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    \textbf{ANA001-0.ax} Analysis (limits) axioms for continuous functions






$x + n_0 = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$n_0 + x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\neg\,x < x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less\_than},\mbox{axiom})$

$( x < y\mbox{ and }y < z ) \ \Rightarrow \ x < z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_than},\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ n_0 < \min(x,y)\qquad\mbox{cnf}(\mbox{axiom\_2}_1,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < x\qquad\mbox{cnf}(\mbox{axiom\_2}_2,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < y\qquad\mbox{cnf}(\mbox{axiom\_2}_3,\mbox{axiom})$

$( x < \mbox{half}(\mbox{xa})\mbox{ and }y < \mbox{half}(\mbox{xa}) ) \ \Rightarrow \ x + y < \mbox{xa}\qquad\mbox{cnf}(\mbox{axiom}_3,\mbox{axiom})$

$|x| + |y| < \mbox{xa} \ \Rightarrow \ |x + y| < \mbox{xa}\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$( x + y ) + z = x + ( y + z )\qquad\mbox{cnf}(\mbox{axiom}_5,\mbox{axiom})$

$x + y = y + x\qquad\mbox{cnf}(\mbox{axiom\_6}_1,\mbox{axiom})$

$n_0 < \mbox{xa} \ \Rightarrow \ n_0 < \mbox{half}(\mbox{xa})\qquad\mbox{cnf}(\mbox{axiom\_6}_2,\mbox{axiom})$

$n_0 < \mbox{xa} \ \Rightarrow \ n_0 < \mbox{half}(\mbox{xa})\qquad\mbox{cnf}(\mbox{axiom}_7,\mbox{axiom})$

$-( x + y ) = -x + -y\qquad\mbox{cnf}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{ANA002-0.ax} Analysis (limits) axioms for continuous functions






$|x + y| \leq |x| + |y|\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_1,\mbox{axiom})$

$|x| + |y| \leq z \ \Rightarrow \ |x + y| \leq z\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_2,\mbox{axiom})$

$x \leq y \ \Rightarrow \ \min(x,y) = x\qquad\mbox{cnf}(\mbox{minimum}_1,\mbox{axiom})$

$\min(x,y) \leq x\qquad\mbox{cnf}(\mbox{minimum}_2,\mbox{axiom})$

$z \leq \min(x,y) \ \Rightarrow \ z \leq x\qquad\mbox{cnf}(\mbox{minimum}_3,\mbox{axiom})$

$x \leq y \ \Rightarrow \ x \leq \min(x,y)\qquad\mbox{cnf}(\mbox{minimum}_4,\mbox{axiom})$

$y \leq x \ \Rightarrow \ \min(x,y) = y\qquad\mbox{cnf}(\mbox{minimum}_5,\mbox{axiom})$

$\min(x,y) \leq y\qquad\mbox{cnf}(\mbox{minimum}_6,\mbox{axiom})$

$z \leq \min(x,y) \ \Rightarrow \ z \leq y\qquad\mbox{cnf}(\mbox{minimum}_7,\mbox{axiom})$

$y \leq x \ \Rightarrow \ y \leq \min(x,y)\qquad\mbox{cnf}(\mbox{minimum}_8,\mbox{axiom})$

$\min(x,y) \leq n_0 \ \Rightarrow \ ( x \leq n_0\mbox{ or }y \leq n_0 )\qquad\mbox{cnf}(\mbox{minimum}_9,\mbox{axiom})$

$\mbox{half}(x) + \mbox{half}(x) = x\qquad\mbox{cnf}(\mbox{half\_plus\_half\_is\_whole},\mbox{axiom})$

$\mbox{half}(x) + \mbox{half}(x) \leq x\qquad\mbox{cnf}(\mbox{half\_plus\_half\_less\_or\_equal\_whole},\mbox{axiom})$

$x \leq \mbox{half}(x) + \mbox{half}(x)\qquad\mbox{cnf}(\mbox{whole\_less\_or\_equal\_half\_plus\_half},\mbox{axiom})$

$( x \leq \mbox{half}(z)\mbox{ and }y \leq \mbox{half}(z) ) \ \Rightarrow \ x + y \leq z\qquad\mbox{cnf}(\mbox{less\_or\_equal\_sum\_of\_halves},\mbox{axiom})$

$\mbox{half}(x) \leq n_0 \ \Rightarrow \ x \leq n_0\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$x \leq y \ \Rightarrow \ x + z \leq y + z\qquad\mbox{cnf}(\mbox{add\_to\_both\_sides\_of\_less\_equal}_1,\mbox{axiom})$

$( x \leq y\mbox{ and }z \leq w ) \ \Rightarrow \ x + z \leq y + w\qquad\mbox{cnf}(\mbox{add\_to\_both\_sides\_of\_less\_equal}_2,\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{commutativity\_of\_less\_or\_equal},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_or\_equal},\mbox{axiom})$

$x + y = y + x\qquad\mbox{cnf}(\mbox{commutativity\_of\_add},\mbox{axiom})$

$x + y \leq y + x\qquad\mbox{cnf}(\mbox{commutativity\_of\_add\_for\_less\_or\_equal},\mbox{axiom})$

$( x + y ) + z = x + ( y + z )\qquad\mbox{cnf}(\mbox{associativity\_of\_add},\mbox{axiom})$

$( x + y ) + z \leq x + ( y + z )\qquad\mbox{cnf}(\mbox{associativity\_of\_add\_for\_less\_or\_equal}_1,\mbox{axiom})$

$x + ( y + z ) \leq ( x + y ) + z\qquad\mbox{cnf}(\mbox{associativity\_of\_add\_for\_less\_or\_equal}_2,\mbox{axiom})$

$x = y \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{equal\_implies\_less\_or\_equal},\mbox{axiom})$

\medskip\textbf{ANA044-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(c_0,\mbox{v\_l}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_times}(\mbox{v\_l}(\mbox{v\_x},\mbox{v\_xa}),\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x},\mbox{v\_xa})),\mbox{t\_b})\ne \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_l}(\mbox{v\_x},\mbox{v\_xa}),\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x},\mbox{v\_xa})),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$( \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_of\_\_nonneg}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{40},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA002-3.p} Intermediate value theorem

If a function f is continuous in a real closed interval
 [a,b], where f(a)$\leq$0 and 0$\leq$f(b), then there exists X such
 that f(X) = 0.




$x \leq x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{totality\_of\_less},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$x \leq q(y,x) \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{interpolation}_1,\mbox{axiom})$

$q(x,y) \leq x \ \Rightarrow \ y \leq x\qquad\mbox{cnf}(\mbox{interpolation}_2,\mbox{axiom})$

$( x \leq h(x)\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ f(x) \leq n_0\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( y \leq x\mbox{ and }f(y) \leq n_0\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ ( f(x) \leq n_0\mbox{ or }y \leq h(x) )\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

$( k(x) \leq x\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ n_0 \leq f(x)\qquad\mbox{cnf}(\mbox{continuity}_3,\mbox{axiom})$

$( x \leq y\mbox{ and }n_0 \leq f(y)\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ ( n_0 \leq f(x)\mbox{ or }k(x) \leq y )\qquad\mbox{cnf}(\mbox{continuity}_4,\mbox{axiom})$

$( x \leq b\mbox{ and }f(x) \leq n_0 ) \ \Rightarrow \ x \leq l\qquad\mbox{cnf}(\mbox{crossover}_1,\mbox{axiom})$

$g(x) \leq b\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover2\_and\_g\_function}_1,\mbox{axiom})$

$f(g(x)) \leq n_0\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover3\_and\_g\_function}_2,\mbox{axiom})$

$g(x) \leq x \ \Rightarrow \ l \leq x\qquad\mbox{cnf}(\mbox{crossover4\_and\_g\_function}_3,\mbox{axiom})$

$a \leq b\qquad\mbox{cnf}(\mbox{the\_interval},\mbox{hypothesis})$

$f(a) \leq n_0\qquad\mbox{cnf}(\mbox{lower\_mapping},\mbox{hypothesis})$

$n_0 \leq f(b)\qquad\mbox{cnf}(\mbox{upper\_mapping},\mbox{hypothesis})$

$f(x) \leq n_0 \ \Rightarrow \ \neg\,n_0 \leq f(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_x\_which\_crosses},\mbox{negated\_conjecture})$

\medskip\textbf{ANA034-2.p} Problem about Big-O notation






$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_c},\mbox{v\_d},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_d},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_mono}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}) = \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a}(\mbox{v\_x}),\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_{17},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{40},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA037-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(c_0,\mbox{v\_f}(\mbox{v\_xa}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_f}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_Orderings\_Omax}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{v\_f}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_8,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_b}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Ole\_\_maxI1}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_right\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

\medskip\textbf{ANA033-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_c},\mbox{v\_d},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_d},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_mono}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a}(\mbox{v\_x}(\mbox{v\_U})),\mbox{v\_b}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA004-1.p} Lemma 2 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{include}(\mbox{'Axioms/ANA001-0.ax'})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|( f(\mbox{fp}_{33}(x)) + -l_1 ) + ( g(\mbox{fp}_{33}(x)) + -l_2 )| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA004-2.p} Lemma 2 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$x + n_0{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$n_0 + x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\neg\,x < x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less\_than},\mbox{axiom})$

$( x < y\mbox{ and }y < z ) \ \Rightarrow \ x < z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_than},\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ n_0 < \min(x,y)\qquad\mbox{cnf}(\mbox{axiom\_2}_1,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < x\qquad\mbox{cnf}(\mbox{axiom\_2}_2,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < y\qquad\mbox{cnf}(\mbox{axiom\_2}_3,\mbox{axiom})$

$( x < \mbox{half}(\mbox{xa})\mbox{ and }y < \mbox{half}(\mbox{xa}) ) \ \Rightarrow \ x + y < \mbox{xa}\qquad\mbox{cnf}(\mbox{axiom}_3,\mbox{axiom})$

$n_0 < \mbox{xa} \ \Rightarrow \ n_0 < \mbox{half}(\mbox{xa})\qquad\mbox{cnf}(\mbox{axiom}_7,\mbox{axiom})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|( f(\mbox{fp}_{33}(x)) + -l_1 ) + ( g(\mbox{fp}_{33}(x)) + -l_2 )| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA003-2.p} Lemma 1 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$x + n_0{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$n_0 + x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\neg\,x < x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less\_than},\mbox{axiom})$

$( x < y\mbox{ and }y < z ) \ \Rightarrow \ x < z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_than},\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ n_0 < \min(x,y)\qquad\mbox{cnf}(\mbox{axiom\_2}_1,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < x\qquad\mbox{cnf}(\mbox{axiom\_2}_2,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < y\qquad\mbox{cnf}(\mbox{axiom\_2}_3,\mbox{axiom})$

$( x < \mbox{half}(\mbox{xa})\mbox{ and }y < \mbox{half}(\mbox{xa}) ) \ \Rightarrow \ x + y < \mbox{xa}\qquad\mbox{cnf}(\mbox{axiom}_3,\mbox{axiom})$

$|x| + |y| < \mbox{xa} \ \Rightarrow \ |x + y| < \mbox{xa}\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$n_0 < \mbox{xa} \ \Rightarrow \ n_0 < \mbox{half}(\mbox{xa})\qquad\mbox{cnf}(\mbox{axiom}_7,\mbox{axiom})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|f(\mbox{fp}_{33}(x)) + -l_1| + |g(\mbox{fp}_{33}(x)) + -l_2| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA015-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_d},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Ofield}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{34},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{46},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{47},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA001-1.p} Attaining minimum (or maximum) value

A continuous function f in a closed real interval [a,b]
 attains its minimum (or maximum) in this interval.




$x \leq x\qquad\mbox{cnf}(\mbox{refelxivity},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{totality},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq x ) \ \Rightarrow \ f(x) \leq f(y)\qquad\mbox{cnf}(\mbox{function},\mbox{hypothesis})$

$( \mbox{lower} \leq x\mbox{ and }x \leq \mbox{upper} ) \ \Rightarrow \ x \in [\mbox{lower},\mbox{upper}]\qquad\mbox{cnf}(\mbox{in\_interval},\mbox{hypothesis})$

$a \leq \mbox{extreme\_point}\qquad\mbox{cnf}(\mbox{interval}_1,\mbox{hypothesis})$

$\mbox{extreme\_point} \leq b\qquad\mbox{cnf}(\mbox{interval}_2,\mbox{hypothesis})$

$x \in [a,\mbox{extreme\_point}] \ \Rightarrow \ f(\mbox{extreme\_point}) \leq f(x)\qquad\mbox{cnf}(\mbox{below\_extreme\_point},\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ ( x \leq \mbox{extreme\_point}\mbox{ or }a \leq q(x) )\qquad\mbox{cnf}(\mbox{q\_function}_1,\mbox{hypothesis})$

$( x \in [a,b]\mbox{ and }f(x) \leq q(x) ) \ \Rightarrow \ x \leq \mbox{extreme\_point}\qquad\mbox{cnf}(\mbox{q\_function}_2,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ ( x \leq \mbox{extreme\_point}\mbox{ or }q(x) \leq x )\qquad\mbox{cnf}(\mbox{q\_function}_3,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ a \leq h(x)\qquad\mbox{cnf}(\mbox{h\_function}_1,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ h(x) \leq b\qquad\mbox{cnf}(\mbox{h\_function}_2,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ f(h(x)) \leq f(x)\qquad\mbox{cnf}(\mbox{h\_function}_3,\mbox{hypothesis})$

$( x \in [a,b]\mbox{ and }y \in [a,b]\mbox{ and }f(y) \leq f(x) ) \ \Rightarrow \ h(x) \leq y\qquad\mbox{cnf}(\mbox{h\_function}_4,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ a \leq k(x)\qquad\mbox{cnf}(\mbox{k\_function}_1,\mbox{hypothesis})$

$x \in [a,b] \ \Rightarrow \ k(x) \leq b\qquad\mbox{cnf}(\mbox{k\_function}_2,\mbox{hypothesis})$

$f(x) \leq f(k(x)) \ \Rightarrow \ \neg\,x \in [a,b]\qquad\mbox{cnf}(\mbox{prove\_something},\mbox{negated\_conjecture})$

\medskip\textbf{ANA004-4.p} Lemma 2 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$|x| + |y| \leq z \ \Rightarrow \ |x + y| \leq z\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_2,\mbox{axiom})$

$z \leq \min(x,y) \ \Rightarrow \ z \leq x\qquad\mbox{cnf}(\mbox{minimum}_3,\mbox{axiom})$

$z \leq \min(x,y) \ \Rightarrow \ z \leq y\qquad\mbox{cnf}(\mbox{minimum}_7,\mbox{axiom})$

$\min(x,y) \leq n_0 \ \Rightarrow \ ( x \leq n_0\mbox{ or }y \leq n_0 )\qquad\mbox{cnf}(\mbox{minimum}_9,\mbox{axiom})$

$( x \leq \mbox{half}(z)\mbox{ and }y \leq \mbox{half}(z) ) \ \Rightarrow \ x + y \leq z\qquad\mbox{cnf}(\mbox{less\_or\_equal\_sum\_of\_halves},\mbox{axiom})$

$\mbox{half}(x) \leq n_0 \ \Rightarrow \ x \leq n_0\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|( f(\mbox{xs}(\delta)) + -f(\mbox{a\_real\_number}) ) + ( g(\mbox{xs}(\delta)) + -g(\mbox{a\_real\_number}) )| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause\_7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{ANA010-2.p} Problem about Big-O notation






$( \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_of\_\_nonneg}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_U}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_{17},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA006-2.p} Analysis (limits) axioms for continuous functions






$\mbox{include}(\mbox{'Axioms/ANA002-0.ax'})$

\medskip\textbf{ANA020-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{tc\_nat}) \ \Rightarrow \ \mbox{v\_x} = \mbox{c\_Suc}(\mbox{c\_minus}(\mbox{v\_x},c_1,\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_NatBin\_OSuc\_\_pred\_H}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_f}(c_0) = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_x} = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA004-5.p} Lemma 2 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$|x + y| \leq |x| + |y|\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_1,\mbox{axiom})$

$\min(x,y) \leq x\qquad\mbox{cnf}(\mbox{minimum}_2,\mbox{axiom})$

$x \leq y \ \Rightarrow \ x \leq \min(x,y)\qquad\mbox{cnf}(\mbox{minimum}_4,\mbox{axiom})$

$\min(x,y) \leq y\qquad\mbox{cnf}(\mbox{minimum}_6,\mbox{axiom})$

$y \leq x \ \Rightarrow \ y \leq \min(x,y)\qquad\mbox{cnf}(\mbox{minimum}_8,\mbox{axiom})$

$( x \leq \mbox{half}(z)\mbox{ and }y \leq \mbox{half}(z) ) \ \Rightarrow \ x + y \leq z\qquad\mbox{cnf}(\mbox{less\_or\_equal\_sum\_of\_halves},\mbox{axiom})$

$\mbox{half}(x) \leq n_0 \ \Rightarrow \ x \leq n_0\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{commutativity\_of\_less\_or\_equal},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_or\_equal},\mbox{axiom})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|( f(\mbox{xs}(\delta)) + -f(\mbox{a\_real\_number}) ) + ( g(\mbox{xs}(\delta)) + -g(\mbox{a\_real\_number}) )| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause\_7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{ANA009-2.p} Problem about Big-O notation






$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_semigroup\_\_add\_\_imp\_\_le}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_plus}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oadd\_\_le\_\_cancel\_\_right}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(\mbox{v\_a},\mbox{c\_uminus}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a}) = c_0\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oright\_\_minus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_lb}(\mbox{v\_U}),\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_plus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{c\_uminus}(\mbox{v\_lb}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_semigroup\_\_add\_\_imp\_\_le}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{37},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_4,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA035-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_c},\mbox{v\_d},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_d},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_mono}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\ne \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA030-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_Orderings\_Omax}(\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_U}),\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),c_0,\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_plus}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$( \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_of\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(\mbox{v\_a},\mbox{c\_uminus}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_1}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_plus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_8}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_minus}(\mbox{v\_a},\mbox{c\_uminus}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Odiff\_\_minus\_\_eq\_\_add}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_uminus}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_plus}(\mbox{c\_uminus}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_uminus}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ominus\_\_add\_\_distrib}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_uminus}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_minus}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ominus\_\_diff\_\_eq}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_uminus}(\mbox{c\_uminus}(\mbox{v\_y},\mbox{t\_a}),\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ominus\_\_minus}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_b}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_y},\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Ole\_\_maxI2}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Olinorder}(\mbox{t\_b})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{v\_z},\mbox{t\_b}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Omin\_\_max\_Obelow\_\_sup\_Oabove\_\_sup\_\_conv}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_1,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_4,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA013-2.p} Problem about Big-O notation






$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Oorder}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

\medskip\textbf{ANA034-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_c},\mbox{v\_d},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_d},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_mono}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}) = \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a}(\mbox{v\_x}),\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA021-2.p} Problem about Big-O notation






$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{tc\_nat}) \ \Rightarrow \ \mbox{v\_x} = \mbox{c\_Suc}(\mbox{c\_minus}(\mbox{v\_x},c_1,\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_NatBin\_OSuc\_\_pred\_H}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_n},\mbox{tc\_nat})\mbox{ or }\mbox{v\_n} = c_0\qquad\mbox{cnf}(\mbox{cls\_Nat\_Onot\_\_gr0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_x}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{ANA043-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_l}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_k}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}))),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_l}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}))),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_left\_\_commute}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{17},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA007-2.p} Problem about Big-O notation






$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Oorder}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_c} = \mbox{c\_times}(c_1,\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_cancel\_\_right1}_2,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

\medskip\textbf{ANA009-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_lb}(\mbox{v\_U}),\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_U}),\mbox{c\_plus}(\mbox{v\_lb}(\mbox{v\_U}),\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_plus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{c\_uminus}(\mbox{v\_lb}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA035-2.p} Problem about Big-O notation






$\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\ne \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{17},\mbox{axiom})$

\medskip\textbf{ANA003-3.p} Lemma 1 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{include}(\mbox{'Axioms/ANA002-0.ax'})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|f(\mbox{xs}(\delta)) + -f(\mbox{a\_real\_number})| + |g(\mbox{xs}(\delta)) + -g(\mbox{a\_real\_number})| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause\_7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{ANA019-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{tc\_nat}) \ \Rightarrow \ \mbox{v\_x} = \mbox{c\_Suc}(\mbox{c\_minus}(\mbox{v\_x},c_1,\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_NatBin\_OSuc\_\_pred\_H}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_f}(c_0) = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_x},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_less}(c_0,\mbox{v\_U},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA018-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_x}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_ca},\mbox{v\_c},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA013-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA039-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_self}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_right\_\_mono}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA016-2.p} Problem about Big-O notation






$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_g}(\mbox{v\_x})\ne \mbox{c\_times}(\mbox{v\_c},\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Omonoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omonoid\_\_mult\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(\mbox{t\_a}) \ \Rightarrow \ ( \mbox{v\_a} = c_0\mbox{ or }\mbox{c\_times}(\mbox{v\_a},\mbox{c\_HOL\_Oinverse}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a}) = c_1 )\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oright\_\_inverse}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Omonoid\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Ofield}_{12},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Ofield}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_0,\mbox{axiom})$

\medskip\textbf{ANA012-2.p} Problem about Big-O notation






$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_1,\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$( \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a})\mbox{ and }\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}) = c_0 ) \ \Rightarrow \ \mbox{v\_a} = c_0\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_eq\_\_0}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Oorder}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(\mbox{t\_a}) \ \Rightarrow \ ( \mbox{v\_a} = c_0\mbox{ or }\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_a},\mbox{t\_a}),\mbox{v\_a},\mbox{t\_a}) = c_1 )\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Ofield\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{47},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{58},\mbox{axiom})$

\medskip\textbf{ANA010-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_U}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA045-2.p} Problem about Big-O notation






$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_c} = \mbox{c\_times}(c_1,\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_cancel\_\_right1}_2,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA023-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocomm\_\_monoid\_\_add\_\_class\_Oaxioms}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_1,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_k}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_k}(\mbox{v\_x}),\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Olinorder}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Orderings\_Olinorder}_4,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{23},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{54},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA007-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA025-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_less}(\mbox{v\_y},\mbox{v\_x},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Olinorder\_\_not\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Olinorder}(\mbox{t\_b})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_b})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_b}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{v\_z},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Omin\_\_max\_Obelow\_\_sup\_Oabove\_\_sup\_\_conv}_2,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_k}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Orderings\_Omax}(\mbox{c\_minus}(\mbox{v\_k}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),c_0,\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_LOrder\_Ojoin\_\_semilorder}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_LOrder\_Ojoin\_\_semilorder}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_LOrder\_Ojoin\_\_semilorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_{15},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA044-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_l}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_A}(\mbox{v\_x}),\mbox{t\_d})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_times}(\mbox{v\_l}(\mbox{v\_x},\mbox{v\_xa}),\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x},\mbox{v\_xa})),\mbox{t\_b})\ne \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_l}(\mbox{v\_x},\mbox{v\_xa}),\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x},\mbox{v\_xa})),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA005-5.p} The sum of two continuous functions is continuous






$\mbox{less\_or\_equalish}(|x + y|,|x| + |y|)\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_1,\mbox{axiom})$

$\mbox{less\_or\_equalish}(x,y) \ \Rightarrow \ \min(x,y){=}x\qquad\mbox{cnf}(\mbox{minimum}_1,\mbox{axiom})$

$\mbox{less\_or\_equalish}(y,x) \ \Rightarrow \ \min(x,y){=}y\qquad\mbox{cnf}(\mbox{minimum}_5,\mbox{axiom})$

$\mbox{half}(x) + \mbox{half}(x){=}x\qquad\mbox{cnf}(\mbox{half\_plus\_half\_is\_whole},\mbox{axiom})$

$\mbox{less\_or\_equalish}(\mbox{half}(x),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(x,n_0)\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$\mbox{less\_or\_equalish}(x,y) \ \Rightarrow \ \mbox{less\_or\_equalish}(x + z,y + z)\qquad\mbox{cnf}(\mbox{add\_to\_both\_sides\_of\_less\_equal}_1,\mbox{axiom})$

$\mbox{less\_or\_equalish}(x,y)\mbox{ or }\mbox{less\_or\_equalish}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_less\_or\_equal},\mbox{axiom})$

$( \mbox{less\_or\_equalish}(x,y)\mbox{ and }\mbox{less\_or\_equalish}(y,z) ) \ \Rightarrow \ \mbox{less\_or\_equalish}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_or\_equal},\mbox{axiom})$

$x + y{=}y + x\qquad\mbox{cnf}(\mbox{commutativity\_of\_add},\mbox{axiom})$

$( x + y ) + z{=}x + ( y + z )\qquad\mbox{cnf}(\mbox{associativity\_of\_add},\mbox{axiom})$

$x{=}y \ \Rightarrow \ \mbox{less\_or\_equalish}(x,y)\qquad\mbox{cnf}(\mbox{equal\_implies\_less\_or\_equal},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetry},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$( x{=}z\mbox{ and }\mbox{less\_or\_equalish}(x,y) ) \ \Rightarrow \ \mbox{less\_or\_equalish}(z,y)\qquad\mbox{cnf}(\mbox{less\_or\_equal\_substitution}_1,\mbox{axiom})$

$x{=}z \ \Rightarrow \ |x|{=}|z|\qquad\mbox{cnf}(\mbox{absolute\_substitution},\mbox{axiom})$

$x{=}z \ \Rightarrow \ x + y{=}z + y\qquad\mbox{cnf}(\mbox{add\_substitution}_1,\mbox{axiom})$

$y{=}z \ \Rightarrow \ x + y{=}x + z\qquad\mbox{cnf}(\mbox{add\_substitution}_2,\mbox{axiom})$

$\mbox{less\_or\_equalish}(\mbox{delta}_1(\varepsilon),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\varepsilon,n_0)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(\mbox{delta}_2(\varepsilon),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\varepsilon,n_0)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|z + -\mbox{a\_real\_number}|,\mbox{delta}_1(\varepsilon)) \ \Rightarrow \ ( \mbox{less\_or\_equalish}(\varepsilon,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|f(z) + -f(\mbox{a\_real\_number})|,\varepsilon) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|z + -\mbox{a\_real\_number}|,\mbox{delta}_2(\varepsilon)) \ \Rightarrow \ ( \mbox{less\_or\_equalish}(\varepsilon,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|g(z) + -g(\mbox{a\_real\_number})|,\varepsilon) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{less\_or\_equalish}(\mbox{epsilon}_0,n_0)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(\delta,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|\mbox{xs}(\delta) + -\mbox{a\_real\_number}|,\delta)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|( f(\mbox{xs}(\delta)) + g(\mbox{xs}(\delta)) ) + ( -f(\mbox{a\_real\_number}) + -g(\mbox{a\_real\_number}) )|,\mbox{epsilon}_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\delta,n_0)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{ANA002-1.p} Intermediate value theorem

If a function f is continuous in a real closed interval
 [a,b], where f(a)$\leq$0 and 0$\leq$f(b), then there exists X such
 that f(X) = 0.




$( \mbox{lower} \leq x\mbox{ and }x \leq \mbox{upper} ) \ \Rightarrow \ x \in [\mbox{lower},\mbox{upper}]\qquad\mbox{cnf}(\mbox{in\_interval},\mbox{axiom})$

$x \leq x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{totality\_of\_less},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$x \leq q(y,x) \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{interpolation}_1,\mbox{axiom})$

$q(x,y) \leq x \ \Rightarrow \ y \leq x\qquad\mbox{cnf}(\mbox{interpolation}_2,\mbox{axiom})$

$( x \leq h(x)\mbox{ and }x \in [a,b] ) \ \Rightarrow \ f(x) \leq n_0\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( y \leq x\mbox{ and }f(y) \leq n_0\mbox{ and }x \in [a,b] ) \ \Rightarrow \ ( f(x) \leq n_0\mbox{ or }y \leq h(x) )\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

$( k(x) \leq x\mbox{ and }x \in [a,b] ) \ \Rightarrow \ n_0 \leq f(x)\qquad\mbox{cnf}(\mbox{continuity}_3,\mbox{axiom})$

$( x \leq y\mbox{ and }n_0 \leq f(y)\mbox{ and }x \in [a,b] ) \ \Rightarrow \ ( n_0 \leq f(x)\mbox{ or }k(x) \leq y )\qquad\mbox{cnf}(\mbox{continuity}_4,\mbox{axiom})$

$( x \leq b\mbox{ and }f(x) \leq n_0 ) \ \Rightarrow \ x \leq l\qquad\mbox{cnf}(\mbox{crossover}_1,\mbox{axiom})$

$g(x) \leq b\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover2\_and\_g\_function}_1,\mbox{axiom})$

$f(g(x)) \leq n_0\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover3\_and\_g\_function}_2,\mbox{axiom})$

$g(x) \leq x \ \Rightarrow \ l \leq x\qquad\mbox{cnf}(\mbox{crossover4\_and\_g\_function}_3,\mbox{axiom})$

$a \leq b\qquad\mbox{cnf}(\mbox{the\_interval},\mbox{hypothesis})$

$f(a) \leq n_0\qquad\mbox{cnf}(\mbox{lower\_mapping},\mbox{hypothesis})$

$n_0 \leq f(b)\qquad\mbox{cnf}(\mbox{upper\_mapping},\mbox{hypothesis})$

$\neg\,n_0 \in [f(x),f(x)]\qquad\mbox{cnf}(\mbox{prove\_there\_is\_x\_which\_crosses},\mbox{negated\_conjecture})$

\medskip\textbf{ANA028-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocomm\_\_monoid\_\_add\_\_class\_Oaxioms}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_minus}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_minus}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_3}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_minus}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_minus}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_5}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_plus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_8}_1,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_minus}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a}) = c_0\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Odiff\_\_self}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_g}(\mbox{v\_U}),\mbox{v\_k}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_k}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{23},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_4,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{54},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA043-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_left\_\_commute}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_l}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_k}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}))),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_l}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_k}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}))),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA017-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA003-1.p} Lemma 1 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{include}(\mbox{'Axioms/ANA001-0.ax'})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|f(\mbox{fp}_{33}(x)) + -l_1| + |g(\mbox{fp}_{33}(x)) + -l_2| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA018-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_x}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_x}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_times}(\mbox{v\_ca},\mbox{v\_c},\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_LOrder\_Ojoin\_\_semilorder}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_LOrder\_Ojoin\_\_semilorder}_1,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_LOrder\_Ojoin\_\_semilorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{35},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

\medskip\textbf{ANA021-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{tc\_nat}) \ \Rightarrow \ \mbox{v\_x} = \mbox{c\_Suc}(\mbox{c\_minus}(\mbox{v\_x},c_1,\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_NatBin\_OSuc\_\_pred\_H}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_f}(c_0) = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_x}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA024-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_a} = \mbox{c\_plus}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_10}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_k}(\mbox{v\_U}),\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_k}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{54},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA041-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_xa},\mbox{v\_xb}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{ANA039-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_U}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_self}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_right\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA042-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{c\_times}(\mbox{v\_x},\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xb}(\mbox{v\_U}),\mbox{v\_A}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_xb}(\mbox{v\_U})),\mbox{t\_c}),\mbox{c\_times}(\mbox{v\_U},\mbox{v\_h}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_xb}(\mbox{v\_U})),\mbox{t\_c}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_c})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA036-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_plus}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_triangle\_\_ineq}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_semigroup\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_c},\mbox{v\_d},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_plus}(\mbox{v\_b},\mbox{v\_d},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oadd\_\_mono}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{class\_OrderedGroup\_Opordered\_\_cancel\_\_ab\_\_semigroup\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_y},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_x},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_plus}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oadd\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_b}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Ole\_\_maxI1}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_b}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_y},\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Ole\_\_maxI2}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_less}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_le\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_right\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Osemiring}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_plus}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_plus}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oring\_\_distrib\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Osemiring}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_plus}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oring\_\_distrib\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_ca},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_plus}(\mbox{v\_a}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{v\_b}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{c\_plus}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{v\_g}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}) \ \Rightarrow \ \neg\,\mbox{c\_less}(c_0,\mbox{v\_U},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA027-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_k}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_g}(\mbox{v\_x}),\mbox{v\_k}(\mbox{v\_x}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocomm\_\_monoid\_\_add\_\_class\_Oaxioms}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_plus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Ocompare\_\_rls\_\_9}_1,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$\mbox{class\_LOrder\_Ojoin\_\_semilorder}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_LOrder\_Ojoin\_\_semilorder}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t) \ \Rightarrow \ \mbox{class\_LOrder\_Ojoin\_\_semilorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}_{15},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{23},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA019-2.p} Problem about Big-O notation






$\mbox{v\_f}(c_0) = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_x},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{c\_Suc}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_less}(c_0,\mbox{v\_U},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_x},\mbox{tc\_nat}) \ \Rightarrow \ \mbox{v\_x} = \mbox{c\_Suc}(\mbox{c\_minus}(\mbox{v\_x},c_1,\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_NatBin\_OSuc\_\_pred\_H}_0,\mbox{axiom})$

$\mbox{c\_less}(c_0,\mbox{v\_n},\mbox{tc\_nat})\mbox{ or }\mbox{v\_n} = c_0\qquad\mbox{cnf}(\mbox{cls\_Nat\_Onot\_\_gr0}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(c_0,\mbox{t\_a}) = c_0\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_eq\_\_0}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_LOrder\_Ojoin\_\_semilorder}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_LOrder\_Ojoin\_\_semilorder}_1,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_LOrder\_Ojoin\_\_semilorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{35},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{40},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA020-2.p} Problem about Big-O notation






$\mbox{v\_f}(c_0) = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(c_0,\mbox{v\_c},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_x} = c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_h}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(c_0,\mbox{t\_a}) = c_0\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_eq\_\_0}_1,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_a},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Omult\_\_nonneg\_\_nonneg}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_cancel\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{40},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

\medskip\textbf{ANA002-2.p} Intermediate value theorem

If a function f is continuous in a real closed interval
 [a,b], where f(a)$\leq$0 and 0$\leq$f(b), then there exists X such
 that f(X) = 0.




$( \mbox{lower} \leq x\mbox{ and }x \leq \mbox{upper} ) \ \Rightarrow \ x \in [\mbox{lower},\mbox{upper}]\qquad\mbox{cnf}(\mbox{in\_interval},\mbox{axiom})$

$x \leq x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{totality\_of\_less},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$x \leq q(y,x) \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{interpolation}_1,\mbox{axiom})$

$q(x,y) \leq x \ \Rightarrow \ y \leq x\qquad\mbox{cnf}(\mbox{interpolation}_2,\mbox{axiom})$

$( x \leq h(x)\mbox{ and }x \in [a,b] ) \ \Rightarrow \ f(x) \leq n_0\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( y \leq x\mbox{ and }f(y) \leq n_0\mbox{ and }x \in [a,b] ) \ \Rightarrow \ ( f(x) \leq n_0\mbox{ or }y \leq h(x) )\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

$( k(x) \leq x\mbox{ and }x \in [a,b] ) \ \Rightarrow \ n_0 \leq f(x)\qquad\mbox{cnf}(\mbox{continuity}_3,\mbox{axiom})$

$( x \leq y\mbox{ and }n_0 \leq f(y)\mbox{ and }x \in [a,b] ) \ \Rightarrow \ ( n_0 \leq f(x)\mbox{ or }k(x) \leq y )\qquad\mbox{cnf}(\mbox{continuity}_4,\mbox{axiom})$

$( x \leq b\mbox{ and }f(x) \leq n_0 ) \ \Rightarrow \ x \leq l\qquad\mbox{cnf}(\mbox{crossover}_1,\mbox{axiom})$

$g(x) \leq b\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover2\_and\_g\_function}_1,\mbox{axiom})$

$f(g(x)) \leq n_0\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover3\_and\_g\_function}_2,\mbox{axiom})$

$g(x) \leq x \ \Rightarrow \ l \leq x\qquad\mbox{cnf}(\mbox{crossover4\_and\_g\_function}_3,\mbox{axiom})$

$a \leq b\qquad\mbox{cnf}(\mbox{the\_interval},\mbox{hypothesis})$

$f(a) \leq n_0\qquad\mbox{cnf}(\mbox{lower\_mapping},\mbox{hypothesis})$

$n_0 \leq f(b)\qquad\mbox{cnf}(\mbox{upper\_mapping},\mbox{hypothesis})$

$\neg\,n_0 \in [f(l),f(l)]\qquad\mbox{cnf}(\mbox{prove\_there\_is\_x\_which\_crosses},\mbox{negated\_conjecture})$

\medskip\textbf{ANA005-3.p} The sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{include}(\mbox{'Axioms/ANA002-0.ax'})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|( f(\mbox{xs}(\delta)) + g(\mbox{xs}(\delta)) ) + ( -f(\mbox{a\_real\_number}) + -g(\mbox{a\_real\_number}) )| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{ANA005-4.p} The sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{less\_or\_equalish}(|x + y|,|x| + |y|)\qquad\mbox{cnf}(\mbox{absolute\_sum\_less\_or\_equal\_sum\_of\_absolutes}_1,\mbox{axiom})$

$\mbox{less\_or\_equalish}(\min(x,y),x)\qquad\mbox{cnf}(\mbox{minimum}_2,\mbox{axiom})$

$\mbox{less\_or\_equalish}(x,y) \ \Rightarrow \ \mbox{less\_or\_equalish}(x,\min(x,y))\qquad\mbox{cnf}(\mbox{minimum}_4,\mbox{axiom})$

$\mbox{less\_or\_equalish}(\min(x,y),y)\qquad\mbox{cnf}(\mbox{minimum}_6,\mbox{axiom})$

$\mbox{less\_or\_equalish}(y,x) \ \Rightarrow \ \mbox{less\_or\_equalish}(y,\min(x,y))\qquad\mbox{cnf}(\mbox{minimum}_8,\mbox{axiom})$

$( \mbox{less\_or\_equalish}(x,\mbox{half}(z))\mbox{ and }\mbox{less\_or\_equalish}(y,\mbox{half}(z)) ) \ \Rightarrow \ \mbox{less\_or\_equalish}(x + y,z)\qquad\mbox{cnf}(\mbox{less\_or\_equal\_sum\_of\_halves},\mbox{axiom})$

$\mbox{less\_or\_equalish}(\mbox{half}(x),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(x,n_0)\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$\mbox{less\_or\_equalish}(x,y)\mbox{ or }\mbox{less\_or\_equalish}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_less\_or\_equal},\mbox{axiom})$

$( \mbox{less\_or\_equalish}(x,y)\mbox{ and }\mbox{less\_or\_equalish}(y,z) ) \ \Rightarrow \ \mbox{less\_or\_equalish}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_or\_equal},\mbox{axiom})$

$x + y{=}y + x\qquad\mbox{cnf}(\mbox{commutativity\_of\_add},\mbox{axiom})$

$( x + y ) + z{=}x + ( y + z )\qquad\mbox{cnf}(\mbox{associativity\_of\_add},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetry},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$( x{=}z\mbox{ and }\mbox{less\_or\_equalish}(x,y) ) \ \Rightarrow \ \mbox{less\_or\_equalish}(z,y)\qquad\mbox{cnf}(\mbox{less\_or\_equal\_substitution}_1,\mbox{axiom})$

$x{=}z \ \Rightarrow \ |x|{=}|z|\qquad\mbox{cnf}(\mbox{absolute\_substitution},\mbox{axiom})$

$x{=}z \ \Rightarrow \ x + y{=}z + y\qquad\mbox{cnf}(\mbox{add\_substitution}_1,\mbox{axiom})$

$y{=}z \ \Rightarrow \ x + y{=}x + z\qquad\mbox{cnf}(\mbox{add\_substitution}_2,\mbox{axiom})$

$\mbox{less\_or\_equalish}(\mbox{delta}_1(\varepsilon),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\varepsilon,n_0)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(\mbox{delta}_2(\varepsilon),n_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\varepsilon,n_0)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|z + -\mbox{a\_real\_number}|,\mbox{delta}_1(\varepsilon)) \ \Rightarrow \ ( \mbox{less\_or\_equalish}(\varepsilon,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|f(z) + -f(\mbox{a\_real\_number})|,\varepsilon) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|z + -\mbox{a\_real\_number}|,\mbox{delta}_2(\varepsilon)) \ \Rightarrow \ ( \mbox{less\_or\_equalish}(\varepsilon,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|g(z) + -g(\mbox{a\_real\_number})|,\varepsilon) )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{less\_or\_equalish}(\mbox{epsilon}_0,n_0)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(\delta,n_0)\mbox{ or }\mbox{less\_or\_equalish}(|\mbox{xs}(\delta) + -\mbox{a\_real\_number}|,\delta)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$\mbox{less\_or\_equalish}(|( f(\mbox{xs}(\delta)) + g(\mbox{xs}(\delta)) ) + ( -f(\mbox{a\_real\_number}) + -g(\mbox{a\_real\_number}) )|,\mbox{epsilon}_0) \ \Rightarrow \ \mbox{less\_or\_equalish}(\delta,n_0)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{negated\_conjecture})$

\medskip\textbf{ANA038-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(c_0,\mbox{v\_g}(\mbox{v\_xa}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{v\_g}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{c\_times}(\mbox{c\_Orderings\_Omax}(\mbox{v\_c},\mbox{v\_ca},\mbox{t\_b}),\mbox{v\_g}(\mbox{v\_xa}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_8,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_b}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_y},\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Ole\_\_maxI2}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_right\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{44},\mbox{axiom})$

\medskip\textbf{ANA016-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_d},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_g}(\mbox{v\_x})\ne \mbox{c\_times}(\mbox{v\_c},\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA045-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA015-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_d},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA017-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_ca},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA029-2.p} Problem about Big-O notation






$\neg\,\mbox{c\_lessequals}(c_0,\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_k}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Orderings\_Omax}(\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_k}(\mbox{v\_x}),\mbox{t\_b}),c_0,\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{c\_minus}(\mbox{v\_f}(\mbox{v\_x}),\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),c_0,\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Odiff\_\_le\_\_0\_\_iff\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),c_0,\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Odiff\_\_le\_\_0\_\_iff\_\_le}_1,\mbox{axiom})$

$\mbox{class\_Orderings\_Olinorder}(\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_less}(\mbox{v\_y},\mbox{v\_x},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Olinorder\_\_not\_\_le}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Olinorder}(\mbox{t\_b})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_b})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_b}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Orderings\_Omax}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_b}),\mbox{v\_z},\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Omin\_\_max\_Obelow\_\_sup\_Oabove\_\_sup\_\_conv}_2,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_z},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oorder\_\_trans}_0,\mbox{axiom})$

$( \mbox{class\_Orderings\_Oorder}(\mbox{t\_a})\mbox{ and }\mbox{c\_less}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_imp\_\_le}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}_{10},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Olinorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{33},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Opordered\_\_ab\_\_group\_\_add}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{54},\mbox{axiom})$

\medskip\textbf{ANA042-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{c\_times}(\mbox{v\_x},\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_xb}(\mbox{v\_U})),\mbox{t\_c}),\mbox{c\_times}(\mbox{v\_U},\mbox{v\_h}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_xb}(\mbox{v\_U})),\mbox{t\_c}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{ANA031-2.p} Problem about Big-O notation






$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{17},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA032-2.p} Problem about Big-O notation






$\mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(c_0,\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Oabs\_\_ge\_\_zero}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{17},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{42},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Olordered\_\_ab\_\_group\_\_abs}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_idom}_{50},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA012-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_1,\mbox{c\_times}(\mbox{v\_U},\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA004-3.p} Lemma 2 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$\mbox{include}(\mbox{'Axioms/ANA002-0.ax'})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|( f(\mbox{xs}(\delta)) + -f(\mbox{a\_real\_number}) ) + ( g(\mbox{xs}(\delta)) + -g(\mbox{a\_real\_number}) )| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause\_7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{ANA014-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA005-1.p} The sum of two continuous functions is continuous






$\mbox{include}(\mbox{'Axioms/ANA001-0.ax'})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|( f(\mbox{fp}_{33}(x)) + g(\mbox{fp}_{33}(x)) ) + -( l_1 + l_2 )| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA032-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA041-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{c\_times}(\mbox{v\_x},\mbox{v\_h}(\mbox{v\_U},\mbox{v\_V}),\mbox{t\_c}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xb},\mbox{v\_A}(\mbox{v\_xa}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(c_0,\mbox{v\_h}(\mbox{v\_xa},\mbox{v\_xb}),\mbox{t\_c})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_c})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA002-4.p} Intermediate value theorem

If a function f is continuous in a real closed interval
 [a,b], where f(a)$\leq$0 and 0$\leq$f(b), then there exists X such
 that f(X) = 0.




$x \leq x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{totality\_of\_less},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$x \leq q(y,x) \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{interpolation}_1,\mbox{axiom})$

$q(x,y) \leq x \ \Rightarrow \ y \leq x\qquad\mbox{cnf}(\mbox{interpolation}_2,\mbox{axiom})$

$( x \leq h(x)\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ f(x) \leq n_0\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( y \leq x\mbox{ and }f(y) \leq n_0\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ ( f(x) \leq n_0\mbox{ or }y \leq h(x) )\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

$( k(x) \leq x\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ n_0 \leq f(x)\qquad\mbox{cnf}(\mbox{continuity}_3,\mbox{axiom})$

$( x \leq y\mbox{ and }n_0 \leq f(y)\mbox{ and }a \leq x\mbox{ and }x \leq b ) \ \Rightarrow \ ( n_0 \leq f(x)\mbox{ or }k(x) \leq y )\qquad\mbox{cnf}(\mbox{continuity}_4,\mbox{axiom})$

$( x \leq b\mbox{ and }f(x) \leq n_0 ) \ \Rightarrow \ x \leq l\qquad\mbox{cnf}(\mbox{crossover}_1,\mbox{axiom})$

$g(x) \leq b\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover2\_and\_g\_function}_1,\mbox{axiom})$

$f(g(x)) \leq n_0\mbox{ or }l \leq x\qquad\mbox{cnf}(\mbox{crossover3\_and\_g\_function}_2,\mbox{axiom})$

$g(x) \leq x \ \Rightarrow \ l \leq x\qquad\mbox{cnf}(\mbox{crossover4\_and\_g\_function}_3,\mbox{axiom})$

$a \leq b\qquad\mbox{cnf}(\mbox{the\_interval},\mbox{hypothesis})$

$f(a) \leq n_0\qquad\mbox{cnf}(\mbox{lower\_mapping},\mbox{hypothesis})$

$n_0 \leq f(b)\qquad\mbox{cnf}(\mbox{upper\_mapping},\mbox{hypothesis})$

$f(l) \leq n_0 \ \Rightarrow \ \neg\,n_0 \leq f(l)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_x\_which\_crosses},\mbox{negated\_conjecture})$

\medskip\textbf{ANA006-1.p} Analysis (limits) axioms for continuous functions






$\mbox{include}(\mbox{'Axioms/ANA001-0.ax'})$

\medskip\textbf{ANA003-4.p} Lemma 1 for the sum of two continuous functions is continuous

A lemma formed by adding in some resolvants and taking out
 the corresponding clauses.




$z \leq \min(x,y) \ \Rightarrow \ z \leq x\qquad\mbox{cnf}(\mbox{minimum}_3,\mbox{axiom})$

$z \leq \min(x,y) \ \Rightarrow \ z \leq y\qquad\mbox{cnf}(\mbox{minimum}_7,\mbox{axiom})$

$\min(x,y) \leq n_0 \ \Rightarrow \ ( x \leq n_0\mbox{ or }y \leq n_0 )\qquad\mbox{cnf}(\mbox{minimum}_9,\mbox{axiom})$

$( x \leq \mbox{half}(z)\mbox{ and }y \leq \mbox{half}(z) ) \ \Rightarrow \ x + y \leq z\qquad\mbox{cnf}(\mbox{less\_or\_equal\_sum\_of\_halves},\mbox{axiom})$

$\mbox{half}(x) \leq n_0 \ \Rightarrow \ x \leq n_0\qquad\mbox{cnf}(\mbox{zero\_and\_half},\mbox{axiom})$

$\mbox{delta}_1(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{hypothesis})$

$\mbox{delta}_2(\varepsilon) \leq n_0 \ \Rightarrow \ \varepsilon \leq n_0\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_1(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|f(z) + -f(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{hypothesis})$

$|z + -\mbox{a\_real\_number}| \leq \mbox{delta}_2(\varepsilon) \ \Rightarrow \ ( \varepsilon \leq n_0\mbox{ or }|g(z) + -g(\mbox{a\_real\_number})| \leq \varepsilon )\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{hypothesis})$

$\neg\,\mbox{epsilon}_0 \leq n_0\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{hypothesis})$

$\delta \leq n_0\mbox{ or }|\mbox{xs}(\delta) + -\mbox{a\_real\_number}| \leq \delta\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{hypothesis})$

$|f(\mbox{xs}(\delta)) + -f(\mbox{a\_real\_number})| + |g(\mbox{xs}(\delta)) + -g(\mbox{a\_real\_number})| \leq \mbox{epsilon}_0 \ \Rightarrow \ \delta \leq n_0\qquad\mbox{cnf}(\mbox{clause\_7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{ANA005-2.p} The sum of two continuous functions is continuous






$x + n_0{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$n_0 + x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\neg\,x < x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less\_than},\mbox{axiom})$

$( x < y\mbox{ and }y < z ) \ \Rightarrow \ x < z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_than},\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ n_0 < \min(x,y)\qquad\mbox{cnf}(\mbox{axiom\_2}_1,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < x\qquad\mbox{cnf}(\mbox{axiom\_2}_2,\mbox{axiom})$

$( n_0 < x\mbox{ and }n_0 < y ) \ \Rightarrow \ \min(x,y) < y\qquad\mbox{cnf}(\mbox{axiom\_2}_3,\mbox{axiom})$

$( x < \mbox{half}(\mbox{xa})\mbox{ and }y < \mbox{half}(\mbox{xa}) ) \ \Rightarrow \ x + y < \mbox{xa}\qquad\mbox{cnf}(\mbox{axiom}_3,\mbox{axiom})$

$n_0 < \mbox{xa} \ \Rightarrow \ n_0 < \mbox{half}(\mbox{xa})\qquad\mbox{cnf}(\mbox{axiom}_7,\mbox{axiom})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{31}(x)\qquad\mbox{cnf}(c_{10},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{31}(x) ) \ \Rightarrow \ |f(y) + -l_1| < x\qquad\mbox{cnf}(c_{11},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ n_0 < \mbox{fp}_{32}(x)\qquad\mbox{cnf}(c_{12},\mbox{negated\_conjecture})$

$( n_0 < x\mbox{ and }|y + -a| < \mbox{fp}_{32}(x) ) \ \Rightarrow \ |g(y) + -l_2| < x\qquad\mbox{cnf}(c_{13},\mbox{negated\_conjecture})$

$n_0 < b\qquad\mbox{cnf}(c_{14},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ |\mbox{fp}_{33}(x) + -a| < x\qquad\mbox{cnf}(c_{15},\mbox{negated\_conjecture})$

$n_0 < x \ \Rightarrow \ \neg\,|( f(\mbox{fp}_{33}(x)) + g(\mbox{fp}_{33}(x)) ) + -( l_1 + l_2 )| < b\qquad\mbox{cnf}(c_{16},\mbox{negated\_conjecture})$

\medskip\textbf{ANA031-1.p} Problem about Big-O notation






$\mbox{include}(\mbox{'Axioms/ANA003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_2}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Oab\_\_semigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_b},\mbox{c\_times}(\mbox{v\_a},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omult\_\_ac\_\_3}_0,\mbox{axiom})$

$( \mbox{class\_Ring\_\_and\_\_Field\_Opordered\_\_semiring}(\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(c_0,\mbox{v\_c},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_times}(\mbox{v\_c},\mbox{v\_a},\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_c},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Opordered\_\_semiring\_\_class\_Omult\_\_left\_\_mono}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_one\_\_times}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Ocomm\_\_monoid\_\_add}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_SetsAndFunctions\_Oelt\_\_set\_\_plus}(c_0,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_SetsAndFunctions\_Oset\_\_zero\_\_plus}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_U}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_c},\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_U}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_b}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{c\_HOL\_Oabs}(\mbox{v\_g}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b}),\mbox{t\_b})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_b})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

\medskip\textbf{ANA014-2.p} Problem about Big-O notation






$\mbox{v\_c}\ne c_0\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{c\_times}(\mbox{v\_U},\mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_c},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_f}(\mbox{v\_x}(\mbox{v\_U})),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(\mbox{t\_a})\qquad\mbox{cnf}(\mbox{tfree\_tcs},\mbox{negated\_conjecture})$

$\mbox{class\_OrderedGroup\_Omonoid\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(c_1,\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Omonoid\_\_mult\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_times}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{v\_c},\mbox{t\_a}) = \mbox{c\_times}(\mbox{v\_a},\mbox{c\_times}(\mbox{v\_b},\mbox{v\_c},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_OrderedGroup\_Osemigroup\_\_mult\_\_class\_Omult\_\_assoc}_0,\mbox{axiom})$

$\mbox{class\_Orderings\_Oorder}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_HOL\_Oabs}(\mbox{c\_times}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_times}(\mbox{c\_HOL\_Oabs}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_HOL\_Oabs}(\mbox{v\_b},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Oabs\_\_mult}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(\mbox{t\_a}) \ \Rightarrow \ ( \mbox{v\_a} = c_0\mbox{ or }\mbox{c\_times}(\mbox{c\_HOL\_Oinverse}(\mbox{v\_a},\mbox{t\_a}),\mbox{v\_a},\mbox{t\_a}) = c_1 )\qquad\mbox{cnf}(\mbox{cls\_Ring\_\_and\_\_Field\_Ofield\_\_class\_Oaxioms\_\_1}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Omonoid\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Ofield}_{12},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t) \ \Rightarrow \ \mbox{class\_OrderedGroup\_Osemigroup\_\_mult}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Ofield}_{21},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Ofield}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_0,\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_idom}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{34},\mbox{axiom})$

$\mbox{class\_Ring\_\_and\_\_Field\_Oordered\_\_field}(t) \ \Rightarrow \ \mbox{class\_Orderings\_Oorder}(t)\qquad\mbox{cnf}(\mbox{clsrel\_Ring\_\_and\_\_Field\_Oordered\_\_field}_{58},\mbox{axiom})$

\medskip
\end{document}