mbox{1.}_0) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI394=1.p} Real: -3.25 not greatereq to 3.25






$\neg\,\mbox{\$greatereq}(-\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI376=1.p} Real: Something greater than 7.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(x,\mbox{7.}_0)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI245=1.p} Rational: Something not equal to 3/4






$\exists x{:}\;\mbox{\$rat}{:}\;x\ne 3/4\qquad\mbox{tff}(\mbox{rat\_not\_equal\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI266=1.p} Rational: Sum 1/2 and -1/4 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(1/2,-1/4) = x\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{21},\mbox{conjecture})$

\medskip\textbf{ARI382=1.p} Real: 0.0 greater than something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{0.}_0,x)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI508=1.p} Mixed: 2 is less than 3.5 coerced to integer






$\mbox{\$less}(2,\mbox{\$to\_int}(\mbox{3.}_5))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI035=1.p} Integer: -4 greater than -6






$\mbox{\$greater}(-4,-6)\qquad\mbox{tff}(\mbox{n4\_greater\_n}_6,\mbox{conjecture})$

\medskip\textbf{ARI734=1.p} Verification example






$\forall i{:}\;\mbox{\$int}{:}\;( \mbox{\$greatereq}(i,0) \ \Rightarrow \ \exists \mbox{res}_1{:}\;\mbox{\$int}{:}\;( \exists \mbox{res1}_2{:}\;\mbox{\$int}{:}\;( \exists i_3{:}\;\mbox{\$int}{:}\;( i_3 = \mbox{\$difference}(i,1)\mbox{ and }\mbox{res1}_2 = \mbox{\$product}(2,i_3) )\mbox{ and }( i = 0 \ \Rightarrow \ \mbox{res}_1 = 0 )\mbox{ and }( i\ne 0 \ \Rightarrow \ \mbox{res}_1 = \mbox{\$sum}(\mbox{res1}_2,2) ) )\mbox{ and }\mbox{res}_1 = \mbox{\$product}(2,i) ) )\qquad\mbox{tff}(\mbox{formula},\mbox{conjecture})$

\medskip\textbf{ARI011=1.p} Integer: Something less than 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(x,0)\qquad\mbox{tff}(\mbox{something\_less}_0,\mbox{conjecture})$

\medskip\textbf{ARI345=1.p} Real: 2.5 less than 3.0






$\mbox{\$less}(\mbox{2.}_5,\mbox{3.}_0)\qquad\mbox{tff}(\mbox{real\_less\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI567=1.p} Real: (-0.25 - (something * -0.05) lesseq to 0.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(\mbox{\$difference}(-\mbox{0.}_{25},\mbox{\$product}(x,-\mbox{0.}_{05})),\mbox{0.}_0)\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI116=1.p} Integer: Product of zero and something is zero






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,0) = 0\qquad\mbox{tff}(\mbox{product\_zero\_identity},\mbox{conjecture})$

\medskip\textbf{ARI338=1.p} Rational: Something less than sum something and 1/1






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(x,1/1) = y\mbox{ and }\mbox{\$less}(x,y) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI009=1.p} Integer: 2 not less than -2






$\neg\,\mbox{\$less}(2,-2)\qquad\mbox{tff}(\mbox{prove\_2\_not\_less\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI283=1.p} Rational: Difference -7/8 and 3/8 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$difference}(-7/8,3/8) = x\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI431=1.p} Real: Difference only 16.05 and 12.05 is 4.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$difference}(x,\mbox{12.}_{05}) = \mbox{4.}_0 \ \Rightarrow \ x = \mbox{16.}_{05} )\qquad\mbox{tff}(\mbox{real\_difference\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI681=1.p} 0 $<$ a * b, 0 $<$ c * d, 0 $<$ a * c imply 0 $<$ b * d






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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

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

$\mbox{\$less}(0,\mbox{\$product}(a,c))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$less}(0,\mbox{\$product}(c,d))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$less}(0,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

$\mbox{\$less}(0,\mbox{\$product}(b,d))\qquad\mbox{tff}(\mbox{conj}_{003},\mbox{conjecture})$

\medskip\textbf{ARI267=1.p} Rational: Sum 1/2 and -1/2 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(1/2,-1/2) = x\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{22},\mbox{conjecture})$

\medskip\textbf{ARI204=1.p} Rational: 1/4 lesseq to 5/12






$\mbox{\$lesseq}(1/4,5/12)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI272=1.p} Rational: Sum something and 0/1 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,0/1) = x\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{27},\mbox{conjecture})$

\medskip\textbf{ARI243=1.p} Rational: Something greatereq to -7/16






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(x,-7/16)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI636=1.p} Example 10






$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$greatereq}(x,y) \ \Rightarrow \ \mbox{\$greatereq}(f(x),f(y)) )\qquad\mbox{tff}(\mbox{f\_non\_decreasing},\mbox{axiom})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(v,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(f(x),u),\mbox{\$sum}(\mbox{\$product}(v,v),f(y)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI721=1.p} floor(X+1.5) $>$ X






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{\$floor}(\mbox{\$sum}(x,\mbox{1.}_5)),x)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI697=1.p} Solve simple system of linear equations






$x_1{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x1\_type},\mbox{type})$

$x_2{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x2\_type},\mbox{type})$

$x_3{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x3\_type},\mbox{type})$

$x_4{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x4\_type},\mbox{type})$

$x_5{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x5\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$product}(-1,\mbox{\$product}(2,x_2)),\mbox{\$product}(5,x_1)),x_3),x_4),x_5) = 0\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(9,x_1),\mbox{\$product}(62,x_2)),\mbox{\$product}(-1,\mbox{\$product}(5,x_3))),\mbox{\$product}(-1,\mbox{\$product}(3,x_4))),\mbox{\$product}(101,x_5)) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(56,x_1),\mbox{\$product}(-1,\mbox{\$product}(34,x_2))),\mbox{\$product}(-1,\mbox{\$product}(11,x_3))),\mbox{\$product}(67,x_4)),\mbox{\$product}(-1,\mbox{\$product}(98,x_5))) = 0\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$remainder\_t}(x_5,2) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI619=1.p} 5 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$int}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{pow}_2(x)\iff ( x = 1\mbox{ or }( \mbox{\$lesseq}(2,x)\mbox{ and }\exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(5)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_5,\mbox{conjecture})$

\medskip\textbf{ARI007=1.p} Integer: -2 less than 2






$\mbox{\$less}(-2,2)\qquad\mbox{tff}(\mbox{n2\_less}_2,\mbox{conjecture})$

\medskip\textbf{ARI680=1.p} Solve system of nonlinear inequalities






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$\mbox{\$greater}(4,\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(3,x),y),\mbox{\$product}(7,a)))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$less}(3,\mbox{\$product}(2,x))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$less}(1,y)\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

$\mbox{\$greater}(0,a)\qquad\mbox{tff}(\mbox{conj}_{003},\mbox{conjecture})$

\medskip\textbf{ARI236=1.p} Rational: -3/4 greatereq to -3/4






$\mbox{\$greatereq}(-3/4,-3/4)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI598=1.p} Either a or 3a+1 is even






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

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

$( p(a)\mbox{ and }p(\mbox{\$sum}(\mbox{\$product}(3,a),1)) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(2,x))\qquad\mbox{tff}(\mbox{a\_or\_3aplus1\_even},\mbox{conjecture})$

\medskip\textbf{ARI294=1.p} Rational: Product 11/2 and something is 121/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(11/2,x) = 121/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI493=1.p} Real: Sum something and iteself is less than -6.5






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(\mbox{\$sum}(x,x),-\mbox{6.}_5)\qquad\mbox{tff}(\mbox{real\_combined\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI289=1.p} Rational: Product -5/2 and 17/5 is -17/2






$\mbox{\$product}(-5/2,17/5) = -17/2\qquad\mbox{tff}(\mbox{rat\_product\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI075=1.p} Integer: Sum something  and -5  is -3






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,-5) = -3\qquad\mbox{tff}(\mbox{sum\_what\_n5\_n}_3,\mbox{conjecture})$

\medskip\textbf{ARI307=1.p} Rational: Product something and -4/5 is 3/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(x,-4/5) = 3/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{24},\mbox{conjecture})$

\medskip\textbf{ARI308=1.p} Rational: -17/25 is - 17/25






$-17/25 = \mbox{\$uminus}(17/25)\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI114=1.p} Integer: Product of something and -5 is -10






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,-5) = -10\qquad\mbox{tff}(\mbox{product\_what\_n5\_n}_{10},\mbox{conjecture})$

\medskip\textbf{ARI678=1.p} Prove that a $>$= 0 and a*a*a $\leq$ 7 imply a*a*a*a $\leq$ 1






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

$\mbox{\$greatereq}(7,\mbox{\$product}(\mbox{\$product}(a,a),a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(0,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$greatereq}(1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI607=1.p} For monotonic f, f(2) $\leq$ f(3) and f(5) $\leq$ f(7), hence the sum






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

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y) \ \Rightarrow \ \mbox{\$lesseq}(f(x),f(y)) ) \ \Rightarrow \ \mbox{\$lesseq}(\mbox{\$sum}(f(2),f(5)),\mbox{\$sum}(f(7),f(3)))\qquad\mbox{tff}(\mbox{f\_mon\_implies\_f2plusf5\_gt\_f7plusf}_3,\mbox{conjecture})$

\medskip\textbf{ARI426=1.p} Real: Difference 10.0 and 5.75 is 4.25






$\mbox{\$difference}(\mbox{10.}_0,\mbox{5.}_{75}) = \mbox{4.}_{25}\qquad\mbox{tff}(\mbox{real\_difference\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI704=1.p} Solve simple system of linear equations






$x_6{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x6\_type},\mbox{type})$

$x_5{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x5\_type},\mbox{type})$

$x_3{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x3\_type},\mbox{type})$

$x_4{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x4\_type},\mbox{type})$

$x_2{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x2\_type},\mbox{type})$

$x_6 = 0\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(2,x_5),\mbox{\$product}(1,x_3)) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(5,x_4),\mbox{\$product}(3,x_2)) = 0\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(7,x_3),\mbox{\$product}(11,x_2)) = 0\qquad\mbox{tff}(\mbox{eq}_4,\mbox{axiom})$

\medskip\textbf{ARI109=1.p} Integer: Product of 5 and -2 is -10






$\mbox{\$product}(5,-2) = -10\qquad\mbox{tff}(\mbox{product\_5\_n2\_n}_{10},\mbox{conjecture})$

\medskip\textbf{ARI677=1.p} Prove that a $>$= 0 and a*a*a $\leq$ 0 imply a*a*a*a = 0






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

$\mbox{\$greatereq}(0,\mbox{\$product}(\mbox{\$product}(a,a),a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(0,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a) = 0\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI287=1.p} Rational: Product 2/5 and 9/1 is 18/5






$\mbox{\$product}(2/5,9/1) = 18/5\qquad\mbox{tff}(\mbox{rat\_product\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI171=1.p} Integer: Not 7 less than sum of 2 and 3






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,3) = x \ \Rightarrow \ \mbox{\$less}(7,x) )\qquad\mbox{tff}(\mbox{anti\_sum\_2\_3\_greater}_7,\mbox{conjecture})$

\medskip\textbf{ARI285=1.p} Rational: Problem 3/8 and 7/10 is 21/80






$\mbox{\$product}(3/8,7/10) = 21/80\qquad\mbox{tff}(\mbox{rat\_product\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI627=1.p} Overflow checking on the rationals

A simple test computing  (2$\wedge$64-1)/2$\wedge$64 + (2$\wedge$64-1)/2$\wedge$64, for which 
 the denominator should get too big for machine int, therefore 
 detecting whether the prover uses arbitrary precision arithmetic.




$\exists x{:}\;\mbox{\$rat}{:}\;( x = \mbox{\$sum}(18446744073709551615/18446744073709551616,18446744073709551615/18446744073709551616)\mbox{ and }( x\ne 2/1 ) )\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI121=1.p} Integer: Commutative product of 6 and 7






$\forall z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( ( \mbox{\$product}(6,7) = z_1\mbox{ and }\mbox{\$product}(7,6) = z_2 ) \ \Rightarrow \ z_1 = z_2 )\qquad\mbox{tff}(\mbox{commutative\_product\_6}_7,\mbox{conjecture})$

\medskip\textbf{ARI686=1.p} Expand and rewrite polynomial






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

$\mbox{\$product}(\mbox{\$product}(5,a),a) = \mbox{\$product}(2,a)\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(a,\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(2,a),\mbox{\$sum}(a,\mbox{\$product}(\mbox{\$product}(3,a),\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(4,a),\mbox{\$sum}(a,\mbox{\$product}(\mbox{\$product}(5,a),\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(6,a),\mbox{\$sum}(a,\mbox{\$product}(\mbox{\$product}(7,a),\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(8,a),\mbox{\$sum}(9,a)))))))))))))))))))) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(2,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a),a),a),a)),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(612,a),\mbox{\$product}(-1,\mbox{\$product}(a,a))),\mbox{\$product}(-2,\mbox{\$product}(\mbox{\$product}(a,a),a))),\mbox{\$product}(2,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a))),\mbox{\$product}(-2,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a),a))),\mbox{\$product}(-2,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a),a),a))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI224=1.p} Rational: -17/25 greater than -29/34






$\mbox{\$greater}(-17/25,-29/34)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI226=1.p} Rational: -29/34 not greater than -17/25






$\neg\,\mbox{\$greater}(-29/34,-17/25)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI105=1.p} Integer: Product of only 2 and 3 is 6






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,3) = 6 \ \Rightarrow \ x = 2 )\qquad\mbox{tff}(\mbox{product\_only\_2\_3}_6,\mbox{conjecture})$

\medskip\textbf{ARI108=1.p} Integer: Product of 2 and -5 is -10






$\mbox{\$product}(2,-5) = -10\qquad\mbox{tff}(\mbox{product\_2\_n5\_n}_{10},\mbox{conjecture})$

\medskip\textbf{ARI183=1.p} Integer: Monotonic function formula 1






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

$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(u,v) \ \Rightarrow \ \mbox{\$less}(f(u),f(v)) ) \ \Rightarrow \ \forall w{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(\mbox{\$sum}(f(w),2)),f(f(w)))\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI343=1.p} Rational: (Something * 16/5) + -3/4 less than -64/5






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(x,16/5),-3/4),-64/5)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI608=1.p} Combining monotonicity and transitivity






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

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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$( \forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y) \ \Rightarrow \ \mbox{\$lesseq}(f(x),f(y)) )\mbox{ and }\mbox{\$lesseq}(a,b)\mbox{ and }\mbox{\$less}(b,c) ) \ \Rightarrow \ \mbox{\$lesseq}(f(a),f(c))\qquad\mbox{tff}(\mbox{f\_mon\_implies\_trans},\mbox{conjecture})$

\medskip\textbf{ARI195=1.p} Rational: Something less than something else






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(x,y)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI060=1.p} Integer: Sum something and 23 is 34






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,23) = 34\qquad\mbox{tff}(\mbox{sum\_something\_23}_{34},\mbox{conjecture})$

\medskip\textbf{ARI463=1.p} Real: - - 0.75 is 0.75






$\mbox{\$uminus}(\mbox{\$uminus}(\mbox{0.}_{75})) = \mbox{0.}_{75}\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI404=1.p} Real: Sum 3.5 and 2.05 is 5.55






$\mbox{\$sum}(\mbox{3.}_5,\mbox{2.}_{05}) = \mbox{5.}_{55}\qquad\mbox{tff}(\mbox{real\_sum\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI443=1.p} Real: Product 5.5 and 5.5 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(\mbox{5.}_5,\mbox{5.}_5) = x\qquad\mbox{tff}(\mbox{real\_product\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI451=1.p} Real: Product 0.05 and -70.4 is -3.52






$\mbox{\$product}(\mbox{0.}_{05},-\mbox{70.}_4) = -\mbox{3.}_{52}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI211=1.p} Rational: -5/8 lesseq to -1/4






$\mbox{\$lesseq}(-5/8,-1/4)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI363=1.p} Real: Something lesseq to something else






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI262=1.p} Rational: Sum 5/12 and -1/6 is 1/4






$\mbox{\$sum}(5/12,-1/6) = 1/4\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI209=1.p} Rational: -3/4 lesseq to -3/4






$\mbox{\$lesseq}(-3/4,-3/4)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI306=1.p} Rational: Product something and -4/5 is -3/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(x,-4/5) = -3/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{23},\mbox{conjecture})$

\medskip\textbf{ARI025=1.p} Integer: Something lesseq to 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(x,0)\qquad\mbox{tff}(\mbox{something\_lesseq}_0,\mbox{conjecture})$

\medskip\textbf{ARI068=1.p} Integer: Sum 5 and -2 is 3






$\mbox{\$sum}(5,-2) = 3\qquad\mbox{tff}(\mbox{sum\_5\_n2}_3,\mbox{conjecture})$

\medskip\textbf{ARI276=1.p} Rational: Difference -1/8 and 3/16 is -5/16






$\mbox{\$difference}(-1/8,3/16) = -5/16\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI373=1.p} Real: 2.5 not greater than 3.0






$\neg\,\mbox{\$greater}(\mbox{2.}_5,\mbox{3.}_0)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI019=1.p} Integer: Something lesseq to something






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{something\_lesseq\_something},\mbox{conjecture})$

\medskip\textbf{ARI282=1.p} Rational: Difference 5/12 and 1/12 is 1/3 in a predicate






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

$p(\mbox{\$difference}(5/12,1/12)) \ \Rightarrow \ p(1/3)\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI164=1.p} Integer: Something is less than sum of something and 1






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,1) = y\mbox{ and }\mbox{\$less}(x,y) )\qquad\mbox{tff}(\mbox{exist\_bigger\_plus\_one},\mbox{conjecture})$

\medskip\textbf{ARI255=1.p} Rational: Sum of 5/16 and something is 1/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(5/16,x) = 1/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI682=1.p} 0 $\leq$ a, a $<$ b imply a+1 $\leq$ a*b + b






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

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

$\mbox{\$less}(a,b)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(0,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(\mbox{\$sum}(a,1),\mbox{\$sum}(\mbox{\$product}(a,b),b))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI082=1.p} Integer: Associative sum






$\forall z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(2,3) = z_1\mbox{ and }\mbox{\$sum}(z_1,6) = z_2\mbox{ and }\mbox{\$sum}(3,6) = z_3\mbox{ and }\mbox{\$sum}(2,z_3) = z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{tff}(\mbox{associative\_sum},\mbox{conjecture})$

\medskip\textbf{ARI353=1.p} Real: 3.25 not less than -3.25






$\neg\,\mbox{\$less}(\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_less\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI347=1.p} Real: 9.53 less than 9.58






$\mbox{\$less}(\mbox{9.}_{53},\mbox{9.}_{58})\qquad\mbox{tff}(\mbox{real\_less\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI589=1.p} There is a number different from Y and Z






$\forall y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;\exists x{:}\;\mbox{\$int}{:}\;( y\ne x\mbox{ and }z\ne x )\qquad\mbox{tff}(\mbox{exists\_X\_noteq\_Y\_Z},\mbox{conjecture})$

\medskip\textbf{ARI497=1.p} Mixed: 17.0 is an integer






$\mbox{\$is\_int}(\mbox{17.}_0)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI693=1.p} Solve simple system of linear equations






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$product}(3,x),\mbox{\$product}(5,a)) = 1\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(7,z),\mbox{\$product}(-1,\mbox{\$product}(17,x))) = 4\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(2,y),\mbox{\$product}(7,x)),\mbox{\$product}(-1,\mbox{\$product}(-1,34))) = 0\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$remainder\_t}(\mbox{\$sum}(z,116),170) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI269=1.p} Rational: Sum something -11/2 is -13/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,-11/2) = -13/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{24},\mbox{conjecture})$

\medskip\textbf{ARI501=1.p} Mixed: 11 is a rational






$\mbox{\$is\_rat}(11)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI072=1.p} Integer: Sum 5 and -2 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(5,-2) = x\qquad\mbox{tff}(\mbox{sum\_5\_n2\_what},\mbox{conjecture})$

\medskip\textbf{ARI097=1.p} Integer: Product of -2 and 3 is -6






$\mbox{\$product}(-2,3) = -6\qquad\mbox{tff}(\mbox{product\_n2\_3\_n}_6,\mbox{conjecture})$

\medskip\textbf{ARI582=1.p} Inequation system is solvable (choose, e.g., Z = X + Y)






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,y) \ \Rightarrow \ \exists z{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(\mbox{\$product}(x,2),z)\mbox{ and }\mbox{\$less}(z,\mbox{\$product}(y,2)) ) )\qquad\mbox{tff}(\mbox{mix\_quant\_ineq\_sys\_solvable}_3,\mbox{conjecture})$

\medskip\textbf{ARI031=1.p} Integer: Something greater than 15






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(x,15)\qquad\mbox{tff}(\mbox{something\_greater}_{15},\mbox{conjecture})$

\medskip\textbf{ARI517=1.p} Mixed: 6/12 coerced to rational is not greater than 3/4






$\neg\,\mbox{\$greater}(\mbox{\$to\_rat}(6/12),3/4)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{22},\mbox{conjecture})$

\medskip\textbf{ARI067=1.p} Integer: Sum 2 and -5 is -3






$\mbox{\$sum}(2,-5) = -3\qquad\mbox{tff}(\mbox{sum\_2\_n5\_n}_3,\mbox{conjecture})$

\medskip\textbf{ARI433=1.p} Real: Difference something and 0.0 is itself






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$difference}(x,\mbox{0.}_0) = x\qquad\mbox{tff}(\mbox{real\_difference\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI070=1.p} Integer: Sum -2 and -5 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(-2,-5) = x\qquad\mbox{tff}(\mbox{sum\_n2\_n5\_what},\mbox{conjecture})$

\medskip\textbf{ARI524=1.p} Mixed: Product 5/12 and 7/10 is rational






$\mbox{\$is\_rat}(\mbox{\$product}(5/12,7/10))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{29},\mbox{conjecture})$

\medskip\textbf{ARI417=1.p} Real: Sum 2.05 and -3.5 is something






$\exists y{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{2.}_{05},-\mbox{3.}_5) = y\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI257=1.p} Rational: Sum of 17/4 and 23/4 is only 10/1






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(17/4,23/4) = x \ \Rightarrow \ x = 10/1 )\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI666=1.p} 5*a + 11*b = 1 implies a*b $\leq$ -37 | a*b $>$= -2






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

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

$\mbox{\$sum}(\mbox{\$product}(5,a),\mbox{\$product}(11,b)) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(-2,\mbox{\$product}(a,b))\mbox{ or }\mbox{\$greatereq}(-37,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI038=1.p} Integer: Something greater than 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(x,0)\qquad\mbox{tff}(\mbox{something\_greater}_0,\mbox{conjecture})$

\medskip\textbf{ARI679=1.p} Prove equivalence of nonlinear inequalities






$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$lesseq}(3,d)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(2,c)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(\mbox{\$product}(2,3),\mbox{\$product}(c,d))\iff \mbox{\$lesseq}(\mbox{\$product}(2,\mbox{\$difference}(d,3)),\mbox{\$product}(c,\mbox{\$difference}(d,3)))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI096=1.p} Integer: Product of 2 and 3 is 6






$\mbox{\$product}(2,3) = 6\qquad\mbox{tff}(\mbox{product\_2\_3}_6,\mbox{conjecture})$

\medskip\textbf{ARI170=1.p} Integer: Not sum is 8 implies difference is 0






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8 \ \Rightarrow \ \mbox{\$difference}(x,y) = 0 )\qquad\mbox{tff}(\mbox{anti\_all\_sum\_same}_8,\mbox{conjecture})$

\medskip\textbf{ARI101=1.p} Integer: Product of something and 11 is 121






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,11) = 121\qquad\mbox{tff}(\mbox{product\_something\_11}_{121},\mbox{conjecture})$

\medskip\textbf{ARI498=1.p} Mixed: 7/12 is not an integer






$\neg\,\mbox{\$is\_int}(7/12)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI662=1.p} Prove that a*a*a*a*a*a*a*a*a*a*a $>$= 1000 and a $>$ 1 are equivalent






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

$\mbox{\$lesseq}(1000,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a),a),a),a),a),a),a))\iff \mbox{\$less}(1,a)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI106=1.p} Integer: Product of 2 and only 3 is 6






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,x) = 6 \ \Rightarrow \ x = 3 )\qquad\mbox{tff}(\mbox{product\_2\_only\_3}_6,\mbox{conjecture})$

\medskip\textbf{ARI367=1.p} Real: 3.25 not lesseq to -3.25






$\neg\,\mbox{\$lesseq}(\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI093=1.p} Integer: Difference with zero is identity






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(x,0) = x\qquad\mbox{tff}(\mbox{diff\_zero\_identity},\mbox{conjecture})$

\medskip\textbf{ARI575=2.p} Inequation system has more than one rational solution






$\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat},z{:}\;\mbox{\$rat}{:}\;( ( \mbox{\$less}(3/1,x)\mbox{ and }\mbox{\$less}(x,y)\mbox{ and }\mbox{\$less}(y,z)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$sum}(x,\mbox{\$product}(2/1,y)),\mbox{\$product}(3/1,z)),35/1) )\iff x = 4/( 1\mbox{ and }( y = 5/( 1\mbox{ and }( z = 6/1 ) ) ) ) )\qquad\mbox{tff}(\mbox{ineq\_sys\_has\_1\_int\_sol},\mbox{conjecture})$

\medskip\textbf{ARI516=1.p} Mixed: 4 coerced to real is greater than 3.2






$\mbox{\$greater}(\mbox{\$to\_real}(4),\mbox{3.}_2)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{21},\mbox{conjecture})$

\medskip\textbf{ARI619=3.p} 5 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$real}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{pow}_2(x)\iff ( x = \mbox{1.}_0\mbox{ or }( \mbox{\$lesseq}(\mbox{2.}_0,x)\mbox{ and }\exists y{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(\mbox{2.}_0,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(\mbox{5.}_0)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_5,\mbox{conjecture})$

\medskip\textbf{ARI581=1.p} Inequation system is solvable (choose, e.g., Y = 8 - X)






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(5,x) \ \Rightarrow \ \exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(y,3)\mbox{ and }\mbox{\$less}(7,\mbox{\$sum}(x,y)) ) )\qquad\mbox{tff}(\mbox{mix\_quant\_ineq\_sys\_solvable}_2,\mbox{conjecture})$

\medskip\textbf{ARI168=1.p} Integer: Not sum is 8 and difference is 1






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8\mbox{ and }\mbox{\$difference}(x,y) = 1 )\qquad\mbox{tff}(\mbox{anti\_exists\_sum\_consecutive}_8,\mbox{conjecture})$

\medskip\textbf{ARI715=1.p} floor(X+0.3) != floor(X)+0.3






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_3))\ne \mbox{\$sum}(\mbox{\$floor}(x),\mbox{0.}_3)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI434=1.p} Real: Difference 5.8 and 0.3 is 5.5 in a predicate






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

$p(\mbox{\$difference}(\mbox{5.}_8,\mbox{0.}_3)) \ \Rightarrow \ p(\mbox{5.}_5)\qquad\mbox{tff}(\mbox{real\_difference\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI522=1.p} Mixed: Sum 3.5 and 3/4 coerced to real is 4.25






$\mbox{\$sum}(\mbox{3.}_5,\mbox{\$to\_real}(3/4)) = \mbox{4.}_{25}\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{27},\mbox{conjecture})$

\medskip\textbf{ARI354=1.p} Real: -3.25 not less than -8.69






$\neg\,\mbox{\$less}(-\mbox{3.}_{25},-\mbox{8.}_{69})\qquad\mbox{tff}(\mbox{real\_less\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI241=1.p} Rational: Something greatereq to 0/1






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(x,0/1)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI407=1.p} Real: Sum 4.25 and something is 10.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{4.}_{25},x) = \mbox{10.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI579=1.p} Inequation system is not solvable over \$int (e.g., X = Y = 1/2)






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,x)\mbox{ and }\mbox{\$less}(0,y)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(3,x),\mbox{\$product}(4,y)),6) )\qquad\mbox{tff}(\mbox{ineq\_sys\_rat\_solvable},\mbox{conjecture})$

\medskip\textbf{ARI437=1.p} Real: Product 3.0 and 2.4 is 7.2






$\mbox{\$product}(\mbox{3.}_0,\mbox{2.}_4) = \mbox{7.}_2\qquad\mbox{tff}(\mbox{real\_product\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI561=1.p} Real: -8.75 less than difference 0.0 and -4.42






$\mbox{\$less}(-\mbox{8.}_{75},\mbox{\$difference}(\mbox{0.}_0,-\mbox{4.}_{42}))\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI543=1.p} Integer: -3 * (14 - 69) less than -76 + 271






$\mbox{\$less}(\mbox{\$product}(-3,\mbox{\$difference}(14,69)),\mbox{\$sum}(\mbox{\$uminus}(76),271))\qquad\mbox{tff}(\mbox{int\_combined\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI311=1.p} Rational: Sum -226/25 and - -226/25 is 0/1






$\mbox{\$sum}(-226/25,\mbox{\$uminus}(-226/25)) = 0/1\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI724=1.p} floor(2*X) $>$= 2*floor(X)






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(\mbox{\$floor}(\mbox{\$product}(\mbox{2.}_0,x)),\mbox{\$product}(\mbox{2.}_0,\mbox{\$floor}(x)))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI073=1.p} Integer: Sum 5 and -5 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(5,-5) = x\qquad\mbox{tff}(\mbox{sum\_5\_n5\_what},\mbox{conjecture})$

\medskip\textbf{ARI375=1.p} Real: 12.8 greater than something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{12.}_8,x)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI281=1.p} Rational: Difference something and 0/1 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$difference}(x,0/1) = x\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI133=1.p} Integer: - - -2 is -2






$\mbox{\$uminus}(\mbox{\$uminus}(-2)) = -2\qquad\mbox{tff}(\mbox{uminus\_uminus\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI651=1.p} Solve simple system of linear inequalities






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

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

$\mbox{\$lesseq}(20,\mbox{\$product}(-1,\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(2,b)))))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(0,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(-5,\mbox{\$product}(-1,\mbox{\$sum}(a,b)))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

\medskip\textbf{ARI265=1.p} Rational: Sum 1/4 and -1/2 is something






$\exists y{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(1/4,-1/2) = y\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{20},\mbox{conjecture})$

\medskip\textbf{ARI712=1.p} Expand and rewrite polynomial






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(a,a),\mbox{\$product}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(a,b),c),\mbox{\$product}(\mbox{\$product}(b,d),a)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(c,c),a),\mbox{\$product}(\mbox{\$product}(d,d),b)),1))),b),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),d),\mbox{\$sum}(2,c)))) = 0\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(d,d),d),b),b),a)),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(2,a),\mbox{\$product}(-1,\mbox{\$product}(a,a))),b),\mbox{\$product}(2,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(d,d),c),b),b),a))),\mbox{\$product}(c,a)),\mbox{\$product}(c,b)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(c,b),a))),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),c),b),a),a))),\mbox{\$product}(2,d)),\mbox{\$product}(d,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(d,b),a))),\mbox{\$product}(c,c)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(d,c),c),b),a),a))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI596=1.p} There is a number in {a,a+1,a-1} that is divisible by 3






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

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

$( p(a)\mbox{ and }p(\mbox{\$sum}(a,1))\mbox{ and }p(\mbox{\$difference}(a,1)) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(3,x))\qquad\mbox{tff}(\mbox{exists\_X\_in\_a\_aplus1\_aminus1\_div}_3,\mbox{conjecture})$

\medskip\textbf{ARI089=1.p} Integer: Difference 7 and 5 is 2






$\mbox{\$difference}(7,5) = 2\qquad\mbox{tff}(\mbox{diff\_7\_5}_2,\mbox{conjecture})$

\medskip\textbf{ARI557=1.p} Rational: Product something and 9/12 is 3/7 means 1/2 less than it






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(x,9/12) = 3/( 7\mbox{ and }\mbox{\$less}(1/2,x) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI588=1.p} If X = 2 then Y $<$ X-1 xor 3-X $\leq$ Y






$\exists x{:}\;\mbox{\$int}{:}\;\forall y{:}\;\mbox{\$int}{:}\;\neg\,\mbox{\$less}(y,\mbox{\$sum}(x,-1))\iff \mbox{\$lesseq}(\mbox{\$sum}(3,\mbox{\$uminus}(x)),y)\qquad\mbox{tff}(\mbox{exists\_X\_complementary\_halflines},\mbox{conjecture})$

\medskip\textbf{ARI647=1.p} 2*a $\leq$ 1 implies 3*a != 3






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

$\neg\,\mbox{\$lesseq}(-1,\mbox{\$product}(-1,\mbox{\$product}(2,a)))\mbox{ or }\mbox{\$product}(3,a)\ne 3\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI310=1.p} Rational: Sum 129/503 and - 129/503 is 0/1






$\mbox{\$sum}(129/503,\mbox{\$uminus}(129/503)) = 0/1\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI177=1.p} Integer: Formula equals 21






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(\mbox{\$product}(4,u),\mbox{\$product}(6,v))\ne 21\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI339=1.p} Rational: Sum of 12/5 and 37/10 is less than 7/1






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(12/5,37/10) = x \ \Rightarrow \ \mbox{\$less}(x,7/1) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI268=1.p} Rational: Sum something and -7/2 is -11/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,-7/2) = -11/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{23},\mbox{conjecture})$

\medskip\textbf{ARI400=1.p} Real: Something is not 20.06






$\exists x{:}\;\mbox{\$real}{:}\;x\ne \mbox{20.}_{06}\qquad\mbox{tff}(\mbox{real\_not\_equal\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI455=1.p} Real: Product 14.25 and -0.08 is something






$\exists y{:}\;\mbox{\$real}{:}\;\mbox{\$product}(\mbox{14.}_{25},-\mbox{0.}_{08}) = y\qquad\mbox{tff}(\mbox{real\_product\_problem}_{20},\mbox{conjecture})$

\medskip\textbf{ARI250=1.p} Rational: Sum of 17/4 and 23/4 is 10/1






$\mbox{\$sum}(17/4,23/4) = 10/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI655=1.p} a $>$= b, c $>$= d imply (a-b)*(c-d) $>$= 0






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$lesseq}(b,a)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(d,c)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(0,\mbox{\$product}(\mbox{\$difference}(a,b),\mbox{\$difference}(c,d)))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI723=1.p} If ceiling(X) = floor(X), then X is an integer






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$ceiling}(x) = \mbox{\$floor}(x) \ \Rightarrow \ \mbox{\$is\_int}(x) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI253=1.p} Rational: Sum of 17/4 and 2/1 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(17/4,2/1) = x\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI278=1.p} Rational: Difference 5/12 and 1/12 is only 1/3






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$difference}(5/12,1/12) = x \ \Rightarrow \ x = 1/3 )\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI641=1.p} Example 22






$m{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{m\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

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

$\mbox{\$less}(a,b)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(x,a)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greatereq}(m,\mbox{\$ceiling}(\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),x)),\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),b))))\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(a,\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{1.}_0,m)),\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),b))),x)\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI555=1.p} Rational: Minus product -18/25 and -1/4 greatereq to -9/50






$\mbox{\$greatereq}(\mbox{\$uminus}(\mbox{\$product}(-18/25,-1/4)),-9/50)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI187=1.p} Integer: Sum of product of 14 and 3, and 8, is 50 in a predicate






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

$p(\mbox{\$sum}(\mbox{\$product}(14,3),8)) \ \Rightarrow \ p(50)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI260=1.p} Rational: Sum -7/2 and -1/2 is -4/1






$\mbox{\$sum}(-7/2,-1/2) = -4/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI047=1.p} Integer: -4 greatereq to -4






$\mbox{\$greatereq}(-4,-4)\qquad\mbox{tff}(\mbox{n4\_greatereq\_n}_4,\mbox{conjecture})$

\medskip\textbf{ARI033=1.p} Integer: Something greater than something






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(x,y)\qquad\mbox{tff}(\mbox{something\_greater\_something},\mbox{conjecture})$

\medskip\textbf{ARI119=1.p} Integer: Product of 4 and 4 is 16






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,y) = 16\mbox{ and }x = 4\mbox{ and }y = 4 )\qquad\mbox{tff}(\mbox{product\_4\_4}_{16},\mbox{conjecture})$

\medskip\textbf{ARI185=1.p} Integer: Positive function formula






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

$\forall u{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(u),1) \ \Rightarrow \ \mbox{\$less}(\mbox{\$difference}(7,\mbox{\$product}(2,f(3))),4)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI398=1.p} Real: Something greatereq to -3.25






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(x,-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI535=1.p} Integer: Stickel's arithmetic challenge






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

$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( p(2,y,\mbox{\$sum}(2,y)) \ \Rightarrow \ p(x,2,\mbox{\$product}(x,2)) )\qquad\mbox{tff}(a,\mbox{conjecture})$

\medskip\textbf{ARI286=1.p} Rational: Problem 3/8 and 5/12 is 5/32






$\mbox{\$product}(3/8,5/12) = 5/32\qquad\mbox{tff}(\mbox{rat\_product\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI568=1.p} Real: Product of sum -2.8 and -3.6, and 0.0 is 0.0






$\mbox{\$product}(\mbox{\$sum}(-\mbox{2.}_8,-\mbox{3.}_6),\mbox{0.}_0) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI602=1.p} If f(X) $>$ X, then 4 $<$ 5 $<$ f(5)






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

$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(x),x) \ \Rightarrow \ \exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(4,y)\mbox{ and }\mbox{\$less}(y,f(5)) )\qquad\mbox{tff}(\mbox{fX\_gt\_X\_implies\_exist\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI512=1.p} Mixed: -2 is lesseq to 2.0 coerced to integer






$\mbox{\$lesseq}(-2,\mbox{\$to\_int}(\mbox{2.}_0))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI571=1.p} Negating and weakening an inequation






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(y,x) \ \Rightarrow \ \mbox{\$less}(\mbox{\$difference}(2,x),\mbox{\$difference}(5,y)) )\qquad\mbox{tff}(\mbox{neg\_weakening\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI507=1.p} Mixed: 39/4 coerced to real is 9.75






$\mbox{\$to\_real}(39/4) = \mbox{9.}_{75}\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI574=1.p} Inequation system has exactly one solution






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(7,\mbox{\$sum}(x,y))\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(x,5),\mbox{\$product}(2,y))\mbox{ and }\mbox{\$lesseq}(y,4) )\iff ( x = 3\mbox{ and }y = 4 ) )\qquad\mbox{tff}(\mbox{ineq\_sys\_has\_1\_sol},\mbox{conjecture})$

\medskip\textbf{ARI544=1.p} Integer: Sum 2 and 3 less than 7






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,3) = x \ \Rightarrow \ \mbox{\$less}(x,7) )\qquad\mbox{tff}(\mbox{int\_combined\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI230=1.p} Rational: 5/12 greatereq to 5/12






$\mbox{\$greatereq}(5/12,5/12)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI117=1.p} Integer: Product with 1 is the identity






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,1) = x\qquad\mbox{tff}(\mbox{product\_1\_identity},\mbox{conjecture})$

\medskip\textbf{ARI188=1.p} Integer: Sum of something and 3 is 5 in a predicate






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

$\forall u{:}\;\mbox{\$int}{:}\;p(\mbox{\$sum}(u,3)) \ \Rightarrow \ p(5)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI041=1.p} Integer: 2 greatereq to 2






$\mbox{\$greatereq}(2,2)\qquad\mbox{tff}(\mbox{prove\_2\_greatereq}_2,\mbox{conjecture})$

\medskip\textbf{ARI726=1.p} There is an integer X such that 0.4*X $>$ 1 and 0.3*X $<$ 1






$\exists x{:}\;\mbox{\$real}{:}\;( \mbox{\$is\_int}(x)\mbox{ and }\mbox{\$greater}(\mbox{\$product}(\mbox{0.}_4,x),\mbox{1.}_0)\mbox{ and }\mbox{\$less}(\mbox{\$product}(\mbox{0.}_3,x),\mbox{1.}_0) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI166=1.p} Integer: 4 is less than the sum of 2 and 3






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,3) = x \ \Rightarrow \ \mbox{\$less}(4,x) )\qquad\mbox{tff}(\mbox{sum\_2\_3\_greater}_4,\mbox{conjecture})$

\medskip\textbf{ARI416=1.p} Real: Sum -2.05 and -3.5 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(-\mbox{2.}_{05},-\mbox{3.}_5) = x\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI113=1.p} Integer: Product of 5 and -2 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(5,-2) = x\qquad\mbox{tff}(\mbox{product\_5\_n2\_what},\mbox{conjecture})$

\medskip\textbf{ARI304=1.p} Rational: Product 15/16 and -4/5 is something






$\exists y{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(15/16,-4/5) = y\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{21},\mbox{conjecture})$

\medskip\textbf{ARI518=1.p} Mixed: Sum of 13.1 coerced to integer and 1 is greatereq to 14






$\mbox{\$greatereq}(\mbox{\$sum}(\mbox{\$to\_int}(\mbox{13.}_1),1),14)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{23},\mbox{conjecture})$

\medskip\textbf{ARI061=1.p} Integer: Sum 23 and something is 34






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(23,x) = 34\qquad\mbox{tff}(\mbox{sum\_23\_something}_{34},\mbox{conjecture})$

\medskip\textbf{ARI550=1.p} Integer: Sum of product something and 32, and -7 less than -128






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(x,32),-7),-128)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI238=1.p} Rational: -1/4 greatereq -5/8






$\mbox{\$greatereq}(-1/4,-5/8)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI674=1.p} Prove that a*a $>$= 4 implies a $>$= 2 | a $\leq$ -2






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

$\mbox{\$lesseq}(4,\mbox{\$product}(a,a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(a,-2)\mbox{ or }\mbox{\$lesseq}(2,a)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI572=1.p} Simple implication between inequations






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(\mbox{\$sum}(y,\mbox{\$uminus}(x)),\mbox{\$sum}(x,\mbox{\$uminus}(y))) \ \Rightarrow \ \mbox{\$less}(y,x) )\qquad\mbox{tff}(\mbox{impl\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI208=1.p} Rational: Something lesseq to something else






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI425=1.p} Real: Difference 9.0 and 4.0 is 5.0






$\mbox{\$difference}(\mbox{9.}_0,\mbox{4.}_0) = \mbox{5.}_0\qquad\mbox{tff}(\mbox{real\_difference\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI722=1.p} If floor(X) = X, then X is an integer






$\forall x{:}\;\mbox{\$real}{:}\;( x = \mbox{\$floor}(x) \ \Rightarrow \ \mbox{\$is\_int}(x) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI384=1.p} Real: Something greater than -32500.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(x,-\mbox{32500.}_0)\qquad\mbox{tff}(\mbox{real\_greater\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI675=1.p} Prove that a*a*a*a + a*a + 10 $>$= 0






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

$\mbox{\$lesseq}(0,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),\mbox{\$product}(a,a)),10))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI190=1.p} Rational: 3/4 less than 7/8






$\mbox{\$less}(3/4,7/8)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI554=1.p} Rational: Product 3/8 and 7/10 lesseq to 23/80






$\mbox{\$lesseq}(\mbox{\$product}(3/8,7/10),23/80)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI661=1.p} Prove that a*a*a*a*a $>$= 40 and a $>$= 3 are equivalent






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

$\mbox{\$lesseq}(40,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a),a))\iff \mbox{\$lesseq}(3,a)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI492=1.p} Real: -0.4 * (72.5 - 113.8) is less than (- 3.04) + 27.1






$\mbox{\$less}(\mbox{\$product}(-\mbox{0.}_4,\mbox{\$difference}(\mbox{72.}_5,\mbox{113.}_8)),\mbox{\$sum}(\mbox{\$uminus}(\mbox{3.}_{04}),\mbox{27.}_1))\qquad\mbox{tff}(\mbox{real\_combined\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI583=1.p} Inequation system is solvable (choose, e.g., W = 3 - X)






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(0,x)\mbox{ and }\mbox{\$lesseq}(0,y)\mbox{ and }\mbox{\$lesseq}(0,z)\mbox{ and }( \mbox{\$less}(x,y)\mbox{ or }\mbox{\$less}(x,z) ) ) \ \Rightarrow \ \exists w{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(\mbox{\$sum}(x,w),4)\mbox{ and }\mbox{\$less}(3,\mbox{\$sum}(\mbox{\$sum}(y,z),w)) ) )\qquad\mbox{tff}(\mbox{mix\_quant\_ineq\_sys\_solvable}_4,\mbox{conjecture})$

\medskip\textbf{ARI462=1.p} Real: Sum -9.04 and - -9.04 is 0.0






$\mbox{\$sum}(-\mbox{9.}_{04},\mbox{\$uminus}(-\mbox{9.}_{04})) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI207=1.p} Rational: 3/16 lesseq to something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(3/16,x)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI337=1.p} Rational: Sum is 36/5 and difference is 0/1






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,y) = 36/( 5\mbox{ and }( \mbox{\$difference}(x,y) = 0/1 ) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI127=1.p} Integer: Not product of only 2 and only 4 is 8






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,y) = 8 \ \Rightarrow \ ( x = 2\mbox{ and }y = 4 ) )\qquad\mbox{tff}(\mbox{anti\_product\_only\_2\_only\_4}_8,\mbox{conjecture})$

\medskip\textbf{ARI200=1.p} Rational: Something less than 0/1






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(x,0/1)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI364=1.p} Real: -3.25 lesseq to -3.25






$\mbox{\$lesseq}(-\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI635=1.p} Example 9






$u{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{u\_type},\mbox{type})$

$v{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{v\_type},\mbox{type})$

$w{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{w\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$s{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{s\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$greatereq}(x,y) \ \Rightarrow \ \mbox{\$greatereq}(f(x),f(y)) )\qquad\mbox{tff}(\mbox{f\_non\_decreasing},\mbox{axiom})$

$\mbox{\$less}(u,v)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(w,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(s,\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{3.}_0),\mbox{\$sum}(w,s)),v)\qquad\mbox{tff}(\mbox{hypothesis}_{03},\mbox{hypothesis})$

$\mbox{\$lesseq}(x,y)\qquad\mbox{tff}(\mbox{hypothesis}_{04},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$sum}(f(x),u),\mbox{\$sum}(\mbox{\$product}(v,v),f(y)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI390=1.p} Real: Something greatereq to something else






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(x,y)\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI491=1.p} Real: 7.5 is less than sum 2.9 and 4.8






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(\mbox{2.}_9,\mbox{4.}_8) = x \ \Rightarrow \ \mbox{\$less}(\mbox{7.}_5,x) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI094=1.p} Integer: Difference 5 and 3 is 2 in a predicate






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

$p(\mbox{\$difference}(5,3)) \ \Rightarrow \ p(2)\qquad\mbox{tff}(\mbox{diff\_5\_3\_2\_predicate},\mbox{conjecture})$

\medskip\textbf{ARI526=1.p} Mixed: (4.05 + 3.6) - 53/20 = 5






$\mbox{\$to\_int}(\mbox{\$difference}(\mbox{\$to\_rat}(\mbox{\$sum}(\mbox{4.}_{05},\mbox{3.}_6)),53/20)) = 5\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{31},\mbox{conjecture})$

\medskip\textbf{ARI408=1.p} Real: Sum 3.5 and 2.05 is not 5.5






$\mbox{\$sum}(\mbox{3.}_5,\mbox{2.}_{05})\ne \mbox{5.}_5\qquad\mbox{tff}(\mbox{real\_sum\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI215=1.p} Rational: Something lesseq to -3/5






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(x,-3/5)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI077=1.p} Integer: Sum something  and -5 is 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,-5) = 0\qquad\mbox{tff}(\mbox{sum\_what\_n5}_0,\mbox{conjecture})$

\medskip\textbf{ARI258=1.p} Rational: Sum only 17/4 and 23/4 is 10/1






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(x,23/4) = 10/1 \ \Rightarrow \ x = 17/4 )\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI702=1.p} Solve a simple system of nonlinear equations






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

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

$\mbox{\$product}(\mbox{\$product}(2,x),y) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(2,y),z) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$product}(a,z) = 2\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(x,y),z) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI054=1.p} Integer: -5 greatereq to something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(-5,x)\qquad\mbox{tff}(\mbox{n5\_greatereq\_something},\mbox{conjecture})$

\medskip\textbf{ARI615=1.p} If Z$\leq$W, then [X-Z,X+Z] is a subset of [X-W,X+W]






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

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(\mbox{\$sum}(y,\mbox{\$uminus}(z)),x)\mbox{ and }\mbox{\$lesseq}(x,\mbox{\$sum}(y,z)) )\iff p(x,y,z) ) \ \Rightarrow \ \forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},w{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(z,w) \ \Rightarrow \ ( p(x,y,z) \ \Rightarrow \ p(x,y,w) ) )\qquad\mbox{tff}(\mbox{interv\_with\_smaller\_radius\_contained},\mbox{conjecture})$

\medskip\textbf{ARI610=1.p} For mon. f, f(b)$<$f(a) =$>$ b$<$a =$>$ b$\leq$a =$>$ 0$\leq$a-b =$>$ f(0)$\leq$f(a-b)






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

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

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

$( \forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y) \ \Rightarrow \ \mbox{\$lesseq}(f(x),f(y)) )\mbox{ and }\mbox{\$less}(f(b),f(a)) ) \ \Rightarrow \ \mbox{\$lesseq}(f(0),f(\mbox{\$sum}(a,\mbox{\$uminus}(b))))\qquad\mbox{tff}(\mbox{f\_mon\_implies\_f\_a\_b}_2,\mbox{conjecture})$

\medskip\textbf{ARI453=1.p} Real: Product 0.05 and 0.0 is 0.0






$\mbox{\$product}(\mbox{0.}_{05},\mbox{0.}_0) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_product\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI295=1.p} Rational: Product 5/8 and 7/8 is not 9/16






$\mbox{\$product}(5/8,7/8)\ne 9/16\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI406=1.p} Real: Sum something and 4.07 is 19.076






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,\mbox{4.}_{07}) = \mbox{19.}_{076}\qquad\mbox{tff}(\mbox{real\_sum\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI511=1.p} Mixed: 5/19 coerced to rational is less than something






$\exists y{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(\mbox{\$to\_rat}(5/19),y)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI181=1.p} Integer: Increasing function applied 3 times






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

$\forall u{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(u),u) \ \Rightarrow \ \mbox{\$greatereq}(f(f(f(6))),9)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI606=1.p} For monotonic f, 2$\leq$5 implies f(2)$\leq$f(5), thus f(f(2)$\leq$f(f(5))






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

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y) \ \Rightarrow \ \mbox{\$lesseq}(f(x),f(y)) ) \ \Rightarrow \ \mbox{\$lesseq}(f(f(2)),f(f(5)))\qquad\mbox{tff}(\mbox{f\_mon\_implies\_ff2\_gt\_ff}_5,\mbox{conjecture})$

\medskip\textbf{ARI542=1.p} Integer: Sum of product -5 and -5, and -25 is 0






$\mbox{\$sum}(\mbox{\$product}(-5,-5),-25) = 0\qquad\mbox{tff}(\mbox{int\_combined\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI288=1.p} Rational: Product 12/5 and 7/1 is 84/5






$\mbox{\$product}(12/5,7/1) = 84/5\qquad\mbox{tff}(\mbox{rat\_product\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI378=1.p} Real: 3.25 greater then -3.25






$\mbox{\$greater}(\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greater\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI184=1.p} Integer: Monotonic function formula 2






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

$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(u,v) \ \Rightarrow \ \mbox{\$less}(f(u),f(v)) ) \ \Rightarrow \ \forall w{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(\mbox{\$sum}(f(w),2)),\mbox{\$sum}(f(f(w)),1))\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI401=1.p} Real: Sum 4.0 and 5.0 is 9.0






$\mbox{\$sum}(\mbox{4.}_0,\mbox{5.}_0) = \mbox{9.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI169=1.p} Integer: Not sum is 8 implies difference is 1






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8 \ \Rightarrow \ \mbox{\$difference}(x,y) = 1 )\qquad\mbox{tff}(\mbox{anti\_all\_sum\_consecutive}_8,\mbox{conjecture})$

\medskip\textbf{ARI397=1.p} Real: -3.25 greatereq to something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(-\mbox{3.}_{25},x)\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI130=1.p} Integer: Sum of 2 and - 2 is 0






$\mbox{\$sum}(2,\mbox{\$uminus}(2)) = 0\qquad\mbox{tff}(\mbox{sum\_2\_uminus\_to}_0,\mbox{conjecture})$

\medskip\textbf{ARI549=1.p} Integer: Product of sum -2 and -3, and 0 is 0






$\mbox{\$product}(\mbox{\$sum}(-2,-3),0) = 0\qquad\mbox{tff}(\mbox{int\_combined\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI562=1.p} Real: Sum 0.52 and 0.6 greater than -0.055






$\mbox{\$greater}(\mbox{\$sum}(\mbox{0.}_{52},\mbox{0.}_6),-\mbox{0.}_{055})\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI110=1.p} Integer: Product of 5 and 0 is 0






$\mbox{\$product}(5,0) = 0\qquad\mbox{tff}(\mbox{product\_5\_0}_0,\mbox{conjecture})$

\medskip\textbf{ARI391=1.p} Real: -3.25 greatereq to -3.25






$\mbox{\$greatereq}(-\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI346=1.p} Real: 3.0 not less than 2.5






$\neg\,\mbox{\$less}(\mbox{3.}_0,\mbox{2.}_5)\qquad\mbox{tff}(\mbox{real\_less\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI445=1.p} Real: Product 5.5 and something is 30.25






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(\mbox{5.}_5,x) = \mbox{30.}_{25}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI509=1.p} Mixed: 7/5 is not less that 1.2 coerced to rational






$\neg\,\mbox{\$less}(7/5,\mbox{\$to\_rat}(\mbox{1.}_2))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI496=1.p} Mixed: 6 is an integer






$\mbox{\$is\_int}(6)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI457=1.p} Real: Product something and -0.08 is -1.14






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(x,-\mbox{0.}_{08}) = -\mbox{1.}_{14}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{22},\mbox{conjecture})$

\medskip\textbf{ARI091=1.p} Integer: Difference only 5 and 2 is 3






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$difference}(x,2) = 3 \ \Rightarrow \ x = 5 )\qquad\mbox{tff}(\mbox{diff\_only\_5\_2}_3,\mbox{conjecture})$

\medskip\textbf{ARI045=1.p} Integer: 15 greatereq to something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(15,x)\qquad\mbox{tff}(\mbox{prove\_15\_greatereq\_something},\mbox{conjecture})$

\medskip\textbf{ARI566=1.p} Real: Product something and 0.75 is 0.42 means 0.5 less than it






$\exists x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(x,\mbox{0.}_{75}) = \mbox{0.}_{42}\mbox{ and }\mbox{\$less}(\mbox{0.}_5,x) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI217=1.p} Rational: 7/8 greater than 3/4






$\mbox{\$greater}(7/8,3/4)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI560=1.p} Real: 0.875 less than sum 0.625 and 0.3125






$\mbox{\$less}(\mbox{0.}_{875},\mbox{\$sum}(\mbox{0.}_{625},\mbox{0.}_{3125}))\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI237=1.p} Rational: 3/4 greatereq to -3/4






$\mbox{\$greatereq}(3/4,-3/4)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI441=1.p} Real: Product -0.075 and -12.0 is 0.9






$\mbox{\$product}(-\mbox{0.}_{075},-\mbox{12.}_0) = \mbox{0.}_9\qquad\mbox{tff}(\mbox{real\_product\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI004=1.p} Integer: Something less than 13






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(x,13)\qquad\mbox{tff}(\mbox{something\_less}_{13},\mbox{conjecture})$

\medskip\textbf{ARI246=1.p} Rational: Sum of 1/2 and 1/4 is 3/4






$\mbox{\$sum}(1/2,1/4) = 3/4\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI683=1.p} Solve system of nonlinear inequalities






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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$g{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{g\_type},\mbox{type})$

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

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

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

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

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$e{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{e\_type},\mbox{type})$

$\mbox{\$lesseq}(\mbox{\$sum}(\mbox{\$product}(a,f),\mbox{\$product}(-1,\mbox{\$product}(a,h))),\mbox{\$sum}(\mbox{\$sum}(b,\mbox{\$product}(c,g)),\mbox{\$product}(-1,\mbox{\$product}(c,h))))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(g,f)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$less}(h,g)\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

$\mbox{\$less}(i,h)\qquad\mbox{tff}(\mbox{conj}_{003},\mbox{axiom})$

$\mbox{\$lesseq}(a,d)\qquad\mbox{tff}(\mbox{conj}_{004},\mbox{axiom})$

$\mbox{\$less}(e,a)\qquad\mbox{tff}(\mbox{conj}_{005},\mbox{axiom})$

$\mbox{\$lesseq}(\mbox{\$sum}(\mbox{\$product}(e,f),\mbox{\$product}(-1,\mbox{\$product}(e,i))),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(b,\mbox{\$product}(c,g)),\mbox{\$product}(-1,\mbox{\$product}(c,h))),\mbox{\$product}(d,h)),\mbox{\$product}(-1,\mbox{\$product}(d,i))))\qquad\mbox{tff}(\mbox{conj}_{006},\mbox{conjecture})$

\medskip\textbf{ARI302=1.p} Rational: Product 1/20 and 0/1 is 0/1






$\mbox{\$product}(1/20,0/1) = 0/1\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI676=1.p} Prove that a*a + 10 $>$= 0






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

$\mbox{\$lesseq}(0,\mbox{\$sum}(\mbox{\$product}(a,a),10))\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI270=1.p} Rational: Sum something and -5/2 is 7/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,-5/2) = 7/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{25},\mbox{conjecture})$

\medskip\textbf{ARI273=1.p} Rational: Difference 7/8 and 3/8 is 1/2






$\mbox{\$difference}(7/8,3/8) = 1/2\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI488=1.p} Real: Sum is 7.2 and difference is 0.0






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(x,y) = \mbox{7.}_2\mbox{ and }\mbox{\$difference}(x,y) = \mbox{0.}_0 )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI630=1.p} Example 2






$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$\mbox{\$greater}(x,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{00},\mbox{hypothesis})$

$\mbox{\$less}(x,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,x),\mbox{1.}_0)),\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,\mbox{\$product}(x,x)),\mbox{1.}_0)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI637=1.p} Example 13






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

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

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

$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;f(\mbox{\$sum}(x,y)) = \mbox{\$product}(f(x),f(y))\qquad\mbox{tff}(\mbox{f\_feature},\mbox{axiom})$

$\mbox{\$greater}(f(a),\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(f(b),\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(f(\mbox{\$sum}(a,b)),\mbox{4.}_0)\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI414=1.p} Real: Sum 5.55 and -3.05 is 2.5






$\mbox{\$sum}(\mbox{5.}_{55},-\mbox{3.}_{05}) = \mbox{2.}_5\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI515=1.p} Mixed: Something is lesseq to 2 coerced to integer






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(x,\mbox{\$to\_int}(2))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{20},\mbox{conjecture})$

\medskip\textbf{ARI193=1.p} Rational: Something less than 9/16






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(x,9/16)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI210=1.p} Rational: -3/4 lesseq to 3/4






$\mbox{\$lesseq}(-3/4,3/4)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI393=1.p} Real: -3.25 greatereq to -8.69






$\mbox{\$greatereq}(-\mbox{3.}_{25},-\mbox{8.}_{69})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI049=1.p} Integer: -3 greatereq to -6






$\mbox{\$greatereq}(-3,-6)\qquad\mbox{tff}(\mbox{n3\_greatereq\_n}_6,\mbox{conjecture})$

\medskip\textbf{ARI613=1.p} There is an X $>$ 3 and a Y $<$ 1 whose sum is 0






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

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

$( \forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(3,x) \ \Rightarrow \ p(x) )\mbox{ and }\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,1) \ \Rightarrow \ q(x) ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( p(x)\mbox{ and }q(y)\mbox{ and }\mbox{\$sum}(x,y) = 0 )\qquad\mbox{tff}(\mbox{interv\_3\_infty\_and\_neginfty\_1\_contain\_compl},\mbox{conjecture})$

\medskip\textbf{ARI625=1.p} There is no enumeration of the reals






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

$\neg\,\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( x = y\mbox{ or }\mbox{\$less}(f(x),f(y))\mbox{ or }\mbox{\$greatereq}(f(x),\mbox{\$sum}(f(y),\mbox{1.}_0)) )\qquad\mbox{tff}(\mbox{not\_ex\_enum\_of\_reals},\mbox{conjecture})$

\medskip\textbf{ARI120=1.p} Integer: Product of X and X is 4






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

$p(2) \ \Rightarrow \ \exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( p(x)\mbox{ and }x\ne y\mbox{ and }\mbox{\$product}(y,y) = 4 )\qquad\mbox{tff}(\mbox{product\_X\_X\_4\_predicate},\mbox{conjecture})$

\medskip\textbf{ARI536=4.p} Rational: Square root of two does not exist






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(x,x) = 2/1\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI180=1.p} Integer: It must be the function of 0






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

$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(u,v) = f(u)\mbox{ and }\mbox{\$difference}(v,f(u)) = 0 ) \ \Rightarrow \ v = f(0) )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI078=1.p} Integer: Sum with zero is the identity






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,0) = x\qquad\mbox{tff}(\mbox{sum\_zero\_identity},\mbox{conjecture})$

\medskip\textbf{ARI301=1.p} Rational: Product 18/25 and -1/4 is -9/50






$\mbox{\$product}(18/25,-1/4) = -9/50\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI694=1.p} Solve simple system of linear equations






$x_0{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x0\_type},\mbox{type})$

$x_1{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x1\_type},\mbox{type})$

$x_2{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x2\_type},\mbox{type})$

$x_3{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x3\_type},\mbox{type})$

$x_4{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x4\_type},\mbox{type})$

$x_5{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x5\_type},\mbox{type})$

$x_6{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x6\_type},\mbox{type})$

$x_7{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x7\_type},\mbox{type})$

$x_8{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x8\_type},\mbox{type})$

$x_9{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x9\_type},\mbox{type})$

$x_0 = \mbox{\$sum}(\mbox{\$product}(5,x_1),1)\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(4,x_1) = \mbox{\$sum}(\mbox{\$product}(5,x_2),1)\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$product}(4,x_2) = \mbox{\$sum}(\mbox{\$product}(5,x_3),1)\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$product}(4,x_3) = \mbox{\$sum}(\mbox{\$product}(5,x_4),1)\qquad\mbox{tff}(\mbox{eq}_4,\mbox{axiom})$

$\mbox{\$product}(4,x_4) = \mbox{\$sum}(\mbox{\$product}(5,x_5),1)\qquad\mbox{tff}(\mbox{eq}_5,\mbox{axiom})$

$\mbox{\$product}(4,x_5) = \mbox{\$sum}(\mbox{\$product}(5,x_6),1)\qquad\mbox{tff}(\mbox{eq}_6,\mbox{axiom})$

$\mbox{\$product}(4,x_6) = \mbox{\$sum}(\mbox{\$product}(5,x_7),1)\qquad\mbox{tff}(\mbox{eq}_7,\mbox{axiom})$

$\mbox{\$product}(4,x_7) = \mbox{\$sum}(\mbox{\$product}(5,x_8),1)\qquad\mbox{tff}(\mbox{eq}_8,\mbox{axiom})$

$\mbox{\$product}(4,x_8) = \mbox{\$sum}(\mbox{\$product}(5,x_9),1)\qquad\mbox{tff}(\mbox{eq}_9,\mbox{axiom})$

$\mbox{\$remainder\_t}(\mbox{\$sum}(x_0,4),1953125) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI428=1.p} Real: Difference 5.55 and 3.5 is 2.05






$\mbox{\$difference}(\mbox{5.}_{55},\mbox{3.}_5) = \mbox{2.}_{05}\qquad\mbox{tff}(\mbox{real\_difference\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI389=1.p} Real: Something greatereq to 11.33






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(x,\mbox{11.}_{33})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI548=1.p} Integer: Difference -25 and product something and 0 lesseq to 0






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(\mbox{\$difference}(-25,\mbox{\$product}(x,-5)),0)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI541=1.p} Integer: Minus product -18 and -4 greatereq -75






$\mbox{\$greatereq}(\mbox{\$uminus}(\mbox{\$product}(-18,-4)),-75)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI427=1.p} Real: Difference 6.25 and 4.25 is 2.0






$\mbox{\$difference}(\mbox{6.}_{25},\mbox{4.}_{25}) = \mbox{2.}_0\qquad\mbox{tff}(\mbox{real\_difference\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI687=1.p} Expand and rewrite polynomial






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(x,x),x),y),\mbox{\$product}(\mbox{\$product}(x,x),y)),\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(3,x),x),x)),\mbox{\$product}(\mbox{\$product}(2,x),x)) = 0\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(2,x),x),x),y),\mbox{\$product}(-1,\mbox{\$product}(x,y))),y),\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(6,x),x),x)),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(2,x),x))),\mbox{\$product}(-1,\mbox{\$product}(3,x))),3) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(3,x),x),y),\mbox{\$product}(\mbox{\$product}(2,x),y)),y),\mbox{\$product}(\mbox{\$product}(9,x),x)),\mbox{\$product}(5,x)),3) = 0\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(2,\mbox{\$product}(\mbox{\$product}(x,x),x)),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$product}(5,x),\mbox{\$product}(5,\mbox{\$product}(x,x))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI446=1.p} Real: Product 5000.0 and 2.5 is not 12000.0






$\mbox{\$product}(\mbox{5000.}_0,\mbox{2.}_5)\ne \mbox{12000.}_0\qquad\mbox{tff}(\mbox{real\_product\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI350=1.p} Real: Something less than something else






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;\mbox{\$less}(x,y)\qquad\mbox{tff}(\mbox{real\_less\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI490=1.p} Real: Sum 2.4 and 3.7 is less than 7.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(\mbox{2.}_4,\mbox{3.}_7) = x \ \Rightarrow \ \mbox{\$less}(x,\mbox{7.}_0) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI614=1.p} There is an X$>$a and a Y$<$1 whose sum is 0 (X = max(a+1,0), Y = -X)






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

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

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

$( \forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(a,x) \ \Rightarrow \ p(x) )\mbox{ and }\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(x,0) \ \Rightarrow \ q(x) ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( p(x)\mbox{ and }q(y)\mbox{ and }\mbox{\$sum}(x,y) = 0 )\qquad\mbox{tff}(\mbox{interv\_a\_infty\_and\_neginfty\_1\_contain\_compl},\mbox{conjecture})$

\medskip\textbf{ARI523=1.p} Mixed: Difference -1/8 and 3/16 is rational






$\mbox{\$is\_rat}(\mbox{\$difference}(-1/8,3/16))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{28},\mbox{conjecture})$

\medskip\textbf{ARI420=1.p} Real: Sum something and -3.5 is -5.55






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,-\mbox{3.}_5) = -\mbox{5.}_{55}\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{20},\mbox{conjecture})$

\medskip\textbf{ARI216=1.p} Rational: -7/16 lesseq to something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(-7/16,x)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI701=1.p} Solve a simple system of nonlinear equations






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

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

$\mbox{\$product}(x,y) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(y,z) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$product}(a,z) = 1\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(z,y),x) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI714=1.p} floor(X+1) = floor(X)+1






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$floor}(\mbox{\$sum}(x,\mbox{1.}_0)) = \mbox{\$sum}(\mbox{\$floor}(x),\mbox{1.}_0)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI621=3.p} 12 is not a power of 2






$\mbox{pow}_2{:}\;\mbox{\$real}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{pow}_2(x)\iff ( x = \mbox{1.}_0\mbox{ or }( \mbox{\$lesseq}(\mbox{2.}_0,x)\mbox{ and }\exists y{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(\mbox{2.}_0,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \neg\,\mbox{pow}_2(\mbox{12.}_0)\qquad\mbox{tff}(\mbox{not\_pow\_of\_2}_{10},\mbox{conjecture})$

\medskip\textbf{ARI595=1.p} There is a number in [a,...,a+2] that is divisible by 3






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

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

$\forall z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(a,z)\mbox{ and }\mbox{\$lesseq}(z,\mbox{\$sum}(a,2)) ) \ \Rightarrow \ p(z) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(3,x))\qquad\mbox{tff}(\mbox{exists\_X\_in\_a\_to\_aplus2\_div}_3,\mbox{conjecture})$

\medskip\textbf{ARI611=1.p} Intervals (5,15) and (8,18) intersect






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

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

$( \forall x{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(5,x)\mbox{ and }\mbox{\$less}(x,15) )\iff p(x) )\mbox{ and }\forall x{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(8,x)\mbox{ and }\mbox{\$less}(x,18) )\iff q(x) ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;( p(x)\mbox{ and }q(x) )\qquad\mbox{tff}(\mbox{interv\_5\_15\_and\_8\_18\_intersect},\mbox{conjecture})$

\medskip\textbf{ARI259=1.p} Rational: Sum 17/4 and only 23/4 is 10/1






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(17/4,x) = 10/1 \ \Rightarrow \ x = 23/4 )\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI186=1.p} Integer: Function of two arguments






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

$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;g(u,v) = g(u,\mbox{\$sum}(v,2)) \ \Rightarrow \ ( g(3,3) = g(3,4) \ \Rightarrow \ g(3,2) = g(3,5) )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI536=3.p} Real: Square root of two exists and is rational






$\exists x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(x,x) = \mbox{2.}_0\mbox{ and }\mbox{\$is\_rat}(x) )\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI533=1.p} Mixed: Product 6.4 and 0.469 is an integer






$\mbox{\$is\_int}(\mbox{\$product}(\mbox{6.}_4,\mbox{0.}_{469}))\qquad\mbox{tff}(\mbox{anti\_mixed\_types\_problem}_{38},\mbox{conjecture})$

\medskip\textbf{ARI037=1.p} Integer: -5 not greater than -3






$\neg\,\mbox{\$greater}(-5,-3)\qquad\mbox{tff}(\mbox{n5\_not\_greater\_n}_3,\mbox{conjecture})$

\medskip\textbf{ARI671=1.p} a*a = 2 is unsatisfiable






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

$\mbox{\$product}(a,a) = 2\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

\medskip\textbf{ARI206=1.p} Rational: Something lesseq to 19/25






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(x,19/25)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI503=1.p} Mixed: 11.33 coerced to integer is an integer






$\mbox{\$is\_int}(\mbox{\$to\_int}(\mbox{11.}_{33}))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI022=1.p} Integer: -4 lesseq to -2






$\mbox{\$lesseq}(-4,-2)\qquad\mbox{tff}(\mbox{n4\_lesseq\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI249=1.p} Rational: Sum of 1/1 and 5/8 is 13/8






$\mbox{\$sum}(1/1,5/8) = 13/8\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI240=1.p} Rational: -5/8 not greatereq to -1/4






$\neg\,\mbox{\$greatereq}(-5/8,-1/4)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI654=1.p} y $>$= 5*x - 1, y $\leq$ 5*x, 5*z $\leq$ y - 1, 5*z $>$= y - 2 are unsat






$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$\mbox{\$greatereq}(y,\mbox{\$difference}(\mbox{\$product}(5,x),1))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(y,\mbox{\$product}(5,x))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(\mbox{\$product}(5,z),\mbox{\$difference}(y,1))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

$\mbox{\$greatereq}(\mbox{\$product}(5,z),\mbox{\$difference}(y,2))\qquad\mbox{tff}(\mbox{conj}_{003},\mbox{axiom})$

\medskip\textbf{ARI369=1.p} Real: Something lesseq to 0.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(x,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI248=1.p} Rational: Sum of 9/16 and 1/2 is 17/16






$\mbox{\$sum}(9/16,1/2) = 17/16\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI638=1.p} Example 14






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

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

$c{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;f(\mbox{\$sum}(x,y)) = \mbox{\$product}(f(x),f(y))\qquad\mbox{tff}(\mbox{f\_feature},\mbox{axiom})$

$\mbox{\$greater}(f(\mbox{\$sum}(a,b)),\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(f(\mbox{\$sum}(c,d)),\mbox{2.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(f(\mbox{\$sum}(a,\mbox{\$sum}(b,\mbox{\$sum}(c,d)))),\mbox{4.}_0)\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI514=1.p} Mixed: -2.4 is lesseq to 2 coerced to real






$\mbox{\$lesseq}(-\mbox{2.}_4,\mbox{\$to\_real}(2))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI293=1.p} Rational: Product something and 11/2 is 121/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(x,11/2) = 121/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI442=1.p} Real: Product 5.5 and 5.5 is 30.25






$\mbox{\$product}(\mbox{5.}_5,\mbox{5.}_5) = \mbox{30.}_{25}\qquad\mbox{tff}(\mbox{real\_product\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI525=1.p} Mixed: ((- 7/15) + 4/15) coerced to integer is greatereq to 0






$\mbox{\$greatereq}(\mbox{\$to\_int}(\mbox{\$sum}(\mbox{\$uminus}(7/15),4/15)),0)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{30},\mbox{conjecture})$

\medskip\textbf{ARI564=1.p} Real: -(-6.48 * -2.25) greatereq to -15.62






$\mbox{\$greatereq}(\mbox{\$uminus}(\mbox{\$product}(-\mbox{6.}_{48},-\mbox{2.}_{25})),-\mbox{15.}_{62})\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI415=1.p} Real: Sum 14.65 and -14.65 is 0.0






$\mbox{\$sum}(\mbox{14.}_{65},-\mbox{14.}_{65}) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI212=1.p} Rational: 3/4 not lesseq to -3/4






$\neg\,\mbox{\$lesseq}(3/4,-3/4)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI020=1.p} Integer: -2 lesseq to -2






$\mbox{\$lesseq}(-2,-2)\qquad\mbox{tff}(\mbox{n2\_lesseq\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI080=1.p} Integer: Sum 4 and 4 is 8






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8\mbox{ and }x = 4\mbox{ and }y = 4 )\qquad\mbox{tff}(\mbox{sum\_4\_4}_8,\mbox{conjecture})$

\medskip\textbf{ARI098=1.p} Integer: Product of -2 and -3 is 6






$\mbox{\$product}(-2,-3) = 6\qquad\mbox{tff}(\mbox{product\_n2\_n3}_6,\mbox{conjecture})$

\medskip\textbf{ARI629=1.p} Example 1






$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$greater}(x,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greatereq}(\mbox{\$product}(\mbox{\$sum}(\mbox{1.}_0,\mbox{\$product}(y,y)),x),\mbox{\$sum}(\mbox{1.}_0,\mbox{\$product}(y,y)))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI065=1.p} Integer: Sum 2 and only 3 is 5






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,x) = 5 \ \Rightarrow \ x = 3 )\qquad\mbox{tff}(\mbox{sum2\_only\_3}_5,\mbox{conjecture})$

\medskip\textbf{ARI716=1.p} floor(X+0.5) $>$= floor(X)






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_5)),\mbox{\$floor}(x))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI006=1.p} Integer: Something less than something






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$less}(x,y)\qquad\mbox{tff}(\mbox{something\_less\_something},\mbox{conjecture})$

\medskip\textbf{ARI182=1.p} Integer: Increasing function in a formula






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

$\forall u{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(u),u) \ \Rightarrow \ \forall v{:}\;\mbox{\$int},w{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(f(\mbox{\$sum}(f(v),w)),\mbox{\$sum}(\mbox{\$sum}(v,w),2))\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI374=1.p} Real: 9.58 greater than 9.53






$\mbox{\$greater}(\mbox{9.}_{58},\mbox{9.}_{53})\qquad\mbox{tff}(\mbox{real\_greater\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI043=1.p} Integer: 6 not greatereq to 7






$\neg\,\mbox{\$greatereq}(6,7)\qquad\mbox{tff}(\mbox{prove\_6\_not\_greatereq}_7,\mbox{conjecture})$

\medskip\textbf{ARI504=1.p} Mixed: 2.05 coerced to rational is a rational






$\mbox{\$is\_rat}(\mbox{\$to\_rat}(\mbox{2.}_{05}))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI402=1.p} Real: Sum 4.25 and 5.75 is 10.0






$\mbox{\$sum}(\mbox{4.}_{25},\mbox{5.}_{75}) = \mbox{10.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI351=1.p} Real: -3.25 less than 3.25






$\mbox{\$less}(-\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_less\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI092=1.p} Integer: Difference 5 and only 3 is 2






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$difference}(5,x) = 2 \ \Rightarrow \ x = 3 )\qquad\mbox{tff}(\mbox{diff\_5\_only\_3}_2,\mbox{conjecture})$

\medskip\textbf{ARI669=1.p} a * b * b * c = 0 and a = 0 | b = 0 | c = 0 are equivalent






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,b),b),c) = 0\iff ( c = 0\mbox{ or }b = 0\mbox{ or }a = 0 )\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI698=1.p} Solve simple system of linear equations, with parameter N






$n{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$x_6{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x6\_type},\mbox{type})$

$x_5{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x5\_type},\mbox{type})$

$x_4{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x4\_type},\mbox{type})$

$x_3{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x3\_type},\mbox{type})$

$x_2{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x2\_type},\mbox{type})$

$x_1{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x1\_type},\mbox{type})$

$n = 1000\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(1,x_6),\mbox{\$product}(n,x_5)),\mbox{\$product}(n,x_4)),\mbox{\$product}(n,x_3)),\mbox{\$product}(n,x_2)),\mbox{\$product}(n,x_1)) = 0\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$product}(n,x_6),x_5),x_4),x_3),x_2),x_1) = 0\qquad\mbox{tff}(\mbox{eq}_3,\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(1,x_5)),x_4),x_3),x_2),x_1) = 0\qquad\mbox{tff}(\mbox{eq}_4,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(1,x_4)),\mbox{\$product}(1,x_3)),\mbox{\$product}(1,x_2)),\mbox{\$product}(1,x_1)) = 0\qquad\mbox{tff}(\mbox{eq}_5,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),x_4),\mbox{\$product}(1,x_3)),\mbox{\$product}(1,x_2)),\mbox{\$product}(1,x_1)) = 0\qquad\mbox{tff}(\mbox{eq}_6,\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$difference}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(0,x_4)),x_3),x_2),x_1) = 0\qquad\mbox{tff}(\mbox{eq}_7,\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(0,x_4)),\mbox{\$product}(1,x_3)),x_2),x_1) = 0\qquad\mbox{tff}(\mbox{eq}_8,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(0,x_4)),\mbox{\$product}(0,x_3)),\mbox{\$product}(1,x_2)),\mbox{\$product}(1,x_1)) = 0\qquad\mbox{tff}(\mbox{eq}_9,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(0,x_4)),\mbox{\$product}(0,x_3)),x_2),\mbox{\$product}(1,x_1)) = 0\qquad\mbox{tff}(\mbox{eq}_{10},\mbox{axiom})$

$\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(n,x_6),\mbox{\$product}(0,x_5)),\mbox{\$product}(0,x_4)),\mbox{\$product}(0,x_3)),\mbox{\$product}(0,x_2)),x_1) = 0\qquad\mbox{tff}(\mbox{eq}_{11},\mbox{axiom})$

$x_6 = 0\mbox{ and }x_5 = 0\mbox{ and }x_4 = 0\mbox{ and }x_3 = 0\mbox{ and }x_2 = 0\mbox{ and }x_1 = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI623=1.p} There is no strictly mon fct from \$rat or \$real to a non-dense set






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

$\neg\,\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$greater}(x,y) \ \Rightarrow \ \mbox{\$greater}(f(x),\mbox{\$sum}(f(y),1/1)) )\qquad\mbox{tff}(\mbox{not\_ex\_mon\_mapping\_rat\_to\_nondense},\mbox{conjecture})$

\medskip\textbf{ARI275=1.p} Rational: Difference 90/117 and 25/117 is 5/9






$\mbox{\$difference}(90/117,25/117) = 5/9\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI366=1.p} Real: -8.69 lesseq to -3.25






$\mbox{\$lesseq}(-\mbox{8.}_{69},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI399=1.p} Real: 14.75 is not 9.69






$\mbox{14.}_{75}\ne \mbox{9.}_{69}\qquad\mbox{tff}(\mbox{real\_not\_equal\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI668=1.p} 11*a + 7*b = 1 and c $>$= a imply a*c $\leq$ 0 | a*c $>$= 4






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$product}(11,a),\mbox{\$product}(7,b)) = 1\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(a,c)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$lesseq}(4,\mbox{\$product}(a,c))\mbox{ or }\mbox{\$greatereq}(0,\mbox{\$product}(a,c))\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI419=1.p} Real: Sum 3.5 and -3.5 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{3.}_5,-\mbox{3.}_5) = x\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI551=1.p} Rational: 7/8 less than sum 5/8 and 5/16






$\mbox{\$less}(7/8,\mbox{\$sum}(5/8,5/16))\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI178=1.p} Integer: It can't be 0






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int},w{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(2,u),v),w) = 10\mbox{ and }\mbox{\$sum}(\mbox{\$sum}(u,\mbox{\$product}(2,v)),w) = 10 ) \ \Rightarrow \ w\ne 0 )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI506=1.p} Mixed: 11/2 is 5.5 coerced to rational






$11/2 = \mbox{\$to\_rat}(\mbox{5.}_5)\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI046=1.p} Integer: Something greatereq to something






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$greatereq}(x,y)\qquad\mbox{tff}(\mbox{something\_greatereq\_something},\mbox{conjecture})$

\medskip\textbf{ARI448=1.p} Real: Product only 7.25 and 4.0 is 29.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(x,\mbox{4.}_0) = \mbox{29.}_0 \ \Rightarrow \ x = \mbox{7.}_{25} )\qquad\mbox{tff}(\mbox{real\_product\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI618=1.p} Absolute value (unusually defined) is idempotent






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

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,f(x))\mbox{ and }\mbox{\$lesseq}(\mbox{\$uminus}(x),f(x))\mbox{ and }( \mbox{\$lesseq}(f(x),x)\mbox{ or }\mbox{\$lesseq}(f(x),\mbox{\$uminus}(x)) ) ) \ \Rightarrow \ \forall x{:}\;\mbox{\$int}{:}\;f(f(x)) = f(x)\qquad\mbox{tff}(\mbox{absolute\_value\_idempotent},\mbox{conjecture})$

\medskip\textbf{ARI263=1.p} Rational: Sum 3/8 and -3/8 is 0/1






$\mbox{\$sum}(3/8,-3/8) = 0/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI663=1.p} Prove that a*b=15 implies a $\leq$ 15 \& a $>$= -15 \& a != 0






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

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

$\mbox{\$product}(a,b) = 15\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$a\ne 0\mbox{ and }\mbox{\$greatereq}(a,-15)\mbox{ and }\mbox{\$lesseq}(a,15)\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI536=1.p} Real: Square root of two exists






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(x,x) = \mbox{2.}_0\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI612=1.p} Interval (8,12) is contained in (5,15)






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

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

$( \forall x{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(5,x)\mbox{ and }\mbox{\$less}(x,15) )\iff p(x) )\mbox{ and }\forall x{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(8,x)\mbox{ and }\mbox{\$less}(x,12) )\iff q(x) ) ) \ \Rightarrow \ \forall x{:}\;\mbox{\$int}{:}\;( q(x) \ \Rightarrow \ p(x) )\qquad\mbox{tff}(\mbox{interv\_8\_12\_subset\_5}_{15},\mbox{conjecture})$

\medskip\textbf{ARI410=1.p} Real: Sum only 4.25 and 5.75 is 10.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(x,\mbox{5.}_{75}) = \mbox{10.}_0 \ \Rightarrow \ x = \mbox{4.}_{25} )\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI422=1.p} Real: Sum something and -2.05 is 3.5






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,-\mbox{2.}_{05}) = \mbox{3.}_5\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{22},\mbox{conjecture})$

\medskip\textbf{ARI692=1.p} Solve simple system of linear equations






$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$product}(2,z),x) = 2\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(2,y),\mbox{\$product}(3,x)) = 1\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

\medskip\textbf{ARI342=1.p} Rational: Sum of something and itself less than -13/2






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(\mbox{\$sum}(x,x),-13/2)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI579=2.p} Inequation system is solvable over \$rat (e.g., X = Y = 1/2)






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$less}(0/1,x)\mbox{ and }\mbox{\$less}(0/1,y)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(3/1,x),\mbox{\$product}(4/1,y)),6/1) )\qquad\mbox{tff}(\mbox{ineq\_sys\_rat\_solvable},\mbox{conjecture})$

\medskip\textbf{ARI387=1.p} Real: 3.25 not greatereq to 7.8






$\neg\,\mbox{\$greatereq}(\mbox{3.}_{25},\mbox{7.}_8)\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI513=1.p} Mixed: 0.5 is lesseq to 1/2 coerced to real






$\mbox{\$lesseq}(\mbox{0.}_5,\mbox{\$to\_real}(1/2))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI234=1.p} Rational: Something greatereq to 3/16






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greatereq}(x,3/16)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI056=1.p} Integer: Something not 12






$\exists x{:}\;\mbox{\$int}{:}\;x\ne 12\qquad\mbox{tff}(\mbox{something\_not}_{12},\mbox{conjecture})$

\medskip\textbf{ARI609=1.p} For mon. f, 0$\leq$a-b =$>$ b$\leq$a =$>$ f(b)$\leq$f(a) =$>$ 0$\leq$f(a)-f(b)






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

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

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

$( \forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y) \ \Rightarrow \ \mbox{\$lesseq}(f(x),f(y)) )\mbox{ and }\mbox{\$lesseq}(0,\mbox{\$sum}(a,\mbox{\$uminus}(b))) ) \ \Rightarrow \ \mbox{\$lesseq}(0,\mbox{\$sum}(f(a),\mbox{\$uminus}(f(b))))\qquad\mbox{tff}(\mbox{f\_mon\_implies\_f\_a\_b}_1,\mbox{conjecture})$

\medskip\textbf{ARI239=1.p} Rational: -3/4 not greatereq to 3/4






$\neg\,\mbox{\$greatereq}(-3/4,3/4)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI090=1.p} Integer: Difference 5 and 3 is only 2






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$difference}(5,3) = x \ \Rightarrow \ x = 2 )\qquad\mbox{tff}(\mbox{diff\_5\_3\_only}_2,\mbox{conjecture})$

\medskip\textbf{ARI305=1.p} Rational: Product 4/5 and -15/16 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$product}(4/5,-15/16) = x\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{22},\mbox{conjecture})$

\medskip\textbf{ARI005=1.p} Integer: 12 less than something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(12,x)\qquad\mbox{tff}(\mbox{prove\_12\_less\_something},\mbox{conjecture})$

\medskip\textbf{ARI123=1.p} Integer: Associative product exists






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( ( \mbox{\$product}(x,y) = z_1\mbox{ and }\mbox{\$product}(z_1,z) = z_2\mbox{ and }\mbox{\$product}(y,z) = z_3\mbox{ and }\mbox{\$product}(x,z_3) = z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{tff}(\mbox{associative\_product\_exists},\mbox{conjecture})$

\medskip\textbf{ARI201=1.p} Rational: Something less than -13/4






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$less}(x,-13/4)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI083=1.p} Integer: Associative sum exists






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$sum}(z_1,z) = z_2\mbox{ and }\mbox{\$sum}(y,z) = z_3\mbox{ and }\mbox{\$sum}(x,z_3) = z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{tff}(\mbox{associative\_sum\_exists},\mbox{conjecture})$

\medskip\textbf{ARI534=1.p} Mixed: Sum 0.5 coerced to integer and 1 is not a rational






$\neg\,\mbox{\$is\_rat}(\mbox{\$sum}(\mbox{\$to\_int}(\mbox{0.}_5),1))\qquad\mbox{tff}(\mbox{anti\_mixed\_types\_problem}_{39},\mbox{conjecture})$

\medskip\textbf{ARI536=2.p} Real: Square root of two exists and is not rational






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(x,x) = \mbox{2.}_0 \ \Rightarrow \ \neg\,\mbox{\$is\_rat}(x) )\qquad\mbox{tff}(\mbox{the},\mbox{conjecture})$

\medskip\textbf{ARI620=1.p} 8 is a power of 2






$\mbox{pow}_2{:}\;\mbox{\$int}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{pow2\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{pow}_2(x)\iff ( x = 1\mbox{ or }( \mbox{\$lesseq}(2,x)\mbox{ and }\exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(2,y) = x\mbox{ and }\mbox{pow}_2(y) ) ) ) ) \ \Rightarrow \ \mbox{pow}_2(8)\qquad\mbox{tff}(\mbox{pow\_of\_2}_8,\mbox{conjecture})$

\medskip\textbf{ARI388=1.p} Real: 14.68 greatereq to something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greatereq}(\mbox{14.}_{68},x)\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI069=1.p} Integer: Sum 5 and -5 is 0






$\mbox{\$sum}(5,-5) = 0\qquad\mbox{tff}(\mbox{sum\_5\_n5}_0,\mbox{conjecture})$

\medskip\textbf{ARI575=3.p} Inequation system has more than one real solution






$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real},z{:}\;\mbox{\$real}{:}\;( ( \mbox{\$less}(\mbox{3.}_0,x)\mbox{ and }\mbox{\$less}(x,y)\mbox{ and }\mbox{\$less}(y,z)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$sum}(x,\mbox{\$product}(\mbox{2.}_0,y)),\mbox{\$product}(\mbox{3.}_0,z)),\mbox{35.}_0) )\iff ( x = \mbox{4.}_0\mbox{ and }y = \mbox{5.}_0\mbox{ and }z = \mbox{6.}_0 ) )\qquad\mbox{tff}(\mbox{ineq\_sys\_has\_1\_int\_sol},\mbox{conjecture})$

\medskip\textbf{ARI540=1.p} Integer: Product 5 an 7 lesseq to 36






$\mbox{\$lesseq}(\mbox{\$product}(5,7),36)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI274=1.p} Rational: Difference 5/12 and 1/2 is -1/12






$\mbox{\$difference}(5/12,1/2) = -1/12\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI412=1.p} Real: Sum -3.5 and -0.5 is -4.0






$\mbox{\$sum}(-\mbox{3.}_5,-\mbox{0.}_5) = -\mbox{4.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI175=1.p} Integer: Formula equals 23






$\exists u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(\mbox{\$product}(3,u),\mbox{\$product}(5,v)) = 23\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI418=1.p} Real: Sum 3.5 and -2.05 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(\mbox{3.}_5,-\mbox{2.}_{05}) = x\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI055=1.p} Integer: 31 not 12






$31\ne 12\qquad\mbox{tff}(\mbox{prove\_31\_not}_{12},\mbox{conjecture})$

\medskip\textbf{ARI464=1.p} Real: - - -70.4 is -70.4






$\mbox{\$uminus}(\mbox{\$uminus}(-\mbox{70.}_4)) = -\mbox{70.}_4\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI299=1.p} Rational: Product -18/25 and -1/4 is 9/50






$\mbox{\$product}(-18/25,-1/4) = 9/50\qquad\mbox{tff}(\mbox{rat\_product\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI385=1.p} Real: 3.25 greatereq to 3.25






$\mbox{\$greatereq}(\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI271=1.p} Rational: Sum something and -12/119 is 0/1






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,-12/119) = 0/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{26},\mbox{conjecture})$

\medskip\textbf{ARI725=1.p} floor(2*X) can be greater than 2*floor(X)






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{\$floor}(\mbox{\$product}(\mbox{2.}_0,x)),\mbox{\$product}(\mbox{2.}_0,\mbox{\$floor}(x)))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI088=1.p} Integer: Sum only 4 and only 4 is 8






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = 8 \ \Rightarrow \ ( x = 4\mbox{ and }y = 4 ) )\qquad\mbox{tff}(\mbox{anti\_sum\_only\_4\_only\_4}_8,\mbox{conjecture})$

\medskip\textbf{ARI379=1.p} Real: -3.25 greater than -8.69






$\mbox{\$greater}(-\mbox{3.}_{25},-\mbox{8.}_{69})\qquad\mbox{tff}(\mbox{real\_greater\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI452=1.p} Real: Product 70.4 and -0.05 is -3.52






$\mbox{\$product}(\mbox{70.}_4,-\mbox{0.}_{05}) = -\mbox{3.}_{52}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{17},\mbox{conjecture})$

\medskip\textbf{ARI213=1.p} Rational: -1/4 not lesseq to -5/8






$\neg\,\mbox{\$lesseq}(-1/4,-5/8)\qquad\mbox{tff}(\mbox{rat\_lesseq\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI510=1.p} Mixed: sum 1/2 and 1/4 is less than 1 coerced to rational






$\mbox{\$less}(\mbox{\$sum}(1/2,1/4),\mbox{\$to\_rat}(1))\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{15},\mbox{conjecture})$

\medskip\textbf{ARI084=1.p} Integer: Sum 2 and 3 is 5 in a predicate






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

$p(\mbox{\$sum}(2,3)) \ \Rightarrow \ p(5)\qquad\mbox{tff}(\mbox{sum\_2\_3\_5\_predicate},\mbox{conjecture})$

\medskip\textbf{ARI733=1.p} Real inequation system has a solution with integer X






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$is\_int}(x)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_5,x),y),\mbox{0.}_{25})\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(\mbox{\$product}(\mbox{4.}_0,x),y),\mbox{30.}_5)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(x,\mbox{\$product}(-\mbox{2.}_5,y)),-\mbox{13.}_0) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI107=1.p} Integer: Product of -2 and -5 is 10






$\mbox{\$product}(-2,-5) = 10\qquad\mbox{tff}(\mbox{product\_n2\_n5}_{10},\mbox{conjecture})$

\medskip\textbf{ARI014=1.p} Integer: 2 lesseq to 2






$\mbox{\$lesseq}(2,2)\qquad\mbox{tff}(\mbox{prove\_2\_lesseq}_2,\mbox{conjecture})$

\medskip\textbf{ARI197=1.p} Rational: -5/8 less than -3/8






$\mbox{\$less}(-5/8,-3/8)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI423=1.p} Real: Sum something and -3500000.0 is 0.0






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,-\mbox{3500000.}_0) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{23},\mbox{conjecture})$

\medskip\textbf{ARI013=1.p} Integer: -2 less than something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(-2,x)\qquad\mbox{tff}(\mbox{n2\_less\_something},\mbox{conjecture})$

\medskip\textbf{ARI570=1.p} Weakening an inequation






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(y,x) \ \Rightarrow \ \mbox{\$less}(y,\mbox{\$sum}(x,3)) )\qquad\mbox{tff}(\mbox{weakening\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI228=1.p} Rational: Something greater than -75/112






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(x,-75/112)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{ARI421=1.p} Real: Sum something and -3.5 is -6.5






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,-\mbox{3.}_5) = -\mbox{6.}_5\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{21},\mbox{conjecture})$

\medskip\textbf{ARI460=1.p} Real: 0.6 is - -0.6






$\mbox{0.}_6 = \mbox{\$uminus}(-\mbox{0.}_6)\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI731=1.p} If X is an integer, then 2*X+3 is an integer






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$is\_int}(x) \ \Rightarrow \ \mbox{\$is\_int}(\mbox{\$sum}(\mbox{\$product}(x,\mbox{2.}_0),\mbox{3.}_0)) )\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI227=1.p} Rational: 0/1 greater than something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(0/1,x)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI573=1.p} Three inequations imply a fourth one






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(1,\mbox{\$sum}(\mbox{\$product}(x,2),\mbox{\$uminus}(y)))\mbox{ and }\mbox{\$lesseq}(1,\mbox{\$sum}(\mbox{\$product}(y,2),\mbox{\$uminus}(z)))\mbox{ and }\mbox{\$lesseq}(1,\mbox{\$sum}(\mbox{\$product}(z,2),\mbox{\$uminus}(x))) ) \ \Rightarrow \ \mbox{\$lesseq}(2,\mbox{\$sum}(\mbox{\$sum}(x,y),z)) )\qquad\mbox{tff}(\mbox{impl\_3\_ineq},\mbox{conjecture})$

\medskip\textbf{ARI719=1.p} floor(X+0.5) can be equal to floor(X-0.3)






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_5)) = \mbox{\$floor}(\mbox{\$sum}(x,-\mbox{0.}_3))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI499=1.p} Mixed: 9.75 is not an integer






$\neg\,\mbox{\$is\_int}(\mbox{9.}_{75})\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI365=1.p} Real: -3.25 lesseq to 3.25






$\mbox{\$lesseq}(-\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI059=1.p} Integer: Sum 23 and 34 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(23,34) = x\qquad\mbox{tff}(\mbox{sum\_23\_34\_something},\mbox{conjecture})$

\medskip\textbf{ARI115=1.p} Integer: Product of something and -5 is 10






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,-5) = 10\qquad\mbox{tff}(\mbox{product\_what\_n5}_{10},\mbox{conjecture})$

\medskip\textbf{ARI699=1.p} Nonlinear inequality reasoning






$z{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{z\_type},\mbox{type})$

$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$greater}(x,0)\qquad\mbox{tff}(\mbox{ineq}_1,\mbox{axiom})$

$\mbox{\$greater}(y,0)\qquad\mbox{tff}(\mbox{ineq}_2,\mbox{axiom})$

$\mbox{\$lesseq}(0,\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(z,z),x),\mbox{\$product}(-1,\mbox{\$product}(y,x))))\qquad\mbox{tff}(\mbox{ineq}_3,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(2,z),z) = y\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

\medskip\textbf{ARI220=1.p} Rational: 13/24 greater than something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(13/24,x)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI639=1.p} Example 15






$\mbox{eps}{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{eps\_type},\mbox{type})$

$c{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$k{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{k\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$n{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{n\_type},\mbox{type})$

$\mbox{\$greatereq}(n,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$less}(n,\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{2.}_0),\mbox{\$product}(k,x)))\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greater}(c,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$greater}(\mbox{eps},\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{03},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{eps},\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{04},\mbox{hypothesis})$

$\mbox{\$less}(\mbox{\$product}(\mbox{\$sum}(\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{3.}_0),\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{3.}_0,c)),\mbox{eps})),\mbox{1.}_0),n),\mbox{\$product}(k,x))\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI528=1.p} Mixed: Mad mixture 1






$\mbox{\$sum}(\mbox{\$to\_int}(\mbox{50.}_{98}),\mbox{\$product}(2,\mbox{\$product}(\mbox{\$to\_int}(11/2),5))) = 100\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{33},\mbox{conjecture})$

\medskip\textbf{ARI435=1.p} Real: Difference -1.28 and 1.0 is something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$difference}(-\mbox{1.}_{28},\mbox{1.}_0) = x\qquad\mbox{tff}(\mbox{real\_difference\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI720=1.p} floor(X+0.5) $>$ floor(X-0.5)






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$greater}(\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_5)),\mbox{\$floor}(\mbox{\$sum}(x,-\mbox{0.}_5)))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI040=1.p} Integer: -5 greater than something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(-5,x)\qquad\mbox{tff}(\mbox{n5\_greater\_something},\mbox{conjecture})$

\medskip\textbf{ARI558=1.p} Rational: -1/4 - (something * -1/20) lesseq to 0/1






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$lesseq}(\mbox{\$difference}(-1/4,\mbox{\$product}(x,-1/20)),0/1)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{18},\mbox{conjecture})$

\medskip\textbf{ARI713=1.p} ceiling is idempotent






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$ceiling}(\mbox{\$ceiling}(x)) = \mbox{\$ceiling}(x)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI657=1.p} Satisfy a $\leq$ -3, a*b $\leq$ 5






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

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

$\mbox{\$lesseq}(a,-3)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$greatereq}(5,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

\medskip\textbf{ARI008=1.p} Integer: -4 less than -2






$\mbox{\$less}(-4,-2)\qquad\mbox{tff}(\mbox{n4\_less\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI708=1.p} Expand and rewrite polynomials






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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

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

$\mbox{\$sum}(a,\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$sum}(c,d),\mbox{\$sum}(d,c)))) = 0\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(\mbox{\$difference}(c,d),\mbox{\$difference}(c,d)) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$sum}(a,b),\mbox{\$product}(-1,\mbox{\$product}(2,\mbox{\$sum}(\mbox{\$product}(c,c),\mbox{\$product}(d,d))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI409=1.p} Real: Sum 4.25 and 5.75 is only 10.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(\mbox{4.}_{25},\mbox{5.}_{75}) = x \ \Rightarrow \ x = \mbox{10.}_0 )\qquad\mbox{tff}(\mbox{real\_sum\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI646=1.p} Simple reasoning about linear inequalities






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

$( \mbox{\$greater}(10,a)\mbox{ and }\mbox{\$lesseq}(0,a) )\iff ( a = 9\mbox{ or }a = 8\mbox{ or }a = 7\mbox{ or }a = 6\mbox{ or }a = 5\mbox{ or }a = 4\mbox{ or }a = 3\mbox{ or }a = 2\mbox{ or }a = 1\mbox{ or }a = 0 )\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI695=1.p} $<$A one line description of the problem$>$






$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(-1,d),\mbox{\$sum}(2,c)),\mbox{\$product}(-1,\mbox{\$product}(d,\mbox{\$difference}(\mbox{\$product}(-1,2),c)))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI642=1.p} Example 23






$m{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{m\_type},\mbox{type})$

$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

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

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

$\forall m{:}\;\mbox{\$real}{:}\;( \mbox{\$greater}(m,\mbox{0.}_0) \ \Rightarrow \ \mbox{\$less}(f(\mbox{\$ceiling}(m)),\mbox{\$sum}(a,\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$ceiling}(m)),\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),b)))) )\qquad\mbox{tff}(\mbox{ceiling\_property},\mbox{axiom})$

$\mbox{\$less}(a,b)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(x,a)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$greatereq}(m,\mbox{\$product}(\mbox{\$quotient}(\mbox{1.}_0,\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),x)),\mbox{\$sum}(\mbox{\$product}(-\mbox{1.}_0,a),b)))\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$less}(f(\mbox{\$ceiling}(m)),x)\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI232=1.p} Rational: 1/4 not greatereq 5/12






$\neg\,\mbox{\$greatereq}(1/4,5/12)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI361=1.p} Real: Something lesseq to 14.68






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(x,\mbox{14.}_{68})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_4,\mbox{conjecture})$

\medskip\textbf{ARI231=1.p} Rational: 5/12 greatereq to 1/4






$\mbox{\$greatereq}(5/12,1/4)\qquad\mbox{tff}(\mbox{rat\_greatereq\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI021=1.p} Integer: -2 lesseq to 2






$\mbox{\$lesseq}(-2,2)\qquad\mbox{tff}(\mbox{n2\_lesseq}_2,\mbox{conjecture})$

\medskip\textbf{ARI537=1.p} Integer: 12 less than sum 5 and 8






$\mbox{\$less}(12,\mbox{\$sum}(5,8))\qquad\mbox{tff}(\mbox{int\_combined\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI592=1.p} If Z $>$ 2, there is an X in the interval (0,Z) different from Y






$\forall y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(2,z) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,x)\mbox{ and }\mbox{\$less}(x,z)\mbox{ and }y\ne x ) )\qquad\mbox{tff}(\mbox{exists\_X\_0\_Z\_noteq\_Y},\mbox{conjecture})$

\medskip\textbf{ARI553=1.p} Rational: Sum 9/17 and 3/5 greater than -8/145






$\mbox{\$greater}(\mbox{\$sum}(9/17,3/5),-8/145)\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI591=1.p} There is an X in the interval (0,3) that is different from Y






$\forall y{:}\;\mbox{\$int}{:}\;\exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,x)\mbox{ and }\mbox{\$less}(x,3)\mbox{ and }y\ne x )\qquad\mbox{tff}(\mbox{exists\_X\_0\_3\_noteq\_Y},\mbox{conjecture})$

\medskip\textbf{ARI063=1.p} Integer: Sum 2 and 3 is only 5






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,3) = x \ \Rightarrow \ x = 5 )\qquad\mbox{tff}(\mbox{sum\_2\_3\_only}_5,\mbox{conjecture})$

\medskip\textbf{ARI085=1.p} Integer: Sum something and another thing is 7, in a predicate






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

$( p(3,4)\mbox{ and }\neg\,p(1,6) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( p(x,y)\mbox{ and }\mbox{\$sum}(x,y) = 7 )\qquad\mbox{tff}(\mbox{sum\_X\_Y\_7\_predicate},\mbox{conjecture})$

\medskip\textbf{ARI344=1.p} Rational: (16/5 * something) + (34/5 * something else) is 273/25






$\exists x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(\mbox{\$product}(16/5,x),\mbox{\$product}(34/5,y)) = 273/25\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI652=1.p} Prove that a + 4 $>$ 2, -2 $\leq$ -3 - a imply a*a = 1






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

$\mbox{\$less}(2,\mbox{\$sum}(a,4))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(-2,\mbox{\$difference}(-3,a))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$product}(a,a) = 1\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{conjecture})$

\medskip\textbf{ARI284=1.p} Rational: Product 1/3 and 3/4 is 1/4






$\mbox{\$product}(1/3,3/4) = 1/4\qquad\mbox{tff}(\mbox{rat\_product\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI665=1.p} a*b $\leq$ -36 | a*b $>$= -2 implies 5*a + 11*b = 1






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

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

$\mbox{\$lesseq}(-2,\mbox{\$product}(a,b))\mbox{ or }\mbox{\$greatereq}(-36,\mbox{\$product}(a,b))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(5,a),\mbox{\$product}(11,b)) = 1\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{conjecture})$

\medskip\textbf{ARI313=1.p} Rational: - - -31/8 is -31/8






$\mbox{\$uminus}(\mbox{\$uminus}(-31/8)) = -31/8\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI264=1.p} Rational: Sum -1/4 and -1/2 is something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(-1/4,-1/2) = x\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{19},\mbox{conjecture})$

\medskip\textbf{ARI172=1.p} Integer: Sum of something and itself is less than -10






$\exists u{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$sum}(u,u),-10)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI640=1.p} Example 18






$x{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

$y{:}\;\mbox{\$real}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

$\mbox{\$greater}(x,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis},\mbox{hypothesis})$

$\mbox{\$greater}(y,\mbox{0.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{01},\mbox{hypothesis})$

$\mbox{\$less}(y,\mbox{1.}_0)\qquad\mbox{tff}(\mbox{hypothesis}_{02},\mbox{hypothesis})$

$\mbox{\$greater}(\mbox{\$product}(x,y),\mbox{\$sum}(x,y))\qquad\mbox{tff}(\mbox{hypothesis}_{03},\mbox{hypothesis})$

$\mbox{\$false}\qquad\mbox{tff}(\mbox{conclusion},\mbox{conjecture})$

\medskip\textbf{ARI380=1.p} Real: -3.25 not greater than 3.25






$\neg\,\mbox{\$greater}(-\mbox{3.}_{25},\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greater\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI689=1.p} Rewrite polynomial using linear equations






$y{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{y\_type},\mbox{type})$

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

$x{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{x\_type},\mbox{type})$

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

$\mbox{\$product}(98164184,y) = a\qquad\mbox{tff}(\mbox{eq}_1,\mbox{axiom})$

$\mbox{\$product}(6472,x) = b\qquad\mbox{tff}(\mbox{eq}_2,\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(1,\mbox{\$product}(\mbox{\$product}(y,x),b)),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(166,\mbox{\$product}(\mbox{\$product}(x,b),a)),\mbox{\$product}(2427,\mbox{\$product}(\mbox{\$product}(x,x),a))),\mbox{\$product}(-2523496,\mbox{\$product}(\mbox{\$product}(y,b),b))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI587=1.p} For X $>$ 1, there is a Y between X+2 and 3X (e.g., Y = 2X + 1)






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(1,x) \ \Rightarrow \ \exists y{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(\mbox{\$sum}(x,2),y)\mbox{ and }\mbox{\$less}(y,\mbox{\$product}(x,3)) ) )\qquad\mbox{tff}(\mbox{exists\_Y\_between\_Xplus2\_and\_3X},\mbox{conjecture})$

\medskip\textbf{ARI280=1.p} Rational: Difference 5/2 and only 5/8 is 15/8






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$difference}(5/2,x) = 15/8 \ \Rightarrow \ x = 5/8 )\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI261=1.p} Rational: Sum 17/4 and -23/4 is -3/2






$\mbox{\$sum}(17/4,-23/4) = -3/2\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI371=1.p} Real: -3.25 lesseq to something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(-\mbox{3.}_{25},x)\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI575=1.p} Inequation system has exactly one integer solution






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$less}(3,x)\mbox{ and }\mbox{\$less}(x,y)\mbox{ and }\mbox{\$less}(y,z)\mbox{ and }\mbox{\$less}(\mbox{\$sum}(\mbox{\$sum}(x,\mbox{\$product}(2,y)),\mbox{\$product}(3,z)),35) )\iff ( x = 4\mbox{ and }y = 5\mbox{ and }z = 6 ) )\qquad\mbox{tff}(\mbox{ineq\_sys\_has\_1\_int\_sol},\mbox{conjecture})$

\medskip\textbf{ARI577=1.p} Inequation system is solvable (e.g., X = 5, Y = 4)






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(4,y)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(y,1),x)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(x,y),10) )\qquad\mbox{tff}(\mbox{ineq\_sys\_solvable}_2,\mbox{conjecture})$

\medskip\textbf{ARI502=1.p} Mixed: 0.08 is a rational






$\mbox{\$is\_rat}(\mbox{0.}_{08})\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI436=1.p} Real: Product 3.0 and 4.0 is 12.0






$\mbox{\$product}(\mbox{3.}_0,\mbox{4.}_0) = \mbox{12.}_0\qquad\mbox{tff}(\mbox{real\_product\_problem}_1,\mbox{conjecture})$

\medskip\textbf{ARI705=1.p} Simple rewriting: d*d = d*d + a implies d*c*a*b*2 = 0






$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

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

$\mbox{\$product}(d,d) = \mbox{\$sum}(\mbox{\$product}(d,d),a)\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(d,c),a),b),2) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI103=1.p} Integer: Product of 2 and 3 is not 5






$\mbox{\$product}(2,3)\ne 5\qquad\mbox{tff}(\mbox{product\_2\_3\_not}_5,\mbox{conjecture})$

\medskip\textbf{ARI403=1.p} Real: Sum 4.25 and 2.0 is 6.25






$\mbox{\$sum}(\mbox{4.}_{25},\mbox{2.}_0) = \mbox{6.}_{25}\qquad\mbox{tff}(\mbox{real\_sum\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI413=1.p} Real: Sum 4.25 and -5.75 is -1.5






$\mbox{\$sum}(\mbox{4.}_{25},-\mbox{5.}_{75}) = -\mbox{1.}_5\qquad\mbox{tff}(\mbox{real\_sum\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI001=1.p} Integer: 2 less than 3






$\mbox{\$less}(2,3)\qquad\mbox{tff}(\mbox{prove\_2\_less}_3,\mbox{conjecture})$

\medskip\textbf{ARI556=1.p} Rational: Sum of product -3/40 and -12/1, and -9/10 is 0/1






$\mbox{\$sum}(\mbox{\$product}(-3/40,-12/1),-9/10) = 0/1\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI649=1.p} a*a = 25, b*b*b = -125, a $<$ 0 imply a = b






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

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

$\mbox{\$less}(a,0)\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$product}(a,a) = 25\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

$\mbox{\$product}(\mbox{\$product}(b,b),b) = -125\qquad\mbox{tff}(\mbox{conj}_{002},\mbox{axiom})$

$a = b\qquad\mbox{tff}(\mbox{conj}_{003},\mbox{conjecture})$

\medskip\textbf{ARI034=1.p} Integer: 4 greater than -4






$\mbox{\$greater}(4,-4)\qquad\mbox{tff}(\mbox{prove\_4\_greater\_n}_4,\mbox{conjecture})$

\medskip\textbf{ARI368=1.p} Real: -3.25 not lesseq to -8.69






$\neg\,\mbox{\$lesseq}(-\mbox{3.}_{25},-\mbox{8.}_{69})\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_{11},\mbox{conjecture})$

\medskip\textbf{ARI111=1.p} Integer: Product of -2 and -5 is something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(-2,-5) = x\qquad\mbox{tff}(\mbox{product\_n2\_n5\_what},\mbox{conjecture})$

\medskip\textbf{ARI505=1.p} Mixed: 17.99 coerced to integer is not 18






$\mbox{\$to\_int}(\mbox{17.}_{99})\ne 18\qquad\mbox{tff}(\mbox{mixed\_types\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI392=1.p} Real: 3.25 greatereq to -3.25






$\mbox{\$greatereq}(\mbox{3.}_{25},-\mbox{3.}_{25})\qquad\mbox{tff}(\mbox{real\_greatereq\_problem}_8,\mbox{conjecture})$

\medskip\textbf{ARI710=1.p} Simple rewriting: b * 1 = a*c*d implies b = d*a*c






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$d{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$\mbox{\$product}(b,1) = \mbox{\$product}(\mbox{\$product}(a,c),d)\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$b = \mbox{\$product}(\mbox{\$product}(d,a),c)\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI191=1.p} Rational: 1/2 not less 1/21






$\neg\,\mbox{\$less}(1/2,1/21)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI279=1.p} Rational: Difference only 1/2 and 3/16 is 5/16






$\forall x{:}\;\mbox{\$rat}{:}\;( \mbox{\$difference}(x,3/16) = 5/16 \ \Rightarrow \ x = 1/2 )\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_7,\mbox{conjecture})$

\medskip\textbf{ARI198=1.p} Rational: 1/4 not less than -1/4






$\neg\,\mbox{\$less}(1/4,-1/4)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI074=1.p} Integer: Sum something and -5 is -7






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,-5) = -7\qquad\mbox{tff}(\mbox{sum\_what\_n5\_n}_7,\mbox{conjecture})$

\medskip\textbf{ARI173=1.p} Integer: Formula less that -12






$\exists u{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$sum}(\mbox{\$product}(u,3),-5),-12)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ARI024=1.p} Integer: -2 not lesseq to -4






$\neg\,\mbox{\$lesseq}(-2,-4)\qquad\mbox{tff}(\mbox{n2\_not\_lesseq\_n}_4,\mbox{conjecture})$

\medskip\textbf{ARI430=1.p} Real: Difference 23.76 and 9.51 is only 14.25






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$difference}(\mbox{23.}_{76},\mbox{9.}_{51}) = x \ \Rightarrow \ x = \mbox{14.}_{25} )\qquad\mbox{tff}(\mbox{real\_difference\_problem}_6,\mbox{conjecture})$

\medskip\textbf{ARI711=1.p} Expand the equation (a+b+c+1)$\wedge$4 = 0






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

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

$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),1),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),1)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),1)),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(a,b),c),1)) = 0\qquad\mbox{tff}(\mbox{eq},\mbox{axiom})$

$\mbox{\$sum}(\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),c),c),\mbox{\$product}(-1,\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$sum}(\mbox{\$difference}(\mbox{\$product}(-1,1),\mbox{\$product}(4,a)),\mbox{\$product}(-6,\mbox{\$product}(a,a))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(a,a),a))),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(a,a),a),a))),\mbox{\$product}(4,b)),\mbox{\$product}(-12,\mbox{\$product}(b,a))),\mbox{\$product}(-6,\mbox{\$product}(b,b))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(b,a),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(b,b),a))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(b,b),b))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(b,a),a),a))),\mbox{\$product}(-6,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(b,b),a),a))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(b,b),b),a))),\mbox{\$product}(-1,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(b,b),b),b))),\mbox{\$product}(4,c)),\mbox{\$product}(-12,\mbox{\$product}(c,a))),\mbox{\$product}(-12,\mbox{\$product}(c,b))),\mbox{\$product}(-6,\mbox{\$product}(c,c))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(c,a),a))),\mbox{\$product}(-24,\mbox{\$product}(\mbox{\$product}(c,b),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(c,b),b))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(c,c),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(c,c),b))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(c,c),c))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,a),a),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,b),a),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,b),b),a))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,b),b),b))),\mbox{\$product}(-6,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),a),a))),\mbox{\$product}(-12,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),b),a))),\mbox{\$product}(-6,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),b),b))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),c),a))),\mbox{\$product}(-4,\mbox{\$product}(\mbox{\$product}(\mbox{\$product}(c,c),c),b))))) = 0\qquad\mbox{tff}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{ARI624=1.p} f(X) cannot always be smaller than avg(f(X-Y),f(X+Y)) - 1






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

$\neg\,\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$greater}(y,0/1) \ \Rightarrow \ \mbox{\$less}(f(x),\mbox{\$sum}(\mbox{\$sum}(\mbox{\$product}(1/2,f(\mbox{\$sum}(x,\mbox{\$uminus}(y)))),\mbox{\$product}(1/2,f(\mbox{\$sum}(x,y)))),-1/1)) )\qquad\mbox{tff}(\mbox{not\_ex\_f\_smaller\_avg\_minus}_1,\mbox{conjecture})$

\medskip\textbf{ARI118=1.p} Integer: Product of something and another thing is the first thing






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,y) = x\qquad\mbox{tff}(\mbox{product\_something\_anotherthing\_firstthing},\mbox{conjecture})$

\medskip\textbf{ARI026=1.p} Integer: Something lesseq to -2






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(x,-2)\qquad\mbox{tff}(\mbox{something\_lesseq\_n}_2,\mbox{conjecture})$

\medskip\textbf{ARI594=1.p} There is a number in [5,...,7] that is divisible by 3






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

$\forall z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(5,z)\mbox{ and }\mbox{\$lesseq}(z,7) ) \ \Rightarrow \ p(z) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$int}{:}\;p(\mbox{\$product}(3,x))\qquad\mbox{tff}(\mbox{exists\_X\_in\_5\_to\_7\_div}_3,\mbox{conjecture})$

\medskip\textbf{ARI076=1.p} Integer: Sum something  and -2 is 3






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,-2) = 3\qquad\mbox{tff}(\mbox{sum\_what\_n2}_3,\mbox{conjecture})$

\medskip\textbf{ARI440=1.p} Real: Product -2.5 and 3.4 is -8.5






$\mbox{\$product}(-\mbox{2.}_5,\mbox{3.}_4) = -\mbox{8.}_5\qquad\mbox{tff}(\mbox{real\_product\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI087=1.p} Integer: Sum something and something is 0






$\exists x{:}\;\mbox{\$int}{:}\;( x\ne 0\mbox{ and }\mbox{\$sum}(x,x) = 0 )\qquad\mbox{tff}(\mbox{anti\_sum\_what\_what}_0,\mbox{conjecture})$

\medskip\textbf{ARI029=1.p} Integer: 3 not greater than 4






$\neg\,\mbox{\$greater}(3,4)\qquad\mbox{tff}(\mbox{prove\_3\_not\_greater}_4,\mbox{conjecture})$

\medskip\textbf{ARI124=1.p} Integer: Product of 5 and 3 is 15 in a predicate






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

$p(\mbox{\$product}(5,3)) \ \Rightarrow \ p(15)\qquad\mbox{tff}(\mbox{product\_5\_3\_15\_predicate},\mbox{conjecture})$

\medskip\textbf{ARI603=1.p} If f(X) $>$ X, then Y = Z + (Y-Z) $<$ Z + f(Y-Z)






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

$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(x),x) \ \Rightarrow \ \forall y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(y,\mbox{\$sum}(z,f(x)))\qquad\mbox{tff}(\mbox{fX\_gt\_X\_implies\_exist\_large\_fX},\mbox{conjecture})$

\medskip\textbf{ARI545=1.p} Integer: Something plus iteself less than -13






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$sum}(x,x),-13)\qquad\mbox{tff}(\mbox{int\_combined\_problem}_9,\mbox{conjecture})$

\medskip\textbf{ARI086=1.p} Integer: Sum 2 and 2 is 5






$\mbox{\$sum}(2,2) = 5\qquad\mbox{tff}(\mbox{anti\_sum\_2\_2}_5,\mbox{conjecture})$

\medskip\textbf{ARI449=1.p} Real: Product 7.25 and only 4.0 is 29.0






$\forall x{:}\;\mbox{\$real}{:}\;( \mbox{\$product}(\mbox{7.}_{25},x) = \mbox{29.}_0 \ \Rightarrow \ x = \mbox{4.}_0 )\qquad\mbox{tff}(\mbox{real\_product\_problem}_{14},\mbox{conjecture})$

\medskip\textbf{ARI131=1.p} Integer: Sum of -2 and - 2 is 0






$\mbox{\$sum}(-2,\mbox{\$uminus}(-2)) = 0\qquad\mbox{tff}(\mbox{sum\_n2\_uminus\_to}_0,\mbox{conjecture})$

\medskip\textbf{ARI650=1.p} a*a $>$= 50 \& a*a $\leq$ 60 are inconsistent






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

$\mbox{\$greatereq}(60,\mbox{\$product}(a,a))\qquad\mbox{tff}(\mbox{conj},\mbox{axiom})$

$\mbox{\$lesseq}(50,\mbox{\$product}(a,a))\qquad\mbox{tff}(\mbox{conj}_{001},\mbox{axiom})$

\medskip\textbf{ARI538=1.p} Integer: -15 less than difference 0 and -15






$\mbox{\$less}(-15,\mbox{\$difference}(0,-15))\qquad\mbox{tff}(\mbox{int\_combined\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI221=1.p} Rational: Something greater than 9/16






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(x,9/16)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI458=1.p} Real: Product something and -0.08 is 1.14






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$product}(x,-\mbox{0.}_{08}) = \mbox{1.}_{14}\qquad\mbox{tff}(\mbox{real\_product\_problem}_{23},\mbox{conjecture})$

\medskip\textbf{ARI565=1.p} Real: (-0.075 * -12.0) + -0.9) is 0.0






$\mbox{\$sum}(\mbox{\$product}(-\mbox{0.}_{075},-\mbox{12.}_0),-\mbox{0.}_9) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_combined\_problem}_{16},\mbox{conjecture})$

\medskip\textbf{ARI718=1.p} floor(X+0.5) can be less than X






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$less}(\mbox{\$floor}(\mbox{\$sum}(x,\mbox{0.}_5)),x)\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{ARI192=1.p} Rational: 1/5 less than 4/15






$\mbox{\$less}(1/5,4/15)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_3,\mbox{conjecture})$

\medskip\textbf{ARI247=1.p} Rational: Sum of 2/5 and 3/5 is 1/1






$\mbox{\$sum}(2/5,3/5) = 1/1\qquad\mbox{tff}(\mbox{rat\_sum\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI590=1.p} There is a positive number different from Y






$\forall y{:}\;\mbox{\$int}{:}\;\exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,x)\mbox{ and }y\ne x )\qquad\mbox{tff}(\mbox{exists\_pos\_X\_noteq\_Y},\mbox{conjecture})$

\medskip\textbf{ARI199=1.p} Rational: -3/8 not less than -5/8






$\neg\,\mbox{\$less}(-3/8,-5/8)\qquad\mbox{tff}(\mbox{rat\_less\_problem}_{10},\mbox{conjecture})$

\medskip\textbf{ARI362=1.p} Real: 11.33 lesseq to something






$\exists x{:}\;\mbox{\$real}{:}\;\mbox{\$lesseq}(\mbox{11.}_{33},x)\qquad\mbox{tff}(\mbox{real\_lesseq\_problem}_5,\mbox{conjecture})$

\medskip\textbf{ARI032=1.p} Integer: 15 greater than something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(15,x)\qquad\mbox{tff}(\mbox{prove\_15\_greater\_something},\mbox{conjecture})$

\medskip\textbf{ARI064=1.p} Integer: Sum only 2 and 3 is 5






$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,3) = 5 \ \Rightarrow \ x = 2 )\qquad\mbox{tff}(\mbox{sum\_only\_2\_3}_5,\mbox{conjecture})$

\medskip\textbf{ARI229=1.p} Rational: -75/112 greater than something






$\exists x{:}\;\mbox{\$rat}{:}\;\mbox{\$greater}(-75/112,x)\qquad\mbox{tff}(\mbox{rat\_greater\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{ARI309=1.p} Rational: 2/3 is - -2/3






$2/3 = \mbox{\$uminus}(-2/3)\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_2,\mbox{conjecture})$

\medskip\textbf{ARI604=1.p} If f(X) $>$ X, then f(-X) $>$ -X, hence -f(-X) $<$ X $<$ f(X)






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

$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(x),x) \ \Rightarrow \ \forall x{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$uminus}(f(\mbox{\$uminus}(x))),f(x))\qquad\mbox{tff}(\mbox{fX\_gt\_X\_implies\_negfnegX\_lt\_fX},\mbox{conjecture})$

\medskip
\end{document}