TPTP Problem File: NUM925+7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925+7 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_1000_fofpt_l192 [Bla11]
% Status : Theorem
% Rating : 0.39 v8.2.0, 0.42 v8.1.0, 0.36 v7.5.0, 0.38 v7.4.0, 0.27 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.39 v7.0.0, 0.43 v6.4.0, 0.46 v6.3.0, 0.42 v6.2.0, 0.44 v6.1.0, 0.63 v6.0.0, 0.48 v5.5.0, 0.63 v5.4.0, 0.57 v5.3.0
% Syntax : Number of formulae : 1251 ( 406 unt; 0 def)
% Number of atoms : 2914 ( 788 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 1859 ( 196 ~; 60 |; 165 &)
% ( 274 <=>;1164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 60 ( 59 usr; 0 prp; 1-2 aty)
% Number of functors : 43 ( 43 usr; 19 con; 0-4 aty)
% Number of variables : 2612 (2590 !; 22 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:34:05
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (97)
fof(tsy_c_Groups_Oabs__class_Oabs_arg1,axiom,
! [B_1_1,X_a] :
( ordere142940540dd_abs(X_a)
=> abs_abs(X_a,ti(X_a,B_1_1)) = abs_abs(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oabs__class_Oabs_res,axiom,
! [B_1_1,X_a] :
( ordere142940540dd_abs(X_a)
=> ti(X_a,abs_abs(X_a,B_1_1)) = abs_abs(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_0_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( group_add(X_a)
=> minus_minus(X_a,ti(X_a,B_1_1),B_2_1) = minus_minus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_0_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( group_add(X_a)
=> minus_minus(X_a,B_1_1,ti(X_a,B_2_1)) = minus_minus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_0_res,axiom,
! [B_1_1,B_2_1,X_a] :
( group_add(X_a)
=> ti(X_a,minus_minus(X_a,B_1_1,B_2_1)) = minus_minus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_1_arg1,axiom,
! [B_1_1,B_2_1] : minus_minus(nat,ti(nat,B_1_1),B_2_1) = minus_minus(nat,B_1_1,B_2_1) ).
fof(tsy_c_Groups_Ominus__class_Ominus_1_arg2,axiom,
! [B_1_1,B_2_1] : minus_minus(nat,B_1_1,ti(nat,B_2_1)) = minus_minus(nat,B_1_1,B_2_1) ).
fof(tsy_c_Groups_Ominus__class_Ominus_1_res,axiom,
! [B_1_1,B_2_1] : ti(nat,minus_minus(nat,B_1_1,B_2_1)) = minus_minus(nat,B_1_1,B_2_1) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_a] :
( one(X_a)
=> ti(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( monoid_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( monoid_add(X_a)
=> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [B_1_1,B_2_1,X_a] :
( monoid_add(X_a)
=> ti(X_a,plus_plus(X_a,B_1_1,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_add(X_a)
=> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_add(X_a)
=> ti(X_a,plus_plus(X_a,B_1_1,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( cancel_semigroup_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( cancel_semigroup_add(X_a)
=> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
! [B_1_1,B_2_1,X_a] :
( cancel_semigroup_add(X_a)
=> ti(X_a,plus_plus(X_a,B_1_1,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_2_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( semiring(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_2_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( semiring(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_2_res,axiom,
! [B_1_1,B_2_1,X_a] :
( semiring(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_3_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( no_zero_divisors(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_3_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( no_zero_divisors(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_3_res,axiom,
! [B_1_1,B_2_1,X_a] :
( no_zero_divisors(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_4_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( mult_zero(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_4_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( mult_zero(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_4_res,axiom,
! [B_1_1,B_2_1,X_a] :
( mult_zero(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_5_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_mult(X_a)
=> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_5_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_mult(X_a)
=> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_5_res,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_semigroup_mult(X_a)
=> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_res,axiom,
! [X_a] :
( zero(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_If_arg1,axiom,
! [X_a,B_1_1,B_2_1,B_3] : if(X_a,ti(bool,B_1_1),B_2_1,B_3) = if(X_a,B_1_1,B_2_1,B_3) ).
fof(tsy_c_If_arg2,axiom,
! [X_a,B_1_1,B_2_1,B_3] : if(X_a,B_1_1,ti(X_a,B_2_1),B_3) = if(X_a,B_1_1,B_2_1,B_3) ).
fof(tsy_c_If_arg3,axiom,
! [X_a,B_1_1,B_2_1,B_3] : if(X_a,B_1_1,B_2_1,ti(X_a,B_3)) = if(X_a,B_1_1,B_2_1,B_3) ).
fof(tsy_c_If_res,axiom,
! [X_a,B_1_1,B_2_1,B_3] : ti(X_a,if(X_a,B_1_1,B_2_1,B_3)) = if(X_a,B_1_1,B_2_1,B_3) ).
fof(tsy_c_IntPrimes_Ozcong_arg1,axiom,
! [B_1_1,B_2_1] : zcong(ti(int,B_1_1),B_2_1) = zcong(B_1_1,B_2_1) ).
fof(tsy_c_IntPrimes_Ozcong_arg2,axiom,
! [B_1_1,B_2_1] : zcong(B_1_1,ti(int,B_2_1)) = zcong(B_1_1,B_2_1) ).
fof(tsy_c_IntPrimes_Ozcong_res,axiom,
! [B_1_1,B_2_1] : ti(fun(int,bool),zcong(B_1_1,B_2_1)) = zcong(B_1_1,B_2_1) ).
fof(tsy_c_IntPrimes_Ozprime_res,axiom,
ti(fun(int,bool),zprime) = zprime ).
fof(tsy_c_Int_OBit0_arg1,hypothesis,
! [B_1_1] : bit0(ti(int,B_1_1)) = bit0(B_1_1) ).
fof(tsy_c_Int_OBit0_res,hypothesis,
! [B_1_1] : ti(int,bit0(B_1_1)) = bit0(B_1_1) ).
fof(tsy_c_Int_OBit1_arg1,hypothesis,
! [B_1_1] : bit1(ti(int,B_1_1)) = bit1(B_1_1) ).
fof(tsy_c_Int_OBit1_res,hypothesis,
! [B_1_1] : ti(int,bit1(B_1_1)) = bit1(B_1_1) ).
fof(tsy_c_Int_OMin_res,axiom,
ti(int,min) = min ).
fof(tsy_c_Int_OPls_res,hypothesis,
ti(int,pls) = pls ).
fof(tsy_c_Int_Onat_arg1,axiom,
! [B_1_1] : nat_1(ti(int,B_1_1)) = nat_1(B_1_1) ).
fof(tsy_c_Int_Onat_res,axiom,
! [B_1_1] : ti(nat,nat_1(B_1_1)) = nat_1(B_1_1) ).
fof(tsy_c_Int_Onumber__class_Onumber__of_arg1,axiom,
! [B_1_1,X_a] :
( number(X_a)
=> number_number_of(X_a,ti(int,B_1_1)) = number_number_of(X_a,B_1_1) ) ).
fof(tsy_c_Int_Onumber__class_Onumber__of_res,axiom,
! [B_1_1,X_a] :
( number(X_a)
=> ti(X_a,number_number_of(X_a,B_1_1)) = number_number_of(X_a,B_1_1) ) ).
fof(tsy_c_Int_Osucc_arg1,axiom,
! [B_1_1] : succ(ti(int,B_1_1)) = succ(B_1_1) ).
fof(tsy_c_Int_Osucc_res,axiom,
! [B_1_1] : ti(int,succ(B_1_1)) = succ(B_1_1) ).
fof(tsy_c_Nat_Osemiring__1__class_Oof__nat_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(nat,X_a),semiring_1_of_nat(X_a)) = semiring_1_of_nat(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less_eq(X_a)) = ord_less_eq(X_a) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg1,axiom,
! [B_1_1,X_a] :
( power(X_a)
=> power_power(X_a,ti(X_a,B_1_1)) = power_power(X_a,B_1_1) ) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [B_1_1,X_a] :
( power(X_a)
=> ti(fun(nat,X_a),power_power(X_a,B_1_1)) = power_power(X_a,B_1_1) ) ).
fof(tsy_c_Residues_OLegendre_arg1,axiom,
! [B_1_1,B_2_1] : legendre(ti(int,B_1_1),B_2_1) = legendre(B_1_1,B_2_1) ).
fof(tsy_c_Residues_OLegendre_arg2,axiom,
! [B_1_1,B_2_1] : legendre(B_1_1,ti(int,B_2_1)) = legendre(B_1_1,B_2_1) ).
fof(tsy_c_Residues_OLegendre_res,axiom,
! [B_1_1,B_2_1] : ti(int,legendre(B_1_1,B_2_1)) = legendre(B_1_1,B_2_1) ).
fof(tsy_c_Residues_OQuadRes_arg1,axiom,
! [B_1_1] : quadRes(ti(int,B_1_1)) = quadRes(B_1_1) ).
fof(tsy_c_Residues_OQuadRes_res,axiom,
! [B_1_1] : ti(fun(int,bool),quadRes(B_1_1)) = quadRes(B_1_1) ).
fof(tsy_c_Rings_Odvd__class_Odvd_res,axiom,
! [X_a] :
( dvd(X_a)
=> ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a) ) ).
fof(tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Ois__sum2sq_res,axiom,
ti(fun(int,bool),twoSqu658283162sum2sq) = twoSqu658283162sum2sq ).
fof(tsy_c_fFalse_res,axiom,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_hAPP_0_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> hAPP(X_a,fun(X_a,bool),ti(fun(X_a,fun(X_a,bool)),B_1_1),B_2_1) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_0_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> hAPP(X_a,fun(X_a,bool),B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_0_res,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ti(fun(X_a,bool),hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_1_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> hAPP(X_a,bool,ti(fun(X_a,bool),B_1_1),B_2_1) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_1_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_1_res,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ti(bool,hAPP(X_a,bool,B_1_1,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_2_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> hAPP(X_a,fun(X_a,bool),ti(fun(X_a,fun(X_a,bool)),B_1_1),B_2_1) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_2_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> hAPP(X_a,fun(X_a,bool),B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_2_res,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> ti(fun(X_a,bool),hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_3_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> hAPP(X_a,bool,ti(fun(X_a,bool),B_1_1),B_2_1) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_3_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_3_res,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> ti(bool,hAPP(X_a,bool,B_1_1,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_4_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> hAPP(nat,X_a,ti(fun(nat,X_a),B_1_1),B_2_1) = hAPP(nat,X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_4_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> hAPP(nat,X_a,B_1_1,ti(nat,B_2_1)) = hAPP(nat,X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_hAPP_4_res,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> ti(X_a,hAPP(nat,X_a,B_1_1,B_2_1)) = hAPP(nat,X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_hBOOL_arg1,axiom,
! [B_1_1] :
( hBOOL(ti(bool,B_1_1))
<=> hBOOL(B_1_1) ) ).
fof(tsy_v_m_res,axiom,
ti(int,m) = m ).
fof(tsy_v_m1_____res,axiom,
ti(int,m1) = m1 ).
fof(tsy_v_n_____res,hypothesis,
ti(nat,n) = n ).
fof(tsy_v_s1_____res,axiom,
ti(int,s1) = s1 ).
fof(tsy_v_s_____res,axiom,
ti(int,s) = s ).
fof(tsy_v_t_____res,axiom,
ti(int,t) = t ).
fof(tsy_v_tn_____res,axiom,
ti(nat,tn) = tn ).
fof(tsy_v_x_____res,axiom,
ti(int,x) = x ).
fof(tsy_v_y_____res,axiom,
ti(int,y) = y ).
%----Relevant facts (995)
fof(fact_0_n1pos,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).
fof(fact_1_t1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) ).
fof(fact_2_sum__power2__eq__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a)
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_3_one__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,power_power(X_a,one_one(X_a)),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a) ) ).
fof(fact_4_zero__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a) ) ).
fof(fact_5_zero__eq__power2,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [A_2] :
( hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_6_add__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,bit1(pls),W)) ) ).
fof(fact_7_add__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1] : plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V_1,bit1(pls))) ) ).
fof(fact_8_one__add__one__is__two,axiom,
! [X_a] :
( number_ring(X_a)
=> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))) ) ).
fof(fact_9_semiring__one__add__one__is__two,axiom,
! [X_a] :
( number_semiring(X_a)
=> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))) ) ).
fof(fact_10_quartic__square__square,axiom,
! [X] : hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit0(bit1(pls))))) ).
fof(fact_11_power__0__left__number__of,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [W] :
( ( number_number_of(nat,W) = zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = one_one(X_a) )
& ( number_number_of(nat,W) != zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = zero_zero(X_a) ) ) ) ).
fof(fact_12_semiring__norm_I110_J,axiom,
! [X_a] :
( number_ring(X_a)
=> one_one(X_a) = number_number_of(X_a,bit1(pls)) ) ).
fof(fact_13_numeral__1__eq__1,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,bit1(pls)) = one_one(X_a) ) ).
fof(fact_14_n0,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),n)) ).
fof(fact_15_zless__linear,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y))
| ti(int,X) = ti(int,Y)
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) ) ).
fof(fact_16_less__number__of__int__code,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,K)),number_number_of(int,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) ) ).
fof(fact_17_plus__numeral__code_I9_J,axiom,
! [V_1,W] : plus_plus(int,number_number_of(int,V_1),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V_1,W)) ).
fof(fact_18_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ) ).
fof(fact_19_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
fof(fact_20_zpower__int,axiom,
! [M,N] : hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N)) ).
fof(fact_21_int__power,axiom,
! [M,N] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N) ).
fof(fact_22_zadd__int__left,axiom,
! [M,N,Z] : plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M),plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),N),Z)) = plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,M,N)),Z) ).
fof(fact_23_zadd__int,axiom,
! [M,N] : plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,M,N)) ).
fof(fact_24_int__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
fof(fact_25_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
fof(fact_26_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
fof(fact_27_int__eq__0__conv,axiom,
! [Na] :
( hAPP(nat,int,semiring_1_of_nat(int),Na) = zero_zero(int)
<=> Na = zero_zero(nat) ) ).
fof(fact_28_int__0,axiom,
hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).
fof(fact_29_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
fof(fact_30_less__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_31_rel__simps_I17_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) ) ).
fof(fact_32_rel__simps_I2_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),pls)) ).
fof(fact_33_less__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_34_rel__simps_I14_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) ) ).
fof(fact_35_zadd__strict__right__mono,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,I_2,K_1)),plus_plus(int,J_1,K_1))) ) ).
fof(fact_36_add__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,V_2) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
=> plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,V_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
=> plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,plus_plus(int,V_1,V_2)) ) ) ) ) ).
fof(fact_37_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
fof(fact_38_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
fof(fact_39_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
fof(fact_40_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [Xa,Ya] :
( number_number_of(X_a,Xa) = number_number_of(X_a,Ya)
<=> ti(int,Xa) = ti(int,Ya) ) ) ).
fof(fact_41_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [Wa,Xa] :
( number_number_of(X_a,Wa) = ti(X_a,Xa)
<=> ti(X_a,Xa) = number_number_of(X_a,Wa) ) ) ).
fof(fact_42_rel__simps_I51_J,axiom,
! [K,L_1] :
( bit1(K) = bit1(L_1)
<=> ti(int,K) = ti(int,L_1) ) ).
fof(fact_43_rel__simps_I48_J,axiom,
! [K,L_1] :
( bit0(K) = bit0(L_1)
<=> ti(int,K) = ti(int,L_1) ) ).
fof(fact_44_even__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_45_zadd__assoc,axiom,
! [Z1,Z2,Z3] : plus_plus(int,plus_plus(int,Z1,Z2),Z3) = plus_plus(int,Z1,plus_plus(int,Z2,Z3)) ).
fof(fact_46_zadd__left__commute,axiom,
! [X,Y,Z] : plus_plus(int,X,plus_plus(int,Y,Z)) = plus_plus(int,Y,plus_plus(int,X,Z)) ).
fof(fact_47_zadd__commute,axiom,
! [Z,W] : plus_plus(int,Z,W) = plus_plus(int,W,Z) ).
fof(fact_48_int__int__eq,axiom,
! [Ma,Na] :
( hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(nat,int,semiring_1_of_nat(int),Na)
<=> Ma = Na ) ).
fof(fact_49_less__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),pls)) ) ) ).
fof(fact_50_less__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Ya)) ) ) ).
fof(fact_51_rel__simps_I12_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_52_less__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_53_rel__simps_I16_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) ) ).
fof(fact_54_rel__simps_I10_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_55_rel__simps_I4_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit0(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) ) ).
fof(fact_56_bin__less__0__simps_I4_J,axiom,
! [Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(Wa)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).
fof(fact_57_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),zero_zero(int))) ).
fof(fact_58_bin__less__0__simps_I3_J,axiom,
! [Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(Wa)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).
fof(fact_59_int__0__less__1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))) ).
fof(fact_60_zless__add1__eq,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int))))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1))
| ti(int,Wa) = ti(int,Z_1) ) ) ).
fof(fact_61_int__less__0__conv,axiom,
! [K_1] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_1)),zero_zero(int))) ).
fof(fact_62_less__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),bit1(pls))) ) ) ).
fof(fact_63_less__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),Ya)) ) ) ).
fof(fact_64_odd__less__0,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))) ) ).
fof(fact_65_double__eq__0__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( plus_plus(X_a,A_2,A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_66_rel__simps_I46_J,axiom,
! [K_1] : bit1(K_1) != pls ).
fof(fact_67_rel__simps_I39_J,axiom,
! [L] : pls != bit1(L) ).
fof(fact_68_rel__simps_I50_J,axiom,
! [K_1,L] : bit1(K_1) != bit0(L) ).
fof(fact_69_rel__simps_I49_J,axiom,
! [K_1,L] : bit0(K_1) != bit1(L) ).
fof(fact_70_rel__simps_I44_J,axiom,
! [K] :
( bit0(K) = pls
<=> ti(int,K) = pls ) ).
fof(fact_71_rel__simps_I38_J,axiom,
! [L_1] :
( pls = bit0(L_1)
<=> pls = ti(int,L_1) ) ).
fof(fact_72_Bit0__Pls,axiom,
bit0(pls) = pls ).
fof(fact_73_Pls__def,axiom,
pls = zero_zero(int) ).
fof(fact_74_int__0__neq__1,axiom,
zero_zero(int) != one_one(int) ).
fof(fact_75_add__Pls__right,axiom,
! [K_1] : plus_plus(int,K_1,pls) = ti(int,K_1) ).
fof(fact_76_add__Pls,axiom,
! [K_1] : plus_plus(int,pls,K_1) = ti(int,K_1) ).
fof(fact_77_add__Bit0__Bit0,axiom,
! [K_1,L] : plus_plus(int,bit0(K_1),bit0(L)) = bit0(plus_plus(int,K_1,L)) ).
fof(fact_78_Bit0__def,axiom,
! [K_1] : bit0(K_1) = plus_plus(int,K_1,K_1) ).
fof(fact_79_zadd__0__right,axiom,
! [Z] : plus_plus(int,Z,zero_zero(int)) = ti(int,Z) ).
fof(fact_80_zadd__0,axiom,
! [Z] : plus_plus(int,zero_zero(int),Z) = ti(int,Z) ).
fof(fact_81_semiring__numeral__0__eq__0,axiom,
! [X_a] :
( number_semiring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_82_number__of__Pls,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_83_semiring__norm_I112_J,axiom,
! [X_a] :
( number_ring(X_a)
=> zero_zero(X_a) = number_number_of(X_a,pls) ) ).
fof(fact_84_add__numeral__0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : plus_plus(X_a,number_number_of(X_a,pls),A_1) = ti(X_a,A_1) ) ).
fof(fact_85_add__numeral__0__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : plus_plus(X_a,A_1,number_number_of(X_a,pls)) = ti(X_a,A_1) ) ).
fof(fact_86_power__eq__0__iff__number__of,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [A_2,Wa] :
( hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,Wa)) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& number_number_of(nat,Wa) != zero_zero(nat) ) ) ) ).
fof(fact_87_add__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W,Z] : plus_plus(X_a,number_number_of(X_a,V_1),plus_plus(X_a,number_number_of(X_a,W),Z)) = plus_plus(X_a,number_number_of(X_a,plus_plus(int,V_1,W)),Z) ) ).
fof(fact_88_add__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W] : plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,V_1,W)) ) ).
fof(fact_89_number__of__add,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W] : number_number_of(X_a,plus_plus(int,V_1,W)) = plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) ) ).
fof(fact_90_add__Bit1__Bit0,axiom,
! [K_1,L] : plus_plus(int,bit1(K_1),bit0(L)) = bit1(plus_plus(int,K_1,L)) ).
fof(fact_91_add__Bit0__Bit1,axiom,
! [K_1,L] : plus_plus(int,bit0(K_1),bit1(L)) = bit1(plus_plus(int,K_1,L)) ).
fof(fact_92_Bit1__def,axiom,
! [K_1] : bit1(K_1) = plus_plus(int,plus_plus(int,one_one(int),K_1),K_1) ).
fof(fact_93_odd__nonzero,axiom,
! [Z] : plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int) ).
fof(fact_94_number__of__int,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [N] : number_number_of(X_a,hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).
fof(fact_95_zero__less__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls))))))
<=> ti(X_a,A_2) != zero_zero(X_a) ) ) ).
fof(fact_96_power2__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a))) ) ).
fof(fact_97_sum__power2__gt__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls)))))))
<=> ( ti(X_a,Xa) != zero_zero(X_a)
| ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).
fof(fact_98_not__sum__power2__lt__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a))) ) ).
fof(fact_99_number__of__Bit0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : number_number_of(X_a,bit0(W)) = plus_plus(X_a,plus_plus(X_a,zero_zero(X_a),number_number_of(X_a,W)),number_number_of(X_a,W)) ) ).
fof(fact_100_number__of__Bit1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : number_number_of(X_a,bit1(W)) = plus_plus(X_a,plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)),number_number_of(X_a,W)) ) ).
fof(fact_101_semiring__numeral__1__eq__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> number_number_of(X_a,bit1(pls)) = one_one(X_a) ) ).
fof(fact_102_mn,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),m1),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).
fof(fact_103_of__nat__less__two__power,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,bit0(bit1(pls)))),N))) ) ).
fof(fact_104_transfer__int__nat__numerals_I3_J,axiom,
number_number_of(int,bit0(bit1(pls))) = hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,bit0(bit1(pls)))) ).
fof(fact_105_transfer__int__nat__numerals_I4_J,axiom,
number_number_of(int,bit1(bit1(pls))) = hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,bit1(bit1(pls)))) ).
fof(fact_106_realpow__two__sum__zero__iff,axiom,
! [Xa,Ya] :
( plus_plus(real,hAPP(nat,real,power_power(real,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Ya),number_number_of(nat,bit0(bit1(pls))))) = zero_zero(real)
<=> ( Xa = zero_zero(real)
& Ya = zero_zero(real) ) ) ).
fof(fact_107_of__nat__0__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).
fof(fact_108_one__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ).
fof(fact_109_power__0__left,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [N] :
( ( N = zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = one_one(X_a) )
& ( N != zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = zero_zero(X_a) ) ) ) ).
fof(fact_110_power__strict__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ) ).
fof(fact_111_zero__less__two,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,one_one(X_a),one_one(X_a)))) ) ).
fof(fact_112_int__gr__induct,axiom,
! [P_1,K,I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_1))
=> ( hBOOL(hAPP(int,bool,P_1,plus_plus(int,K,one_one(int))))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I))
=> ( hBOOL(hAPP(int,bool,P_1,I))
=> hBOOL(hAPP(int,bool,P_1,plus_plus(int,I,one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,P_1,I_1)) ) ) ) ).
fof(fact_113_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) ).
fof(fact_114_tn0,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),tn)) ).
fof(fact_115_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_116_real__zero__not__eq__one,axiom,
zero_zero(real) != one_one(real) ).
fof(fact_117_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).
fof(fact_118_not__add__less1,axiom,
! [I_2,J_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,J_1)),I_2)) ).
fof(fact_119_not__add__less2,axiom,
! [J_1,I_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,J_1,I_2)),I_2)) ).
fof(fact_120_number__of__is__id,axiom,
! [K_1] : number_number_of(int,K_1) = ti(int,K_1) ).
fof(fact_121_nat__neq__iff,axiom,
! [Ma,Na] :
( Ma != Na
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ) ).
fof(fact_122_nat__add__commute,axiom,
! [M,N] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
fof(fact_123_nat__add__left__commute,axiom,
! [X,Y,Z] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
fof(fact_124_nat__add__assoc,axiom,
! [M,N,K_1] : plus_plus(nat,plus_plus(nat,M,N),K_1) = plus_plus(nat,M,plus_plus(nat,N,K_1)) ).
fof(fact_125_nat__add__left__cancel,axiom,
! [K,Ma,Na] :
( plus_plus(nat,K,Ma) = plus_plus(nat,K,Na)
<=> Ma = Na ) ).
fof(fact_126_nat__add__right__cancel,axiom,
! [Ma,K,Na] :
( plus_plus(nat,Ma,K) = plus_plus(nat,Na,K)
<=> Ma = Na ) ).
fof(fact_127_nat__add__left__cancel__less,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,K,Ma)),plus_plus(nat,K,Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).
fof(fact_128_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
fof(fact_129_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).
fof(fact_130_less__not__refl2,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> M != N ) ).
fof(fact_131_less__not__refl3,axiom,
! [S_1,T_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_1),T_2))
=> S_1 != T_2 ) ).
fof(fact_132_trans__less__add1,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),plus_plus(nat,J_1,M))) ) ).
fof(fact_133_trans__less__add2,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),plus_plus(nat,M,J_1))) ) ).
fof(fact_134_add__less__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,K_1))) ) ).
fof(fact_135_add__less__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,L))) ) ) ).
fof(fact_136_less__add__eq__less,axiom,
! [M,N,K_1,L] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L))
=> ( plus_plus(nat,M,L) = plus_plus(nat,K_1,N)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_137_add__lessD1,axiom,
! [I_2,J_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,J_1)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),K_1)) ) ).
fof(fact_138_nat__less__cases,axiom,
! [P_1,Ma,Na] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> ( ( Ma = Na
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) ) ) ) ).
fof(fact_139_gr0I,axiom,
! [N] :
( N != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_140_gr__implies__not0,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> N != zero_zero(nat) ) ).
fof(fact_141_nat__power__less__imp__less,axiom,
! [M,N,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,I_2),M)),hAPP(nat,nat,power_power(nat,I_2),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_142_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_143_nat__zero__less__power__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa))
| Na = zero_zero(nat) ) ) ).
fof(fact_144_add__gr__0,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),plus_plus(nat,Ma,Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).
fof(fact_145_neq0__conv,axiom,
! [Na] :
( Na != zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
fof(fact_146_not__less0,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_147_zero__less__power__nat__eq,axiom,
! [Xa,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na)))
<=> ( Na = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).
fof(fact_148_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ).
fof(fact_149_zero__less__power__nat__eq__number__of,axiom,
! [Xa,Wa] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),number_number_of(nat,Wa))))
<=> ( number_number_of(nat,Wa) = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).
fof(fact_150_zless__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).
fof(fact_151_less__nat__number__of,axiom,
! [Va,V_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,Va)),number_number_of(nat,V_3)))
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_3)) )
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_3)) ) ) ).
fof(fact_152_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_153_add__eq__self__zero,axiom,
! [M,N] :
( plus_plus(nat,M,N) = M
=> N = zero_zero(nat) ) ).
fof(fact_154_add__is__0,axiom,
! [Ma,Na] :
( plus_plus(nat,Ma,Na) = zero_zero(nat)
<=> ( Ma = zero_zero(nat)
& Na = zero_zero(nat) ) ) ).
fof(fact_155_Nat_Oadd__0__right,axiom,
! [M] : plus_plus(nat,M,zero_zero(nat)) = M ).
fof(fact_156_plus__nat_Oadd__0,axiom,
! [N] : plus_plus(nat,zero_zero(nat),N) = N ).
fof(fact_157_power__one__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat)) = ti(X_a,A_1) ) ).
fof(fact_158_of__nat__eq__iff,axiom,
! [X_a] :
( semiring_char_0(X_a)
=> ! [Ma,Na] :
( hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma) = hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)
<=> Ma = Na ) ) ).
fof(fact_159_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Xa,Ya] :
( hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya)
<=> Xa = Ya ) ).
fof(fact_160_int__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya)) )
& ( ~ hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya)) ) ) ).
fof(fact_161_less__0__number__of,axiom,
! [Va] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,Va)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)) ) ).
fof(fact_162_zero__less__int__conv,axiom,
! [Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
fof(fact_163_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_164_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_165_field__power__not__zero,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [N,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(nat,X_a,power_power(X_a,A_1),N) != zero_zero(X_a) ) ) ).
fof(fact_166_power__one,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [N] : hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N) = one_one(X_a) ) ).
fof(fact_167_of__nat__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).
fof(fact_168_less__imp__of__nat__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ) ).
fof(fact_169_of__nat__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_170_of__nat__add,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),plus_plus(nat,M,N)) = plus_plus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ).
fof(fact_171_of__nat__1,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a) ) ).
fof(fact_172_of__nat__power,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N) ) ).
fof(fact_173_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).
fof(fact_174_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),X),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,X,Y)) ).
fof(fact_175_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N] : hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),X)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,X),N)) ).
fof(fact_176_pos__add__strict,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [B_1,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C))) ) ) ) ).
fof(fact_177_not__one__less__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_178_zero__less__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_179_zero__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_180_less__add__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),plus_plus(X_a,A_1,one_one(X_a)))) ) ).
fof(fact_181_power__inject__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_2))
=> ( hAPP(nat,X_a,power_power(X_a,A_2),Ma) = hAPP(nat,X_a,power_power(X_a,A_2),Na)
<=> Ma = Na ) ) ) ).
fof(fact_182_power__strict__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Xa,Ya,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,B),Xa)),hAPP(nat,X_a,power_power(X_a,B),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ) ) ).
fof(fact_183_power__less__imp__less__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).
fof(fact_184_power__strict__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,A_1),N_2))) ) ) ) ).
fof(fact_185_power__eq__0__iff,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [A_2,Na] :
( hAPP(nat,X_a,power_power(X_a,A_2),Na) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& Na != zero_zero(nat) ) ) ) ).
fof(fact_186_of__nat__less__0__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),zero_zero(X_a))) ) ).
fof(fact_187_power__0,axiom,
! [X_a] :
( power(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_188_of__nat__0,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a) ) ).
fof(fact_189_pos2,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_190_zero__less__imp__eq__int,axiom,
! [K_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1))
=> ? [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
& ti(int,K_1) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ) ).
fof(fact_191_less__imp__add__positive,axiom,
! [I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> ? [K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2))
& plus_plus(nat,I_2,K_2) = J_1 ) ) ).
fof(fact_192_exp__eq__1,axiom,
! [Xa,Na] :
( hAPP(nat,nat,power_power(nat,Xa),Na) = one_one(nat)
<=> ( Xa = one_one(nat)
| Na = zero_zero(nat) ) ) ).
fof(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,power_power(X_a,X),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_194_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_2,A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_195_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_196_add__pos__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1))) ) ) ) ).
fof(fact_197_add__neg__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_198_zero__reorient,axiom,
! [X_a] :
( zero(X_a)
=> ! [Xa] :
( zero_zero(X_a) = ti(X_a,Xa)
<=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).
fof(fact_199_add__right__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B_1,A_1,C] :
( plus_plus(X_a,B_1,A_1) = plus_plus(X_a,C,A_1)
=> ti(X_a,B_1) = ti(X_a,C) ) ) ).
fof(fact_200_add__imp__eq,axiom,
! [X_a] :
( cancel146912293up_add(X_a)
=> ! [A_1,B_1,C] :
( plus_plus(X_a,A_1,B_1) = plus_plus(X_a,A_1,C)
=> ti(X_a,B_1) = ti(X_a,C) ) ) ).
fof(fact_201_add__left__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_1,B_1,C] :
( plus_plus(X_a,A_1,B_1) = plus_plus(X_a,A_1,C)
=> ti(X_a,B_1) = ti(X_a,C) ) ) ).
fof(fact_202_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C,D_1] : plus_plus(X_a,plus_plus(X_a,A_1,B_1),plus_plus(X_a,C,D_1)) = plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1)) ) ).
fof(fact_203_add__right__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B,A_2,C_1] :
( plus_plus(X_a,B,A_2) = plus_plus(X_a,C_1,A_2)
<=> ti(X_a,B) = ti(X_a,C_1) ) ) ).
fof(fact_204_add__left__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_2,B,C_1] :
( plus_plus(X_a,A_2,B) = plus_plus(X_a,A_2,C_1)
<=> ti(X_a,B) = ti(X_a,C_1) ) ) ).
fof(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C] : plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B_1) ) ).
fof(fact_206_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ! [A_1,B_1,C] : plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,A_1,plus_plus(X_a,B_1,C)) ) ).
fof(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C] : plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,A_1,plus_plus(X_a,B_1,C)) ) ).
fof(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : plus_plus(X_a,A_1,plus_plus(X_a,C,D_1)) = plus_plus(X_a,plus_plus(X_a,A_1,C),D_1) ) ).
fof(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : plus_plus(X_a,A_1,plus_plus(X_a,C,D_1)) = plus_plus(X_a,C,plus_plus(X_a,A_1,D_1)) ) ).
fof(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C] : plus_plus(X_a,A_1,C) = plus_plus(X_a,C,A_1) ) ).
fof(fact_211_one__reorient,axiom,
! [X_a] :
( one(X_a)
=> ! [Xa] :
( one_one(X_a) = ti(X_a,Xa)
<=> ti(X_a,Xa) = one_one(X_a) ) ) ).
fof(fact_212_add__0__iff,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [B,A_2] :
( ti(X_a,B) = plus_plus(X_a,B,A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_213_add_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_215_add__0__right,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_216_double__zero__sym,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( zero_zero(X_a) = plus_plus(X_a,A_2,A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_217_add__0,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_219_add__0__left,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_220_add__less__imp__less__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ).
fof(fact_221_add__less__imp__less__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ).
fof(fact_222_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1))) ) ) ) ).
fof(fact_223_add__strict__left__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))) ) ) ).
fof(fact_224_add__strict__right__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))) ) ) ).
fof(fact_225_add__less__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C_1,A_2)),plus_plus(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) ) ) ).
fof(fact_226_add__less__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) ) ) ).
fof(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,power_power(X_a,X),one_one(nat)) = ti(X_a,X) ) ).
fof(fact_228_nat__power__eq__0__iff,axiom,
! [Ma,Na] :
( hAPP(nat,nat,power_power(nat,Ma),Na) = zero_zero(nat)
<=> ( Na != zero_zero(nat)
& Ma = zero_zero(nat) ) ) ).
fof(fact_229_realpow__pos__nth__unique,axiom,
! [A_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
& hAPP(nat,real,power_power(real,X_1),N) = A_1
& ! [Y_1] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
& hAPP(nat,real,power_power(real,Y_1),N) = A_1 )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_230_realpow__pos__nth,axiom,
! [A_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [R] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
& hAPP(nat,real,power_power(real,R),N) = A_1 ) ) ) ).
fof(fact_231_tpos,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),t)) ).
fof(fact_232_nat__number__of__add__1,axiom,
! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> plus_plus(nat,number_number_of(nat,V_1),one_one(nat)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> plus_plus(nat,number_number_of(nat,V_1),one_one(nat)) = number_number_of(nat,succ(V_1)) ) ) ).
fof(fact_233_nat__1__add__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> plus_plus(nat,one_one(nat),number_number_of(nat,V_1)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> plus_plus(nat,one_one(nat),number_number_of(nat,V_1)) = number_number_of(nat,succ(V_1)) ) ) ).
fof(fact_234_zadd__power3,axiom,
! [A_1,B_1] : hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),B_1)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit1(bit1(pls))))) ).
fof(fact_235_zadd__power2,axiom,
! [A_1,B_1] : hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B_1)),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_236_tn,axiom,
tn = minus_minus(nat,nat_1(t),one_one(nat)) ).
fof(fact_237__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0_,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),minus_minus(nat,nat_1(t),one_one(nat)))) ).
fof(fact_238_le__iff__diff__le__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,A_2,B)),zero_zero(X_a))) ) ) ).
fof(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1] : times_times(X_a,A_1,B_1) = times_times(X_a,B_1,A_1) ) ).
fof(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,Lx,Ry)) ) ).
fof(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),Ry) ) ).
fof(fact_242_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_1,B_1,C] : times_times(X_a,times_times(X_a,A_1,B_1),C) = times_times(X_a,A_1,times_times(X_a,B_1,C)) ) ).
fof(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,Lx,times_times(X_a,Ly,Rx)) ) ).
fof(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,times_times(X_a,Lx,Rx),Ly) ) ).
fof(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,Lx,times_times(X_a,Ly,times_times(X_a,Rx,Ry))) ) ).
fof(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,times_times(X_a,Lx,Ly),Ry)) ) ).
fof(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),times_times(X_a,Ly,Ry)) ) ).
fof(fact_248_diff__eq__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_2,B,C_1,D] :
( minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D)
=> ( ti(X_a,A_2) = ti(X_a,B)
<=> ti(X_a,C_1) = ti(X_a,D) ) ) ) ).
fof(fact_249_diff__eq__diff__less__eq,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B,C_1,D] :
( minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D)) ) ) ) ).
fof(fact_250_semiring__mult__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_2,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2))
=> times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,times_times(int,V_1,V_2)) ) ) ) ).
fof(fact_251_zle__refl,axiom,
! [W] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W)) ).
fof(fact_252_zmult__commute,axiom,
! [Z,W] : times_times(int,Z,W) = times_times(int,W,Z) ).
fof(fact_253_zle__linear,axiom,
! [Z,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ).
fof(fact_254_zmult__assoc,axiom,
! [Z1,Z2,Z3] : times_times(int,times_times(int,Z1,Z2),Z3) = times_times(int,Z1,times_times(int,Z2,Z3)) ).
fof(fact_255_zle__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K_1)) ) ) ).
fof(fact_256_zle__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z))
=> ti(int,Z) = ti(int,W) ) ) ).
fof(fact_257_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ) ).
fof(fact_258_number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W] : number_number_of(X_a,times_times(int,V_1,W)) = times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) ) ).
fof(fact_259_arith__simps_I32_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W] : times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V_1,W)) ) ).
fof(fact_260_mult__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W,Z] : times_times(X_a,number_number_of(X_a,V_1),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V_1,W)),Z) ) ).
fof(fact_261_eq__nat__nat__iff,axiom,
! [Z_3,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_3))
=> ( nat_1(Z_1) = nat_1(Z_3)
<=> ti(int,Z_1) = ti(int,Z_3) ) ) ) ).
fof(fact_262_right__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [V_1,B_1,C] : times_times(X_b,number_number_of(X_b,V_1),minus_minus(X_b,B_1,C)) = minus_minus(X_b,times_times(X_b,number_number_of(X_b,V_1),B_1),times_times(X_b,number_number_of(X_b,V_1),C)) ) ).
fof(fact_263_left__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [A_1,B_1,V_1] : times_times(X_b,minus_minus(X_b,A_1,B_1),number_number_of(X_b,V_1)) = minus_minus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1))) ) ).
fof(fact_264_number__of__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W] : number_number_of(X_a,minus_minus(int,V_1,W)) = minus_minus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) ) ).
fof(fact_265_of__nat__mult,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),times_times(nat,M,N)) = times_times(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ).
fof(fact_266_of__nat__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).
fof(fact_267_diff__commute,axiom,
! [I_2,J_1,K_1] : minus_minus(nat,minus_minus(nat,I_2,J_1),K_1) = minus_minus(nat,minus_minus(nat,I_2,K_1),J_1) ).
fof(fact_268_nat__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> nat_1(Xa) = nat_1(if(int,P_1,Xa,Ya)) )
& ( ~ hBOOL(P_1)
=> nat_1(Ya) = nat_1(if(int,P_1,Xa,Ya)) ) ) ).
fof(fact_269_split__mult__neg__le,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B_1,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ).
fof(fact_270_split__mult__pos__le,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [B_1,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a))) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1))) ) ) ).
fof(fact_271_mult__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_272_mult__mono_H,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_273_mult__left__mono__neg,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [C,B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_274_mult__right__mono__neg,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [C,B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C))) ) ) ) ).
fof(fact_275_comm__mult__left__mono,axiom,
! [X_a] :
( ordere1490568538miring(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_276_mult__left__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_277_mult__right__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C))) ) ) ) ).
fof(fact_278_mult__nonpos__nonpos,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1))) ) ) ) ).
fof(fact_279_mult__nonpos__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_280_mult__nonneg__nonpos2,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,B_1,A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_281_mult__nonneg__nonpos,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_282_transfer__nat__int__relations_I1_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
=> ( nat_1(Xa) = nat_1(Ya)
<=> ti(int,Xa) = ti(int,Ya) ) ) ) ).
fof(fact_283_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,X,Y))) ) ) ).
fof(fact_284_mult__nonneg__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1))) ) ) ) ).
fof(fact_285_le__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1)),D)) ) ) ).
fof(fact_286_eq__add__iff1,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( plus_plus(X_a,times_times(X_a,A_2,E_1),C_1) = plus_plus(X_a,times_times(X_a,B,E_1),D)
<=> plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1) = ti(X_a,D) ) ) ).
fof(fact_287_le__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))) ) ) ).
fof(fact_288_eq__add__iff2,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( plus_plus(X_a,times_times(X_a,A_2,E_1),C_1) = plus_plus(X_a,times_times(X_a,B,E_1),D)
<=> ti(X_a,C_1) = plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D) ) ) ).
fof(fact_289_mult__diff__mult,axiom,
! [X_a] :
( ring(X_a)
=> ! [X,Y,A_1,B_1] : minus_minus(X_a,times_times(X_a,X,Y),times_times(X_a,A_1,B_1)) = plus_plus(X_a,times_times(X_a,X,minus_minus(X_a,Y,B_1)),times_times(X_a,minus_minus(X_a,X,A_1),B_1)) ) ).
fof(fact_290_mult__le__0__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_2,B)),zero_zero(X_a)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) ) ) ) ) ).
fof(fact_291_zero__le__mult__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_2,B)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) ) ) ) ).
fof(fact_292_all__nat,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
<=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))) ) ) ).
fof(fact_293_ex__nat,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
<=> ? [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))) ) ) ).
fof(fact_294_zero__le__square,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_1,A_1))) ) ).
fof(fact_295_less__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1)),D)) ) ) ).
fof(fact_296_less__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,C_1,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))) ) ) ).
fof(fact_297_real__squared__diff__one__factored,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [X] : minus_minus(X_a,times_times(X_a,X,X),one_one(X_a)) = times_times(X_a,plus_plus(X_a,X,one_one(X_a)),minus_minus(X_a,X,one_one(X_a))) ) ).
fof(fact_298_mult__left__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) ) ) ) ).
fof(fact_299_mult__right__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) ) ) ) ).
fof(fact_300_mult__less__imp__less__left,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ) ).
fof(fact_301_mult__left__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ) ).
fof(fact_302_mult__less__imp__less__right,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ) ).
fof(fact_303_mult__right__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [A_1,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ) ).
fof(fact_304_mult__le__less__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_305_mult__less__le__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_306_mult__strict__mono_H,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_307_mult__strict__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ) ) ).
fof(fact_308_mult__le__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_2)) ) ) ) ).
fof(fact_309_mult__le__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) ) ) ) ).
fof(fact_310_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y)))) ) ).
fof(fact_311_sum__squares__le__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya))),zero_zero(X_a)))
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_312_mult__right__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,X,Y)),X)) ) ) ) ) ).
fof(fact_313_mult__left__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,Y,X)),X)) ) ) ) ) ).
fof(fact_314_nat__le__0,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
=> nat_1(Z) = zero_zero(nat) ) ).
fof(fact_315_nat__0__iff,axiom,
! [I_1] :
( nat_1(I_1) = zero_zero(nat)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),zero_zero(int))) ) ).
fof(fact_316_power__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,A_1),N_2))) ) ) ) ).
fof(fact_317_int__nat__eq,axiom,
! [Z] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)) = ti(int,Z) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)) = zero_zero(int) ) ) ).
fof(fact_318_int__eq__iff,axiom,
! [Ma,Z_1] :
( hAPP(nat,int,semiring_1_of_nat(int),Ma) = ti(int,Z_1)
<=> ( Ma = nat_1(Z_1)
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) ) ) ).
fof(fact_319_nat__0__le,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)) = ti(int,Z) ) ).
fof(fact_320_right__minus__eq,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_2,B] :
( minus_minus(X_a,A_2,B) = zero_zero(X_a)
<=> ti(X_a,A_2) = ti(X_a,B) ) ) ).
fof(fact_321_eq__iff__diff__eq__0,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_2,B] :
( ti(X_a,A_2) = ti(X_a,B)
<=> minus_minus(X_a,A_2,B) = zero_zero(X_a) ) ) ).
fof(fact_322_diff__self,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : minus_minus(X_a,A_1,A_1) = zero_zero(X_a) ) ).
fof(fact_323_diff__0__right,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : minus_minus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_324_diff__eq__diff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B,C_1,D] :
( minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D)) ) ) ) ).
fof(fact_325_add__diff__add,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,C,B_1,D_1] : minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1)) = plus_plus(X_a,minus_minus(X_a,A_1,B_1),minus_minus(X_a,C,D_1)) ) ).
fof(fact_326_add__diff__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B_1] : minus_minus(X_a,plus_plus(X_a,A_1,B_1),B_1) = ti(X_a,A_1) ) ).
fof(fact_327_diff__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B_1] : plus_plus(X_a,minus_minus(X_a,A_1,B_1),B_1) = ti(X_a,A_1) ) ).
fof(fact_328_mult__zero__left,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a) ) ).
fof(fact_329_mult__zero__right,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_330_mult__eq__0__iff,axiom,
! [X_a] :
( ring_n68954251visors(X_a)
=> ! [A_2,B] :
( times_times(X_a,A_2,B) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
| ti(X_a,B) = zero_zero(X_a) ) ) ) ).
fof(fact_331_no__zero__divisors,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [B_1,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B_1) != zero_zero(X_a)
=> times_times(X_a,A_1,B_1) != zero_zero(X_a) ) ) ) ).
fof(fact_332_divisors__zero,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [A_1,B_1] :
( times_times(X_a,A_1,B_1) = zero_zero(X_a)
=> ( ti(X_a,A_1) = zero_zero(X_a)
| ti(X_a,B_1) = zero_zero(X_a) ) ) ) ).
fof(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a) ) ).
fof(fact_335_diffs0__imp__equal,axiom,
! [M,N] :
( minus_minus(nat,M,N) = zero_zero(nat)
=> ( minus_minus(nat,N,M) = zero_zero(nat)
=> M = N ) ) ).
fof(fact_336_diff__self__eq__0,axiom,
! [M] : minus_minus(nat,M,M) = zero_zero(nat) ).
fof(fact_337_minus__nat_Odiff__0,axiom,
! [M] : minus_minus(nat,M,zero_zero(nat)) = M ).
fof(fact_338_diff__0__eq__0,axiom,
! [N] : minus_minus(nat,zero_zero(nat),N) = zero_zero(nat) ).
fof(fact_339_comm__semiring__class_Odistrib,axiom,
! [X_a] :
( comm_semiring(X_a)
=> ! [A_1,B_1,C] : times_times(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C)) ) ).
fof(fact_340_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E,B_1,C] : plus_plus(X_a,times_times(X_a,A_1,E),plus_plus(X_a,times_times(X_a,B_1,E),C)) = plus_plus(X_a,times_times(X_a,plus_plus(X_a,A_1,B_1),E),C) ) ).
fof(fact_341_crossproduct__eq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [Wa,Ya,Xa,Z_1] :
( plus_plus(X_a,times_times(X_a,Wa,Ya),times_times(X_a,Xa,Z_1)) = plus_plus(X_a,times_times(X_a,Wa,Z_1),times_times(X_a,Xa,Ya))
<=> ( ti(X_a,Wa) = ti(X_a,Xa)
| ti(X_a,Ya) = ti(X_a,Z_1) ) ) ) ).
fof(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,B_1] : plus_plus(X_a,times_times(X_a,A_1,M),times_times(X_a,B_1,M)) = times_times(X_a,plus_plus(X_a,A_1,B_1),M) ) ).
fof(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C] : times_times(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C)) ) ).
fof(fact_344_crossproduct__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C_1,D,A_2,B] :
( ( ti(X_a,A_2) != ti(X_a,B)
& ti(X_a,C_1) != ti(X_a,D) )
<=> plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B,D)) != plus_plus(X_a,times_times(X_a,A_2,D),times_times(X_a,B,C_1)) ) ) ).
fof(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Z] : times_times(X_a,X,plus_plus(X_a,Y,Z)) = plus_plus(X_a,times_times(X_a,X,Y),times_times(X_a,X,Z)) ) ).
fof(fact_346_nat__int,axiom,
! [N] : nat_1(hAPP(nat,int,semiring_1_of_nat(int),N)) = N ).
fof(fact_347_mult_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_349_mult__1__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_350_mult__1,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_352_mult__1__left,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1) ) ).
fof(fact_353_add__le__imp__le__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) ) ) ).
fof(fact_354_add__le__imp__le__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) ) ) ).
fof(fact_355_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1))) ) ) ) ).
fof(fact_356_add__left__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))) ) ) ).
fof(fact_357_add__right__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))) ) ) ).
fof(fact_358_add__le__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C_1,A_2)),plus_plus(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) ) ) ).
fof(fact_359_add__le__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) ) ) ).
fof(fact_360_less__imp__diff__less,axiom,
! [N,J_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,J_1,N)),K_1)) ) ).
fof(fact_361_diff__less__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,L,N)),minus_minus(nat,L,M))) ) ) ).
fof(fact_362_power__commutes,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N] : times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N),A_1) = times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)) ) ).
fof(fact_363_power__mult__distrib,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1,B_1,N] : hAPP(nat,X_a,power_power(X_a,times_times(X_a,A_1,B_1)),N) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N),hAPP(nat,X_a,power_power(X_a,B_1),N)) ) ).
fof(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Q] : hAPP(nat,X_a,power_power(X_a,times_times(X_a,X,Y)),Q) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),Q),hAPP(nat,X_a,power_power(X_a,Y),Q)) ) ).
fof(fact_365_diff__add__inverse2,axiom,
! [M,N] : minus_minus(nat,plus_plus(nat,M,N),N) = M ).
fof(fact_366_diff__add__inverse,axiom,
! [N,M] : minus_minus(nat,plus_plus(nat,N,M),N) = M ).
fof(fact_367_diff__diff__left,axiom,
! [I_2,J_1,K_1] : minus_minus(nat,minus_minus(nat,I_2,J_1),K_1) = minus_minus(nat,I_2,plus_plus(nat,J_1,K_1)) ).
fof(fact_368_Nat_Odiff__cancel,axiom,
! [K_1,M,N] : minus_minus(nat,plus_plus(nat,K_1,M),plus_plus(nat,K_1,N)) = minus_minus(nat,M,N) ).
fof(fact_369_diff__cancel2,axiom,
! [M,K_1,N] : minus_minus(nat,plus_plus(nat,M,K_1),plus_plus(nat,N,K_1)) = minus_minus(nat,M,N) ).
fof(fact_370_mult__Pls,axiom,
! [W] : times_times(int,pls,W) = pls ).
fof(fact_371_mult__Bit0,axiom,
! [K_1,L] : times_times(int,bit0(K_1),L) = bit0(times_times(int,K_1,L)) ).
fof(fact_372_rel__simps_I34_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit1(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) ) ).
fof(fact_373_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit1(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_374_rel__simps_I19_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),pls)) ).
fof(fact_375_rel__simps_I31_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) ) ).
fof(fact_376_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit0(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_377_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).
fof(fact_378_zmult__1,axiom,
! [Z] : times_times(int,one_one(int),Z) = ti(int,Z) ).
fof(fact_379_zmult__1__right,axiom,
! [Z] : times_times(int,Z,one_one(int)) = ti(int,Z) ).
fof(fact_380_zless__le,axiom,
! [Z_1,Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),Wa))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),Wa))
& ti(int,Z_1) != ti(int,Wa) ) ) ).
fof(fact_381_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times(int,W,plus_plus(int,Z1,Z2)) = plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2)) ).
fof(fact_382_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W)) ).
fof(fact_383_zadd__left__mono,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,K_1,I_2)),plus_plus(int,K_1,J_1))) ) ).
fof(fact_384_times__numeral__code_I5_J,axiom,
! [V_1,W] : times_times(int,number_number_of(int,V_1),number_number_of(int,W)) = number_number_of(int,times_times(int,V_1,W)) ).
fof(fact_385_less__eq__number__of__int__code,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,K)),number_number_of(int,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) ) ).
fof(fact_386_convex__bound__le,axiom,
! [X_a] :
( linord1278240602ring_1(X_a)
=> ! [V_1,U_1,Y,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V_1))
=> ( plus_plus(X_a,U_1,V_1) = one_one(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,U_1,X),times_times(X_a,V_1,Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_387_transfer__nat__int__relations_I2_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ) ) ).
fof(fact_388_nat__less__eq__zless,axiom,
! [Z_1,Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) ) ) ).
fof(fact_389_le__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)) ) ) ).
fof(fact_390_le__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),pls)) ) ) ).
fof(fact_391_power__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ) ).
fof(fact_392_power__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Xa,Ya,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,B),Xa)),hAPP(nat,X_a,power_power(X_a,B),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ) ) ).
fof(fact_393_power__le__imp__le__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
fof(fact_394_nat__eq__iff2,axiom,
! [Ma,Wa] :
( Ma = nat_1(Wa)
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> Ma = zero_zero(nat) ) ) ) ).
fof(fact_395_nat__eq__iff,axiom,
! [Wa,Ma] :
( nat_1(Wa) = Ma
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> Ma = zero_zero(nat) ) ) ) ).
fof(fact_396_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> plus_plus(nat,nat_1(X),nat_1(Y)) = nat_1(plus_plus(int,X,Y)) ) ) ).
fof(fact_397_nat__add__distrib,axiom,
! [Z_2,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_2))
=> nat_1(plus_plus(int,Z,Z_2)) = plus_plus(nat,nat_1(Z),nat_1(Z_2)) ) ) ).
fof(fact_398_int__eq__iff__number__of,axiom,
! [Ma,Va] :
( hAPP(nat,int,semiring_1_of_nat(int),Ma) = number_number_of(int,Va)
<=> ( Ma = nat_1(number_number_of(int,Va))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,Va))) ) ) ).
fof(fact_399_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,nat,power_power(nat,nat_1(X)),N) = nat_1(hAPP(nat,int,power_power(int,X),N)) ) ).
fof(fact_400_nat__power__eq,axiom,
! [N,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> nat_1(hAPP(nat,int,power_power(int,Z),N)) = hAPP(nat,nat,power_power(nat,nat_1(Z)),N) ) ).
fof(fact_401_convex__bound__lt,axiom,
! [X_a] :
( linord626643107strict(X_a)
=> ! [V_1,U_1,Y,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V_1))
=> ( plus_plus(X_a,U_1,V_1) = one_one(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,U_1,X),times_times(X_a,V_1,Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_402_nat__less__iff,axiom,
! [Ma,Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),Ma))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),Ma))) ) ) ).
fof(fact_403_le__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),Ya)) ) ) ).
fof(fact_404_le__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),bit1(pls))) ) ) ).
fof(fact_405_realpow__minus__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [X,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),minus_minus(nat,N,one_one(nat))),X) = hAPP(nat,X_a,power_power(X_a,X),N) ) ) ).
fof(fact_406_less__iff__diff__less__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),minus_minus(X_a,A_2,B)),zero_zero(X_a))) ) ) ).
fof(fact_407_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = nat_1(zero_zero(int)) ).
fof(fact_408_nat__0,axiom,
nat_1(zero_zero(int)) = zero_zero(nat) ).
fof(fact_409_not__square__less__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,A_1)),zero_zero(X_a))) ) ).
fof(fact_410_mult__less__cancel__right__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_2)) ) ) ) ) ).
fof(fact_411_mult__less__cancel__left__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C_1,A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_2)) ) ) ) ) ).
fof(fact_412_mult__less__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) ) ) ) ).
fof(fact_413_mult__pos__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1))) ) ) ) ).
fof(fact_414_mult__pos__neg,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_415_mult__pos__neg2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,B_1,A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_416_zero__less__mult__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1)) ) ) ) ).
fof(fact_417_zero__less__mult__pos2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,B_1,A_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1)) ) ) ) ).
fof(fact_418_mult__less__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_2)) ) ) ) ).
fof(fact_419_mult__neg__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_420_mult__neg__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1))) ) ) ) ).
fof(fact_421_mult__strict__right__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C))) ) ) ) ).
fof(fact_422_mult__strict__left__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_423_comm__mult__strict__left__mono,axiom,
! [X_a] :
( linord893533164strict(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_424_mult__strict__right__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,C))) ) ) ) ).
fof(fact_425_mult__strict__left__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_1))) ) ) ) ).
fof(fact_426_nat__number__of,axiom,
! [W] : nat_1(number_number_of(int,W)) = number_number_of(nat,W) ).
fof(fact_427_nat__number__of__def,axiom,
! [V_1] : number_number_of(nat,V_1) = nat_1(number_number_of(int,V_1)) ).
fof(fact_428_sum__squares__eq__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Xa,Ya] :
( plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya)) = zero_zero(X_a)
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_429_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D_1,A_1,B_1,R_1] :
( ti(X_a,R_1) != zero_zero(X_a)
=> ( ( ti(X_a,A_1) = ti(X_a,B_1)
& ti(X_a,C) != ti(X_a,D_1) )
=> plus_plus(X_a,A_1,times_times(X_a,R_1,C)) != plus_plus(X_a,B_1,times_times(X_a,R_1,D_1)) ) ) ) ).
fof(fact_430_transfer__nat__int__numerals_I2_J,axiom,
one_one(nat) = nat_1(one_one(int)) ).
fof(fact_431_less__1__mult,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,M] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),times_times(X_a,M,N))) ) ) ) ).
fof(fact_432_zero__le__double__add__iff__zero__le__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,A_2,A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_433_double__add__le__zero__iff__single__add__le__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_434_add__nonneg__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1))) ) ) ) ).
fof(fact_435_add__nonneg__eq__0__iff,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [Ya,Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ya))
=> ( plus_plus(X_a,Xa,Ya) = zero_zero(X_a)
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_436_add__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),plus_plus(X_a,A_1,C))) ) ) ) ).
fof(fact_437_add__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),plus_plus(X_a,A_1,C))) ) ) ) ).
fof(fact_438_add__nonpos__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_439_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1))) ) ) ) ).
fof(fact_440_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1))) ) ) ) ).
fof(fact_441_not__one__le__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_442_zero__le__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_443_right__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [V_1,B_1,C] : times_times(X_b,number_number_of(X_b,V_1),plus_plus(X_b,B_1,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V_1),B_1),times_times(X_b,number_number_of(X_b,V_1),C)) ) ).
fof(fact_444_left__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [A_1,B_1,V_1] : times_times(X_b,plus_plus(X_b,A_1,B_1),number_number_of(X_b,V_1)) = plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1))) ) ).
fof(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M] : plus_plus(X_a,M,M) = times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),M) ) ).
fof(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M,A_1] : plus_plus(X_a,M,times_times(X_a,A_1,M)) = times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M) ) ).
fof(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M] : plus_plus(X_a,times_times(X_a,A_1,M),M) = times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M) ) ).
fof(fact_448_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [Va,Wa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,Wa)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Wa)),number_number_of(X_a,Va))) ) ) ).
fof(fact_449_zero__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_450_power__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B_1),N))) ) ) ) ).
fof(fact_451_zero__less__diff,axiom,
! [Na,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),minus_minus(nat,Na,Ma)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).
fof(fact_452_diff__less,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,M,N)),M)) ) ) ).
fof(fact_453_diff__add__0,axiom,
! [N,M] : minus_minus(nat,N,plus_plus(nat,N,M)) = zero_zero(nat) ).
fof(fact_454_one__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_455_less__diff__conv,axiom,
! [I_1,J,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),minus_minus(nat,J,K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_1,K)),J)) ) ).
fof(fact_456_add__diff__inverse,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> plus_plus(nat,N,minus_minus(nat,M,N)) = M ) ).
fof(fact_457_zero__le__imp__of__nat,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M))) ) ).
fof(fact_458_of__nat__0__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ).
fof(fact_459_power__add,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N] : hAPP(nat,X_a,power_power(X_a,A_1),plus_plus(nat,M,N)) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M),hAPP(nat,X_a,power_power(X_a,A_1),N)) ) ).
fof(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_2,Q] : times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),P_2),hAPP(nat,X_a,power_power(X_a,X),Q)) = hAPP(nat,X_a,power_power(X_a,X),plus_plus(nat,P_2,Q)) ) ).
fof(fact_461_rel__simps_I22_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_462_rel__simps_I32_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) ) ).
fof(fact_463_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_464_rel__simps_I21_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit0(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_465_rel__simps_I27_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)) ) ).
fof(fact_466_pos__zmult__pos,axiom,
! [B_1,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),times_times(int,A_1,B_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1)) ) ) ).
fof(fact_467_zmult__zless__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,K_1,I_2)),times_times(int,K_1,J_1))) ) ) ).
fof(fact_468_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))) ).
fof(fact_469_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,X,Y))) ) ) ).
fof(fact_470_of__nat__double,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),X))) = plus_plus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(X)),hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(X))) ) ) ).
fof(fact_471_zadd__zless__mono,axiom,
! [Z_2,Z,W_2,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,W_2,Z_2)),plus_plus(int,W,Z))) ) ) ).
fof(fact_472_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z))) ).
fof(fact_473_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(hAPP(int,bool,P_1,X_1)) )
<=> ? [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).
fof(fact_474_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(hAPP(int,bool,P_1,X_1)) )
<=> ! [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).
fof(fact_475_zero__zle__int,axiom,
! [N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N))) ).
fof(fact_476_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,X),N))) ) ).
fof(fact_477_zle__iff__zadd,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1))
<=> ? [N_1] : ti(int,Z_1) = plus_plus(int,Wa,hAPP(nat,int,semiring_1_of_nat(int),N_1)) ) ).
fof(fact_478_zpower__zadd__distrib,axiom,
! [X,Y,Z] : hAPP(nat,int,power_power(int,X),plus_plus(nat,Y,Z)) = times_times(int,hAPP(nat,int,power_power(int,X),Y),hAPP(nat,int,power_power(int,X),Z)) ).
fof(fact_479_semiring__1__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> plus_plus(X_a,one_one(X_a),number_number_of(X_a,V_1)) = number_number_of(X_a,succ(V_1)) ) ) ).
fof(fact_480_semiring__number__of__add__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a)) = number_number_of(X_a,succ(V_1)) ) ) ).
fof(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N] : hAPP(nat,X_a,power_power(X_a,X),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),N),hAPP(nat,X_a,power_power(X_a,X),N)) ) ).
fof(fact_482_add__number__of__diff1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W,C] : plus_plus(X_a,number_number_of(X_a,V_1),minus_minus(X_a,number_number_of(X_a,W),C)) = minus_minus(X_a,number_number_of(X_a,plus_plus(int,V_1,W)),C) ) ).
fof(fact_483_nat__mono__iff,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) ) ) ).
fof(fact_484_zless__nat__conj,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1)))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) ) ) ).
fof(fact_485_not__sum__squares__lt__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y))),zero_zero(X_a))) ) ).
fof(fact_486_sum__squares__gt__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya))))
<=> ( ti(X_a,Xa) != zero_zero(X_a)
| ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).
fof(fact_487_zless__nat__eq__int__zless,axiom,
! [Ma,Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),nat_1(Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z_1)) ) ).
fof(fact_488_add__pos__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1))) ) ) ) ).
fof(fact_489_add__nonneg__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1))) ) ) ) ).
fof(fact_490_add__strict__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C))) ) ) ) ).
fof(fact_491_add__strict__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C))) ) ) ) ).
fof(fact_492_add__neg__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_493_add__nonpos__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_494_power__less__imp__less__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B_1),N)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) ) ) ) ).
fof(fact_495_power__gt1__lemma,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)))) ) ) ).
fof(fact_496_power__less__power__Suc,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)))) ) ) ).
fof(fact_497_mult__numeral__1__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : times_times(X_a,A_1,number_number_of(X_a,bit1(pls))) = ti(X_a,A_1) ) ).
fof(fact_498_mult__numeral__1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : times_times(X_a,number_number_of(X_a,bit1(pls)),A_1) = ti(X_a,A_1) ) ).
fof(fact_499_nat__diff__split__asm,axiom,
! [P_1,A_2,B] :
( hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B)))
<=> ~ ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B))
& ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
| ? [D_3] :
( A_2 = plus_plus(nat,B,D_3)
& ~ hBOOL(hAPP(nat,bool,P_1,D_3)) ) ) ) ).
fof(fact_500_nat__diff__split,axiom,
! [P_1,A_2,B] :
( hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B)))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B))
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
& ! [D_3] :
( A_2 = plus_plus(nat,B,D_3)
=> hBOOL(hAPP(nat,bool,P_1,D_3)) ) ) ) ).
fof(fact_501_eq__0__number__of,axiom,
! [Va] :
( zero_zero(nat) = number_number_of(nat,Va)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).
fof(fact_502_eq__number__of__0,axiom,
! [Va] :
( number_number_of(nat,Va) = zero_zero(nat)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).
fof(fact_503_rel__simps_I5_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit1(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_504_rel__simps_I29_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_505_rel__simps_I15_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit1(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) ) ).
fof(fact_506_less__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit1(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_507_rel__simps_I33_J,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit0(L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) ) ).
fof(fact_508_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit0(K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_509_mult__Bit1,axiom,
! [K_1,L] : times_times(int,bit1(K_1),L) = plus_plus(int,bit0(times_times(int,K_1,L)),L) ).
fof(fact_510_pos__zmult__eq__1__iff,axiom,
! [Na,Ma] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
=> ( times_times(int,Ma,Na) = one_one(int)
<=> ( ti(int,Ma) = one_one(int)
& ti(int,Na) = one_one(int) ) ) ) ).
fof(fact_511_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ).
fof(fact_512_int__le__0__conv,axiom,
! [Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),zero_zero(int)))
<=> Na = zero_zero(nat) ) ).
fof(fact_513_succ__Pls,axiom,
succ(pls) = bit1(pls) ).
fof(fact_514_succ__Bit0,axiom,
! [K_1] : succ(bit0(K_1)) = bit1(K_1) ).
fof(fact_515_succ__Bit1,axiom,
! [K_1] : succ(bit1(K_1)) = bit0(succ(K_1)) ).
fof(fact_516_zle__add1__eq__le,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)) ) ).
fof(fact_517_add1__zle__eq,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,Wa,one_one(int))),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) ) ).
fof(fact_518_zless__imp__add1__zle,axiom,
! [W,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,W,one_one(int))),Z)) ) ).
fof(fact_519_succ__def,axiom,
! [K_1] : succ(K_1) = plus_plus(int,K_1,one_one(int)) ).
fof(fact_520_power2__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X,Y] : hAPP(nat,X_a,power_power(X_a,minus_minus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = minus_minus(X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X),Y)) ) ).
fof(fact_521_zero__less__nat__eq,axiom,
! [Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat_1(Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ).
fof(fact_522_transfer__nat__int__numerals_I4_J,axiom,
number_number_of(nat,bit1(bit1(pls))) = nat_1(number_number_of(int,bit1(bit1(pls)))) ).
fof(fact_523_split__nat,axiom,
! [P_1,I_1] :
( hBOOL(hAPP(nat,bool,P_1,nat_1(I_1)))
<=> ( ! [N_1] :
( ti(int,I_1) = hAPP(nat,int,semiring_1_of_nat(int),N_1)
=> hBOOL(hAPP(nat,bool,P_1,N_1)) )
& ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int)))
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ) ).
fof(fact_524_power__Suc__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N))),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ).
fof(fact_525_double__number__of__Bit0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),number_number_of(X_a,W)) = number_number_of(X_a,bit0(W)) ) ).
fof(fact_526_power__eq__imp__eq__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B_1] :
( hAPP(nat,X_a,power_power(X_a,A_1),N) = hAPP(nat,X_a,power_power(X_a,B_1),N)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ti(X_a,A_1) = ti(X_a,B_1) ) ) ) ) ) ).
fof(fact_527_power3__eq__cube,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls)))) = times_times(X_a,times_times(X_a,A_1,A_1),A_1) ) ).
fof(fact_528_semiring__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_2,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2))
=> plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,plus_plus(int,V_1,V_2)) ) ) ) ).
fof(fact_529_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit1(bit1(pls))))) ).
fof(fact_530_le__imp__0__less,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),Z))) ) ).
fof(fact_531_zmult__zless__mono2__lemma,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,hAPP(nat,int,semiring_1_of_nat(int),K_1),I_2)),times_times(int,hAPP(nat,int,semiring_1_of_nat(int),K_1),J_1))) ) ) ).
fof(fact_532_number__of__succ,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [K_1] : number_number_of(X_a,succ(K_1)) = plus_plus(X_a,one_one(X_a),number_number_of(X_a,K_1)) ) ).
fof(fact_533_add__Bit1__Bit1,axiom,
! [K_1,L] : plus_plus(int,bit1(K_1),bit1(L)) = bit0(plus_plus(int,K_1,succ(L))) ).
fof(fact_534_transfer__nat__int__numerals_I3_J,axiom,
number_number_of(nat,bit0(bit1(pls))) = nat_1(number_number_of(int,bit0(bit1(pls)))) ).
fof(fact_535_mult__2__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls)))) = plus_plus(X_a,Z,Z) ) ).
fof(fact_536_semiring__mult__2__right,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [Z] : times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls)))) = plus_plus(X_a,Z,Z) ) ).
fof(fact_537_mult__2,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z) ) ).
fof(fact_538_semiring__mult__2,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [Z] : times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z) ) ).
fof(fact_539_power__strict__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B_1),N))) ) ) ) ) ).
fof(fact_540_power2__eq__square,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,A_1,A_1) ) ).
fof(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : times_times(X_a,X,X) = hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_542_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(bit1(pls))))) ).
fof(fact_543_cube__square,axiom,
! [A_1] : times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))) ).
fof(fact_544_power2__ge__self,axiom,
! [X] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls)))))) ).
fof(fact_545_power2__eq__square__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [W] : hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_b,number_number_of(X_b,W),number_number_of(X_b,W)) ) ).
fof(fact_546_zero__le__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))))) ) ).
fof(fact_547_power2__le__imp__le,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ) ).
fof(fact_548_power2__eq__imp__eq,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ) ).
fof(fact_549_of__nat__number__of__lemma,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = number_number_of(X_a,V_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = zero_zero(X_a) ) ) ) ).
fof(fact_550_power2__sum,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [X,Y] : hAPP(nat,X_a,power_power(X_a,plus_plus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X),Y)) ) ).
fof(fact_551_power2__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).
fof(fact_552_sum__power2__ge__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))))) ) ).
fof(fact_553_sum__power2__le__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a)))
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_554_int__pos__lt__two__imp__zero__or__one,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),number_number_of(int,bit0(bit1(pls)))))
=> ( ti(int,X) = zero_zero(int)
| ti(int,X) = one_one(int) ) ) ) ).
fof(fact_555_Euler_Oaux__1,axiom,
! [A_1,P_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_2))
=> hAPP(nat,int,power_power(int,A_1),nat_1(P_2)) = times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),minus_minus(nat,nat_1(P_2),one_one(nat)))) ) ).
fof(fact_556_self__quotient__aux2,axiom,
! [R_1,Q,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( ti(int,A_1) = plus_plus(int,R_1,times_times(int,A_1,Q))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q),one_one(int))) ) ) ) ).
fof(fact_557_self__quotient__aux1,axiom,
! [R_1,Q,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( ti(int,A_1) = plus_plus(int,R_1,times_times(int,A_1,Q))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),A_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q)) ) ) ) ).
fof(fact_558_realpow__num__eq__if,axiom,
! [X_a] :
( power(X_a)
=> ! [M,N] :
( ( N = zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,M),N) = one_one(X_a) )
& ( N != zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,M),N) = times_times(X_a,M,hAPP(nat,X_a,power_power(X_a,M),minus_minus(nat,N,one_one(nat)))) ) ) ) ).
fof(fact_559_smaller_I2_J,axiom,
~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
=> ~ hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))))) ) ).
fof(fact_560_q__pos__lemma,axiom,
! [B_2,Q_1,R_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,times_times(int,B_2,Q_1),R_2)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_1)) ) ) ) ).
fof(fact_561_q__neg__lemma,axiom,
! [B_2,Q_1,R_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),zero_zero(int))) ) ) ) ).
fof(fact_562_nQ1,axiom,
~ hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)))))) ).
fof(fact_563_le0,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
fof(fact_564_p0,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_565_t__l__p,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),t),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_566_qf1pt,axiom,
hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t))) ).
fof(fact_567_IH,axiom,
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))))) ) ).
fof(fact_568__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A,axiom,
~ ! [X_1,Y_1] : plus_plus(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls))))) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))) ).
fof(fact_569_t,axiom,
plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t) ).
fof(fact_570_p,axiom,
hBOOL(hAPP(int,bool,zprime,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_571__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_,axiom,
( t = one_one(int)
=> ? [X_1,Y_1] : plus_plus(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) ) ).
fof(fact_572_le__refl,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).
fof(fact_573_le__square,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),times_times(nat,M,M))) ).
fof(fact_574_le__cube,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),times_times(nat,M,times_times(nat,M,M)))) ).
fof(fact_575_nat__mult__commute,axiom,
! [M,N] : times_times(nat,M,N) = times_times(nat,N,M) ).
fof(fact_576_nat__le__linear,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).
fof(fact_577_nat__mult__assoc,axiom,
! [M,N,K_1] : times_times(nat,times_times(nat,M,N),K_1) = times_times(nat,M,times_times(nat,N,K_1)) ).
fof(fact_578_eq__imp__le,axiom,
! [M,N] :
( M = N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_579_mult__le__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,I_2,K_1)),times_times(nat,J_1,K_1))) ) ).
fof(fact_580_mult__le__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,K_1,I_2)),times_times(nat,K_1,J_1))) ) ).
fof(fact_581_le__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K_1)) ) ) ).
fof(fact_582_le__antisym,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> M = N ) ) ).
fof(fact_583_mult__le__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,I_2,K_1)),times_times(nat,J_1,L))) ) ) ).
fof(fact_584_real__le__eq__diff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,Xa,Ya)),zero_zero(real))) ) ).
fof(fact_585_mult__le__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,Ma,K)),times_times(nat,Na,K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).
fof(fact_586_mult__le__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).
fof(fact_587_real__mult__le__cancel__iff1,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,Xa,Z_1)),times_times(real,Ya,Z_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ).
fof(fact_588_real__mult__le__cancel__iff2,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,Z_1,Xa)),times_times(real,Z_1,Ya)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ).
fof(fact_589_zdiff__int,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,M,N)) ) ).
fof(fact_590_diff__bin__simps_I1_J,axiom,
! [K_1] : minus_minus(int,K_1,pls) = ti(int,K_1) ).
fof(fact_591_diff__bin__simps_I7_J,axiom,
! [K_1,L] : minus_minus(int,bit0(K_1),bit0(L)) = bit0(minus_minus(int,K_1,L)) ).
fof(fact_592_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times(int,minus_minus(int,Z1,Z2),W) = minus_minus(int,times_times(int,Z1,W),times_times(int,Z2,W)) ).
fof(fact_593_zdiff__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times(int,W,minus_minus(int,Z1,Z2)) = minus_minus(int,times_times(int,W,Z1),times_times(int,W,Z2)) ).
fof(fact_594_mult__0,axiom,
! [N] : times_times(nat,zero_zero(nat),N) = zero_zero(nat) ).
fof(fact_595_mult__0__right,axiom,
! [M] : times_times(nat,M,zero_zero(nat)) = zero_zero(nat) ).
fof(fact_596_mult__is__0,axiom,
! [Ma,Na] :
( times_times(nat,Ma,Na) = zero_zero(nat)
<=> ( Ma = zero_zero(nat)
| Na = zero_zero(nat) ) ) ).
fof(fact_597_mult__cancel1,axiom,
! [K,Ma,Na] :
( times_times(nat,K,Ma) = times_times(nat,K,Na)
<=> ( Ma = Na
| K = zero_zero(nat) ) ) ).
fof(fact_598_mult__cancel2,axiom,
! [Ma,K,Na] :
( times_times(nat,Ma,K) = times_times(nat,Na,K)
<=> ( Ma = Na
| K = zero_zero(nat) ) ) ).
fof(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_2,Q] : hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,X),P_2)),Q) = hAPP(nat,X_a,power_power(X_a,X),times_times(nat,P_2,Q)) ) ).
fof(fact_600_power__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N] : hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,M,N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N) ) ).
fof(fact_601_less__eq__nat_Osimps_I1_J,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
fof(fact_602_le__0__eq,axiom,
! [Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat)))
<=> Na = zero_zero(nat) ) ).
fof(fact_603_real__mult__left__cancel,axiom,
! [A_2,B,C_1] :
( C_1 != zero_zero(real)
=> ( times_times(real,C_1,A_2) = times_times(real,C_1,B)
<=> A_2 = B ) ) ).
fof(fact_604_real__mult__right__cancel,axiom,
! [A_2,B,C_1] :
( C_1 != zero_zero(real)
=> ( times_times(real,A_2,C_1) = times_times(real,B,C_1)
<=> A_2 = B ) ) ).
fof(fact_605_less__or__eq__imp__le,axiom,
! [M,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| M = N )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_606_le__neq__implies__less,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( M != N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_607_less__imp__le__nat,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_608_le__eq__less__or__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
| Ma = Na ) ) ).
fof(fact_609_nat__less__le,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
& Ma != Na ) ) ).
fof(fact_610_real__less__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_611_less__eq__real__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_612_add__mult__distrib,axiom,
! [M,N,K_1] : times_times(nat,plus_plus(nat,M,N),K_1) = plus_plus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1)) ).
fof(fact_613_add__mult__distrib2,axiom,
! [K_1,M,N] : times_times(nat,K_1,plus_plus(nat,M,N)) = plus_plus(nat,times_times(nat,K_1,M),times_times(nat,K_1,N)) ).
fof(fact_614_add__leE,axiom,
! [M,K_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)) ) ) ).
fof(fact_615_add__leD1,axiom,
! [M,K_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_616_add__leD2,axiom,
! [M,K_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)) ) ).
fof(fact_617_add__le__mono,axiom,
! [K_1,L,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,L))) ) ) ).
fof(fact_618_add__le__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,K_1))) ) ).
fof(fact_619_trans__le__add2,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),plus_plus(nat,M,J_1))) ) ).
fof(fact_620_trans__le__add1,axiom,
! [M,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),plus_plus(nat,J_1,M))) ) ).
fof(fact_621_nat__add__left__cancel__le,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,K,Ma)),plus_plus(nat,K,Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).
fof(fact_622_le__iff__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
<=> ? [K_2] : Na = plus_plus(nat,Ma,K_2) ) ).
fof(fact_623_le__add1,axiom,
! [N,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),plus_plus(nat,N,M))) ).
fof(fact_624_le__add2,axiom,
! [N,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),plus_plus(nat,M,N))) ).
fof(fact_625_nat__mult__eq__one,axiom,
! [Na,Ma] :
( times_times(nat,Na,Ma) = one_one(nat)
<=> ( Na = one_one(nat)
& Ma = one_one(nat) ) ) ).
fof(fact_626_nat__mult__eq__1__iff,axiom,
! [Ma,Na] :
( times_times(nat,Ma,Na) = one_one(nat)
<=> ( Ma = one_one(nat)
& Na = one_one(nat) ) ) ).
fof(fact_627_nat__mult__1__right,axiom,
! [N] : times_times(nat,N,one_one(nat)) = N ).
fof(fact_628_nat__1__eq__mult__iff,axiom,
! [Ma,Na] :
( one_one(nat) = times_times(nat,Ma,Na)
<=> ( Ma = one_one(nat)
& Na = one_one(nat) ) ) ).
fof(fact_629_nat__mult__1,axiom,
! [N] : times_times(nat,one_one(nat),N) = N ).
fof(fact_630_diff__mult__distrib,axiom,
! [M,N,K_1] : times_times(nat,minus_minus(nat,M,N),K_1) = minus_minus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1)) ).
fof(fact_631_diff__mult__distrib2,axiom,
! [K_1,M,N] : times_times(nat,K_1,minus_minus(nat,M,N)) = minus_minus(nat,times_times(nat,K_1,M),times_times(nat,K_1,N)) ).
fof(fact_632_real__mult__1,axiom,
! [Z] : times_times(real,one_one(real),Z) = Z ).
fof(fact_633_le__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ) ).
fof(fact_634_Nat_Odiff__diff__eq,axiom,
! [N,K_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
=> minus_minus(nat,minus_minus(nat,M,K_1),minus_minus(nat,N,K_1)) = minus_minus(nat,M,N) ) ) ).
fof(fact_635_eq__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
=> ( minus_minus(nat,Ma,K) = minus_minus(nat,Na,K)
<=> Ma = Na ) ) ) ).
fof(fact_636_diff__diff__cancel,axiom,
! [I_2,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N))
=> minus_minus(nat,N,minus_minus(nat,N,I_2)) = I_2 ) ).
fof(fact_637_diff__le__mono,axiom,
! [L,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,M,L)),minus_minus(nat,N,L))) ) ).
fof(fact_638_diff__le__mono2,axiom,
! [L,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,L,N)),minus_minus(nat,L,M))) ) ).
fof(fact_639_Nat_Odiff__le__self,axiom,
! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,M,N)),M)) ).
fof(fact_640_real__add__mult__distrib,axiom,
! [Z1,Z2,W] : times_times(real,plus_plus(real,Z1,Z2),W) = plus_plus(real,times_times(real,Z1,W),times_times(real,Z2,W)) ).
fof(fact_641_zpower__zpower,axiom,
! [X,Y,Z] : hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),Y)),Z) = hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z)) ).
fof(fact_642_real__add__left__mono,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,Z,X)),plus_plus(real,Z,Y))) ) ).
fof(fact_643_diff__bin__simps_I10_J,axiom,
! [K_1,L] : minus_minus(int,bit1(K_1),bit1(L)) = bit0(minus_minus(int,K_1,L)) ).
fof(fact_644_diff__bin__simps_I9_J,axiom,
! [K_1,L] : minus_minus(int,bit1(K_1),bit0(L)) = bit1(minus_minus(int,K_1,L)) ).
fof(fact_645_diff__bin__simps_I3_J,axiom,
! [L] : minus_minus(int,pls,bit0(L)) = bit0(minus_minus(int,pls,L)) ).
fof(fact_646_Euler_Oaux1,axiom,
! [A_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),A_1))
=> ( ti(int,X) != minus_minus(int,A_1,one_one(int))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),minus_minus(int,A_1,one_one(int)))) ) ) ) ).
fof(fact_647_less__bin__lemma,axiom,
! [K,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,K,L_1)),zero_zero(int))) ) ).
fof(fact_648_mult__less__mono2,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K_1,I_2)),times_times(nat,K_1,J_1))) ) ) ).
fof(fact_649_mult__less__mono1,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,I_2,K_1)),times_times(nat,J_1,K_1))) ) ) ).
fof(fact_650_mult__less__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,Ma,K)),times_times(nat,Na,K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).
fof(fact_651_mult__less__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).
fof(fact_652_nat__0__less__mult__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),times_times(nat,Ma,Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).
fof(fact_653_not__real__square__gt__zero,axiom,
! [Xa] :
( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,Xa,Xa)))
<=> Xa = zero_zero(real) ) ).
fof(fact_654_real__mult__less__mono2,axiom,
! [X,Y,Z] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,Z,X)),times_times(real,Z,Y))) ) ) ).
fof(fact_655_real__mult__order,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,X,Y))) ) ) ).
fof(fact_656_real__mult__less__iff1,axiom,
! [Xa,Ya,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,Xa,Z_1)),times_times(real,Ya,Z_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ) ).
fof(fact_657_mult__eq__self__implies__10,axiom,
! [M,N] :
( M = times_times(nat,M,N)
=> ( N = one_one(nat)
| M = zero_zero(nat) ) ) ).
fof(fact_658_diff__is__0__eq,axiom,
! [Ma,Na] :
( minus_minus(nat,Ma,Na) = zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).
fof(fact_659_diff__is__0__eq_H,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> minus_minus(nat,M,N) = zero_zero(nat) ) ).
fof(fact_660_real__two__squares__add__zero__iff,axiom,
! [Xa,Ya] :
( plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya)) = zero_zero(real)
<=> ( Xa = zero_zero(real)
& Ya = zero_zero(real) ) ) ).
fof(fact_661_diff__less__mono,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,A_1,C)),minus_minus(nat,B_1,C))) ) ) ).
fof(fact_662_less__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ) ).
fof(fact_663_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : times_times(int,hAPP(nat,int,semiring_1_of_nat(int),X),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,X,Y)) ).
fof(fact_664_zmult__int,axiom,
! [M,N] : times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N)) ).
fof(fact_665_int__mult,axiom,
! [M,N] : hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N)) = times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) ).
fof(fact_666_diff__diff__right,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> minus_minus(nat,I_2,minus_minus(nat,J_1,K_1)) = minus_minus(nat,plus_plus(nat,I_2,K_1),J_1) ) ).
fof(fact_667_le__diff__conv,axiom,
! [J,K,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,J,K)),I_1))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),plus_plus(nat,I_1,K))) ) ).
fof(fact_668_le__add__diff,axiom,
! [M,K_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),minus_minus(nat,plus_plus(nat,N,M),K_1))) ) ).
fof(fact_669_le__add__diff__inverse,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> plus_plus(nat,N,minus_minus(nat,M,N)) = M ) ).
fof(fact_670_add__diff__assoc,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> plus_plus(nat,I_2,minus_minus(nat,J_1,K_1)) = minus_minus(nat,plus_plus(nat,I_2,J_1),K_1) ) ).
fof(fact_671_le__diff__conv2,axiom,
! [I_1,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),minus_minus(nat,J,K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_1,K)),J)) ) ) ).
fof(fact_672_le__add__diff__inverse2,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> plus_plus(nat,minus_minus(nat,M,N),N) = M ) ).
fof(fact_673_le__imp__diff__is__add,axiom,
! [K,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( minus_minus(nat,J,I_1) = K
<=> J = plus_plus(nat,K,I_1) ) ) ).
fof(fact_674_diff__add__assoc,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> minus_minus(nat,plus_plus(nat,I_2,J_1),K_1) = plus_plus(nat,I_2,minus_minus(nat,J_1,K_1)) ) ).
fof(fact_675_add__diff__assoc2,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> plus_plus(nat,minus_minus(nat,J_1,K_1),I_2) = minus_minus(nat,plus_plus(nat,J_1,I_2),K_1) ) ).
fof(fact_676_diff__add__assoc2,axiom,
! [I_2,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> minus_minus(nat,plus_plus(nat,J_1,I_2),K_1) = plus_plus(nat,minus_minus(nat,J_1,K_1),I_2) ) ).
fof(fact_677_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ).
fof(fact_678_zle__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).
fof(fact_679_is__mult__sum2sq,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X))
=> ( hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,Y))
=> hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,X,Y))) ) ) ).
fof(fact_680_zle__diff1__eq,axiom,
! [Wa,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),minus_minus(int,Z_1,one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) ) ).
fof(fact_681_le__nat__number__of,axiom,
! [Va,V_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,Va)),number_number_of(nat,V_3)))
<=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),V_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ) ).
fof(fact_682_of__nat__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),minus_minus(nat,M,N)) = minus_minus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ) ).
fof(fact_683_transfer__nat__int__relations_I3_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ) ) ).
fof(fact_684_nat__diff__distrib,axiom,
! [Z,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),Z))
=> nat_1(minus_minus(int,Z,Z_2)) = minus_minus(nat,nat_1(Z),nat_1(Z_2)) ) ) ).
fof(fact_685_nat__mult__distrib,axiom,
! [Z_2,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> nat_1(times_times(int,Z,Z_2)) = times_times(nat,nat_1(Z),nat_1(Z_2)) ) ).
fof(fact_686_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> times_times(nat,nat_1(X),nat_1(Y)) = nat_1(times_times(int,X,Y)) ) ) ).
fof(fact_687_mult__eq__if,axiom,
! [N,M] :
( ( M = zero_zero(nat)
=> times_times(nat,M,N) = zero_zero(nat) )
& ( M != zero_zero(nat)
=> times_times(nat,M,N) = plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N)) ) ) ).
fof(fact_688_nat__le__eq__zle,axiom,
! [Z_1,Wa] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Wa))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),nat_1(Wa)),nat_1(Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)) ) ) ).
fof(fact_689_power__eq__if,axiom,
! [P_2,M] :
( ( M = zero_zero(nat)
=> hAPP(nat,nat,power_power(nat,P_2),M) = one_one(nat) )
& ( M != zero_zero(nat)
=> hAPP(nat,nat,power_power(nat,P_2),M) = times_times(nat,P_2,hAPP(nat,nat,power_power(nat,P_2),minus_minus(nat,M,one_one(nat)))) ) ) ).
fof(fact_690_power__even__eq,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N] : hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N)),number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_691_nat__mult__2,axiom,
! [Z] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
fof(fact_692_nat__mult__2__right,axiom,
! [Z] : times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z) ).
fof(fact_693_nat__number__of__mult__left,axiom,
! [V_2,K_1,V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> times_times(nat,number_number_of(nat,V_1),times_times(nat,number_number_of(nat,V_2),K_1)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> times_times(nat,number_number_of(nat,V_1),times_times(nat,number_number_of(nat,V_2),K_1)) = times_times(nat,number_number_of(nat,times_times(int,V_1,V_2)),K_1) ) ) ).
fof(fact_694_mult__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,times_times(int,V_1,V_2)) ) ) ).
fof(fact_695_four__x__squared,axiom,
! [X] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,real,power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X)),number_number_of(nat,bit0(bit1(pls)))) ).
fof(fact_696_two__realpow__ge__one,axiom,
! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,power_power(real,number_number_of(real,bit0(bit1(pls)))),N))) ).
fof(fact_697_Euler_Oaux2,axiom,
! [B_1,A_1,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_1),A_1)) ) ) ) ).
fof(fact_698_zspecial__product,axiom,
! [A_1,B_1] : times_times(int,plus_plus(int,A_1,B_1),minus_minus(int,A_1,B_1)) = minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_699_even__power__le__0__imp__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,K_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,number_number_of(nat,bit0(bit1(pls))),K_1))),zero_zero(X_a)))
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_700_zero__le__even__power_H,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)))) ) ).
fof(fact_701_diff__square,axiom,
! [X,Y] : minus_minus(nat,hAPP(nat,nat,power_power(nat,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,nat,power_power(nat,Y),number_number_of(nat,bit0(bit1(pls))))) = times_times(nat,plus_plus(nat,X,Y),minus_minus(nat,X,Y)) ).
fof(fact_702_zdiff__power3,axiom,
! [A_1,B_1] : hAPP(nat,int,power_power(int,minus_minus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls)))) = minus_minus(int,plus_plus(int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),B_1)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit1(bit1(pls))))) ).
fof(fact_703_zdiff__power2,axiom,
! [A_1,B_1] : hAPP(nat,int,power_power(int,minus_minus(int,A_1,B_1)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B_1)),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_704_zdiv__mono2__neg__lemma,axiom,
! [B_1,Q,R_1,B_2,Q_1,R_2] :
( plus_plus(int,times_times(int,B_1,Q),R_1) = plus_plus(int,times_times(int,B_2,Q_1),R_2)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),B_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q)) ) ) ) ) ) ) ).
fof(fact_705_unique__quotient__lemma__neg,axiom,
! [B_1,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,B_1,Q_1),R_2)),plus_plus(int,times_times(int,B_1,Q),R_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),R_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),R_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q),Q_1)) ) ) ) ) ).
fof(fact_706_zdiv__mono2__lemma,axiom,
! [B_1,Q,R_1,B_2,Q_1,R_2] :
( plus_plus(int,times_times(int,B_1,Q),R_1) = plus_plus(int,times_times(int,B_2,Q_1),R_2)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,times_times(int,B_2,Q_1),R_2)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),B_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q),Q_1)) ) ) ) ) ) ) ).
fof(fact_707_unique__quotient__lemma,axiom,
! [B_1,Q_1,R_2,Q,R_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,B_1,Q_1),R_2)),plus_plus(int,times_times(int,B_1,Q),R_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q)) ) ) ) ) ).
fof(fact_708_xy,axiom,
plus_plus(int,hAPP(nat,int,power_power(int,x),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,y),number_number_of(nat,bit0(bit1(pls))))) = times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))) ).
fof(fact_709_Int2_Oaux__1,axiom,
! [P_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2))
=> minus_minus(nat,nat_1(P_2),number_number_of(nat,bit0(bit1(pls)))) = nat_1(minus_minus(int,P_2,number_number_of(int,bit0(bit1(pls))))) ) ).
fof(fact_710_Int2_Oaux__2,axiom,
! [P_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat_1(minus_minus(int,P_2,number_number_of(int,bit0(bit1(pls))))))) ) ).
fof(fact_711__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_,axiom,
~ ! [T_1] : plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),one_one(int)) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T_1) ).
fof(fact_712_real__le__refl,axiom,
! [W] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W)) ).
fof(fact_713_real__mult__commute,axiom,
! [Z,W] : times_times(real,Z,W) = times_times(real,W,Z) ).
fof(fact_714_real__le__linear,axiom,
! [Z,W] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W))
| hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z)) ) ).
fof(fact_715_real__mult__assoc,axiom,
! [Z1,Z2,Z3] : times_times(real,times_times(real,Z1,Z2),Z3) = times_times(real,Z1,times_times(real,Z2,Z3)) ).
fof(fact_716_real__le__trans,axiom,
! [K_1,I_2,J_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_2),J_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_1),K_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_2),K_1)) ) ) ).
fof(fact_717_real__le__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z))
=> Z = W ) ) ).
fof(fact_718_zprime__2,axiom,
hBOOL(hAPP(int,bool,zprime,number_number_of(int,bit0(bit1(pls))))) ).
fof(fact_719_Int2_Oaux1,axiom,
! [A_1,B_1,C] :
( minus_minus(int,A_1,B_1) = ti(int,C)
=> ti(int,A_1) = plus_plus(int,C,B_1) ) ).
fof(fact_720__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_L_A1_096,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),one_one(int)))) ).
fof(fact_721_real__sum__squared__expand,axiom,
! [X,Y] : hAPP(nat,real,power_power(real,plus_plus(real,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(real,plus_plus(real,hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),Y)) ).
fof(fact_722__096s_____A_094_A2_A_N_A_N1_A_061_As_____A_094_A2_A_L_A1_096,axiom,
minus_minus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)) = plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),one_one(int)) ).
fof(fact_723_dvd__0__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),zero_zero(X_a))) ) ).
fof(fact_724__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_N_A_N1_096,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),minus_minus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)))) ).
fof(fact_725__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) = one_one(int) ).
fof(fact_726_zdvd__zdiffD,axiom,
! [K_1,M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),minus_minus(int,M,N)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M)) ) ) ).
fof(fact_727_zprime__zdvd__zmult__better,axiom,
! [M,N,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),times_times(int,M,N)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),M))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)) ) ) ) ).
fof(fact_728_dvd__mult__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,B_1)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),C)) ) ) ).
fof(fact_729_dvd__mult__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,B_1)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) ) ) ).
fof(fact_730_dvdI,axiom,
! [X_a] :
( dvd(X_a)
=> ! [A_1,B_1,K_1] :
( A_1 = times_times(X_a,B_1,K_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),A_1)) ) ) ).
fof(fact_731_mult__dvd__mono,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,D_1,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1))) ) ) ) ).
fof(fact_732_dvd__mult,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [B_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,B_1,C))) ) ) ).
fof(fact_733_dvd__mult2,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,B_1,C))) ) ) ).
fof(fact_734_dvd__triv__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,B_1,A_1))) ) ).
fof(fact_735_dvd__triv__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,A_1,B_1))) ) ).
fof(fact_736_dvd__diff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),minus_minus(X_a,Y,Z))) ) ) ) ).
fof(fact_737_rel__simps_I24_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),min)) ).
fof(fact_738_zdvd__bounds,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),M)) ) ) ).
fof(fact_739_zprime__zdvd__power,axiom,
! [A_1,N,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,A_1),N)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),A_1)) ) ) ).
fof(fact_740_dvd__mult__cancel__left,axiom,
! [X_a] :
( idom(X_a)
=> ! [C_1,A_2,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))
<=> ( ti(X_a,C_1) = zero_zero(X_a)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)) ) ) ) ).
fof(fact_741_dvd__mult__cancel__right,axiom,
! [X_a] :
( idom(X_a)
=> ! [A_2,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1)))
<=> ( ti(X_a,C_1) = zero_zero(X_a)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)) ) ) ) ).
fof(fact_742_le__imp__power__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_743_dvd__power__le,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N,M,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X),N)),hAPP(nat,X_a,power_power(X_a,Y),M))) ) ) ) ).
fof(fact_744_power__le__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M,A_1,N,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),B_1)) ) ) ) ).
fof(fact_745_zdvd__not__zless,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),N))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).
fof(fact_746_zdvd__mult__cancel,axiom,
! [K_1,M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K_1,M)),times_times(int,K_1,N)))
=> ( ti(int,K_1) != zero_zero(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N)) ) ) ).
fof(fact_747_zdvd__antisym__nonneg,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))
=> ti(int,M) = ti(int,N) ) ) ) ) ).
fof(fact_748_zdvd__reduce,axiom,
! [K,Na,Ma] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),plus_plus(int,Na,times_times(int,K,Ma))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)) ) ).
fof(fact_749_zdvd__period,axiom,
! [C_1,Xa,Ta,A_2,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),D))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),plus_plus(int,Xa,Ta)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),plus_plus(int,plus_plus(int,Xa,times_times(int,C_1,D)),Ta))) ) ) ).
fof(fact_750_rel__simps_I9_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit1(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_751_rel__simps_I13_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) ) ).
fof(fact_752_rel__simps_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),pls)) ).
fof(fact_753_rel__simps_I3_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),min)) ).
fof(fact_754_rel__simps_I8_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit0(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_755_rel__simps_I26_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit1(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)) ) ).
fof(fact_756_rel__simps_I30_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_757_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),zero_zero(int))) ).
fof(fact_758_rel__simps_I23_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),pls)) ).
fof(fact_759_rel__simps_I20_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),min)) ).
fof(fact_760_rel__simps_I28_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_761_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
fof(fact_762_dvd__trans,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) ) ) ) ).
fof(fact_763_dvd__refl,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),A_1)) ) ).
fof(fact_764_rel__simps_I7_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),min)) ).
fof(fact_765_rel__simps_I42_J,axiom,
! [L] : min != bit0(L) ).
fof(fact_766_rel__simps_I45_J,axiom,
! [K_1] : bit0(K_1) != min ).
fof(fact_767_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_768_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_769_Bit1__Min,axiom,
bit1(min) = min ).
fof(fact_770_rel__simps_I43_J,axiom,
! [L_1] :
( min = bit1(L_1)
<=> min = ti(int,L_1) ) ).
fof(fact_771_rel__simps_I47_J,axiom,
! [K] :
( bit1(K) = min
<=> ti(int,K) = min ) ).
fof(fact_772_dvd__power__same,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X),N)),hAPP(nat,X_a,power_power(X_a,Y),N))) ) ) ).
fof(fact_773_one__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),one_one(X_a)),A_1)) ) ).
fof(fact_774_dvd__add,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),plus_plus(X_a,B_1,C))) ) ) ) ).
fof(fact_775_dvd__0__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1))
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_776_succ__Min,axiom,
succ(min) = pls ).
fof(fact_777_diff__bin__simps_I2_J,axiom,
! [K_1] : minus_minus(int,K_1,min) = succ(K_1) ).
fof(fact_778_zdvd__imp__le,axiom,
! [Z,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),N))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),N)) ) ) ).
fof(fact_779_zpower__zdvd__prop1,axiom,
! [P_2,Y,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,Y),N))) ) ) ).
fof(fact_780_rel__simps_I25_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_781_rel__simps_I11_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_782_diff__bin__simps_I4_J,axiom,
! [L] : minus_minus(int,pls,bit1(L)) = bit1(minus_minus(int,min,L)) ).
fof(fact_783_diff__bin__simps_I6_J,axiom,
! [L] : minus_minus(int,min,bit1(L)) = bit0(minus_minus(int,min,L)) ).
fof(fact_784_diff__bin__simps_I5_J,axiom,
! [L] : minus_minus(int,min,bit0(L)) = bit1(minus_minus(int,min,L)) ).
fof(fact_785_zmult__eq__1__iff,axiom,
! [Ma,Na] :
( times_times(int,Ma,Na) = one_one(int)
<=> ( ( ti(int,Ma) = one_one(int)
& ti(int,Na) = one_one(int) )
| ( ti(int,Ma) = number_number_of(int,min)
& ti(int,Na) = number_number_of(int,min) ) ) ) ).
fof(fact_786_pos__zmult__eq__1__iff__lemma,axiom,
! [M,N] :
( times_times(int,M,N) = one_one(int)
=> ( ti(int,M) = one_one(int)
| ti(int,M) = number_number_of(int,min) ) ) ).
fof(fact_787_zprime__power__zdvd__cancel__right,axiom,
! [N,A_1,B_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),times_times(int,A_1,B_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),A_1)) ) ) ) ).
fof(fact_788_zprime__power__zdvd__cancel__left,axiom,
! [N,B_1,A_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),times_times(int,A_1,B_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),B_1)) ) ) ) ).
fof(fact_789_dvd__power,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
| ti(X_a,X) = one_one(X_a) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(nat,X_a,power_power(X_a,X),N))) ) ) ).
fof(fact_790_zpower__zdvd__prop2,axiom,
! [Y,N,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,Y),N)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y)) ) ) ) ).
fof(fact_791_power__m1__even,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [N] : hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = one_one(X_a) ) ).
fof(fact_792__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
hBOOL(hAPP(int,bool,quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),number_number_of(int,min))) ).
fof(fact_793_s,axiom,
hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_794__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ hBOOL(hAPP(int,bool,quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),number_number_of(int,min)))
=> legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) != one_one(int) ) ).
fof(fact_795_s1,axiom,
hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,s1),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_796_s0p,axiom,
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),s))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),s),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& hBOOL(hAPP(int,bool,zcong(s1,s),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ) ).
fof(fact_797__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1] : ~ hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,S1),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_798_divides__mul__r,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,A_1,C)),times_times(nat,B_1,C))) ) ).
fof(fact_799_divides__mul__l,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,C,A_1)),times_times(nat,C,B_1))) ) ).
fof(fact_800_dvd__diff__nat,axiom,
! [N,K_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N))) ) ) ).
fof(fact_801_zcong__eq__zdvd__prop,axiom,
! [Xa,P_3] :
( hBOOL(hAPP(int,bool,zcong(Xa,zero_zero(int)),P_3))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),Xa)) ) ).
fof(fact_802_zcong__zero__equiv__div,axiom,
! [A_2,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,zero_zero(int)),Ma))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_2)) ) ).
fof(fact_803_nat__dvd__not__less,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).
fof(fact_804_divides__ge,axiom,
! [A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> ( B_1 = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B_1)) ) ) ).
fof(fact_805_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,K)),times_times(nat,Na,K)))
<=> ( K = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).
fof(fact_806_dvd__diffD,axiom,
! [K_1,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)) ) ) ) ).
fof(fact_807_dvd__diffD1,axiom,
! [K_1,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N)) ) ) ) ).
fof(fact_808_divides__rev,axiom,
! [A_1,N,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A_1),N)),hAPP(nat,nat,power_power(nat,B_1),N)))
=> ( N != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) ) ) ).
fof(fact_809_divides__exp2,axiom,
! [X,Y,N] :
( N != zero_zero(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X),N)),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_810_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) ) ).
fof(fact_811_zdvd__int,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) ) ).
fof(fact_812_divides__exp,axiom,
! [N,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X),N)),hAPP(nat,nat,power_power(nat,Y),N))) ) ).
fof(fact_813_nat__dvd__1__iff__1,axiom,
! [Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat)))
<=> Ma = one_one(nat) ) ).
fof(fact_814_dvd__reduce,axiom,
! [K,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),plus_plus(nat,Na,K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)) ) ).
fof(fact_815_divides__add__revr,axiom,
! [B_1,D_1,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),plus_plus(nat,A_1,B_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),B_1)) ) ) ).
fof(fact_816_zcong__id,axiom,
! [M] : hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M)) ).
fof(fact_817_zcong__shift,axiom,
! [C,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> hBOOL(hAPP(int,bool,zcong(plus_plus(int,A_1,C),plus_plus(int,B_1,C)),M)) ) ).
fof(fact_818_zcong__eq__trans,axiom,
! [D_1,C,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ( ti(int,B_1) = ti(int,C)
=> ( hBOOL(hAPP(int,bool,zcong(C,D_1),M))
=> hBOOL(hAPP(int,bool,zcong(A_1,D_1),M)) ) ) ) ).
fof(fact_819_zcong__zpower,axiom,
! [Z,X,Y,M] :
( hBOOL(hAPP(int,bool,zcong(X,Y),M))
=> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),Z),hAPP(nat,int,power_power(int,Y),Z)),M)) ) ).
fof(fact_820_zcong__less__eq,axiom,
! [M,Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,zcong(X,Y),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M))
=> ti(int,X) = ti(int,Y) ) ) ) ) ) ) ).
fof(fact_821_zcong__not__zero,axiom,
! [M,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
=> ~ hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),M)) ) ) ).
fof(fact_822_zcong__zmult__prop1,axiom,
! [C_1,D,A_2,B,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),Ma))
=> ( hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,A_2,D)),Ma))
<=> hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,B,D)),Ma)) ) ) ).
fof(fact_823_zcong__zmult__prop2,axiom,
! [C_1,D,A_2,B,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),Ma))
=> ( hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,D,A_2)),Ma))
<=> hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,D,B)),Ma)) ) ) ).
fof(fact_824_Int2_Ozcong__zero,axiom,
! [M,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
=> ( hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),M))
=> ti(int,X) = zero_zero(int) ) ) ) ).
fof(fact_825_zcong__zmult__prop3,axiom,
! [Y,X,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( ~ hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2))
=> ( ~ hBOOL(hAPP(int,bool,zcong(Y,zero_zero(int)),P_2))
=> ~ hBOOL(hAPP(int,bool,zcong(times_times(int,X,Y),zero_zero(int)),P_2)) ) ) ) ).
fof(fact_826_dvd__imp__le,axiom,
! [K_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)) ) ) ).
fof(fact_827_dvd__mult__cancel,axiom,
! [K_1,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K_1,M)),times_times(nat,K_1,N)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
fof(fact_828_zcong__zprime__prod__zero,axiom,
! [B_1,A_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,B_1),zero_zero(int)),P_2))
=> ( hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2))
| hBOOL(hAPP(int,bool,zcong(B_1,zero_zero(int)),P_2)) ) ) ) ) ).
fof(fact_829_zcong__zprime__prod__zero__contra,axiom,
! [B_1,A_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( ( ~ hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2))
& ~ hBOOL(hAPP(int,bool,zcong(B_1,zero_zero(int)),P_2)) )
=> ~ hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,B_1),zero_zero(int)),P_2)) ) ) ) ).
fof(fact_830_divides__div__not,axiom,
! [X,Q,N,R_1] :
( X = plus_plus(nat,times_times(nat,Q,N),R_1)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),N))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),X)) ) ) ) ).
fof(fact_831_dvd__mult__cancel1,axiom,
! [Na,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,Na)),Ma))
<=> Na = one_one(nat) ) ) ).
fof(fact_832_dvd__mult__cancel2,axiom,
! [Na,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Na,Ma)),Ma))
<=> Na = one_one(nat) ) ) ).
fof(fact_833_transfer__nat__int__relations_I4_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Xa)),nat_1(Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)) ) ) ) ).
fof(fact_834_power__dvd__imp__le,axiom,
! [I_2,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,I_2),M)),hAPP(nat,nat,power_power(nat,I_2),N)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_835_nat__dvd__iff,axiom,
! [Z_1,Ma] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Z_1)),Ma))
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> Ma = zero_zero(nat) ) ) ) ).
fof(fact_836_zcong__neg__1__impl__ne__1,axiom,
! [X,P_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2))
=> ( hBOOL(hAPP(int,bool,zcong(X,number_number_of(int,min)),P_2))
=> ~ hBOOL(hAPP(int,bool,zcong(X,one_one(int)),P_2)) ) ) ).
fof(fact_837_Euler_Oaux____1,axiom,
! [Y,X,P_2] :
( ~ hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2))
=> ( hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))),X),P_2))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y)) ) ) ).
fof(fact_838_divides__cases,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M))
=> ( M = zero_zero(nat)
| M = N
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)),M)) ) ) ).
fof(fact_839_Legendre__1mod4,axiom,
! [M] :
( hBOOL(hAPP(int,bool,zprime,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))))
=> legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))) = one_one(int) ) ).
fof(fact_840__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J,axiom,
hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,s1),number_number_of(nat,bit0(bit1(pls))))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ).
fof(fact_841__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& hBOOL(hAPP(int,bool,zcong(s1,X_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& ! [Y_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& hBOOL(hAPP(int,bool,zcong(s1,Y_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) )
=> ti(int,Y_1) = ti(int,X_1) ) ) ).
fof(fact_842_dvd_Oorder__refl,axiom,
! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ).
fof(fact_843__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
~ ! [S] :
~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),S),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))
& hBOOL(hAPP(int,bool,zcong(s1,S),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) ) ).
fof(fact_844_dvd_Oless__asym,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_845_dvd_Oless__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_846_dvd_Oless__asym_H,axiom,
! [A_1,B_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) ) ) ).
fof(fact_847_dvd_Oless__le__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_848_dvd_Oord__less__eq__trans,axiom,
! [C,A_1,B_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) )
=> ( B_1 = C
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) ) ) ) ).
fof(fact_849_dvd_Oless__imp__triv,axiom,
! [P_1,Xa,Ya] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) )
=> hBOOL(P_1) ) ) ).
fof(fact_850_dvd_Oless__imp__not__eq2,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> Y != X ) ).
fof(fact_851_dvd_Oless__imp__not__eq,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> X != Y ) ).
fof(fact_852_dvd_Oless__imp__not__less,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_853_dvd_Oless__imp__le,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).
fof(fact_854_dvd_Oless__not__sym,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_855_dvd_Oless__imp__neq,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> X != Y ) ).
fof(fact_856_dvd_Ole__less__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_857_dvd_Oord__eq__less__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B_1)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) ) ) ) ).
fof(fact_858_dvd_Oorder__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z)) ) ) ).
fof(fact_859_dvd_Oantisym,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
=> X = Y ) ) ).
fof(fact_860_dvd__antisym,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M))
=> M = N ) ) ).
fof(fact_861_dvd_Oord__le__eq__trans,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> ( B_1 = C
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) ) ) ).
fof(fact_862_dvd_Oord__eq__le__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),C))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) ) ) ).
fof(fact_863_dvd_Ole__neq__trans,axiom,
! [A_1,B_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> ( A_1 != B_1
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) ) ) ) ).
fof(fact_864_dvd_Ole__imp__less__or__eq,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
| X = Y ) ) ).
fof(fact_865_dvd_Oantisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_866_dvd_Oeq__refl,axiom,
! [X,Y] :
( X = Y
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).
fof(fact_867_dvd_Oneq__le__trans,axiom,
! [A_1,B_1] :
( A_1 != B_1
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) ) ) ) ).
fof(fact_868_dvd_Oless__le__not__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) ) ) ).
fof(fact_869_dvd_Oless__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_870_dvd_Ole__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
| Xa = Ya ) ) ).
fof(fact_871_dvd_Oeq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) ) ) ).
fof(fact_872_dvd_Oless__irrefl,axiom,
! [X] :
~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ) ).
fof(fact_873_divides__antisym,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
<=> Xa = Ya ) ).
fof(fact_874_neg__one__power__eq__mod__m,axiom,
! [J_1,K_1,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M))
=> ( hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1),hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1)),M))
=> hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1) = hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1) ) ) ).
fof(fact_875_one__not__neg__one__mod__m,axiom,
! [M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M))
=> ~ hBOOL(hAPP(int,bool,zcong(one_one(int),number_number_of(int,min)),M)) ) ).
fof(fact_876_neg__one__power,axiom,
! [N] :
( hAPP(nat,int,power_power(int,number_number_of(int,min)),N) = one_one(int)
| hAPP(nat,int,power_power(int,number_number_of(int,min)),N) = number_number_of(int,min) ) ).
fof(fact_877_Legendre__def,axiom,
! [A_1,P_2] :
( ( hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2))
=> legendre(A_1,P_2) = zero_zero(int) )
& ( ~ hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2))
=> ( ( hBOOL(hAPP(int,bool,quadRes(P_2),A_1))
=> legendre(A_1,P_2) = one_one(int) )
& ( ~ hBOOL(hAPP(int,bool,quadRes(P_2),A_1))
=> legendre(A_1,P_2) = number_number_of(int,min) ) ) ) ) ).
fof(fact_878_QuadRes__def,axiom,
! [Ma,Xa] :
( hBOOL(hAPP(int,bool,quadRes(Ma),Xa))
<=> ? [Y_1] : hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls)))),Xa),Ma)) ) ).
fof(fact_879_Little__Fermat,axiom,
! [X,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),X))
=> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),nat_1(minus_minus(int,P_2,one_one(int)))),one_one(int)),P_2)) ) ) ).
fof(fact_880_zcong__square__zless,axiom,
! [A_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_2))
=> ( hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,A_1),one_one(int)),P_2))
=> ( ti(int,A_1) = one_one(int)
| ti(int,A_1) = minus_minus(int,P_2,one_one(int)) ) ) ) ) ) ).
fof(fact_881_zcong__trans,axiom,
! [C,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(B_1,C),M))
=> hBOOL(hAPP(int,bool,zcong(A_1,C),M)) ) ) ).
fof(fact_882_zcong__refl,axiom,
! [K_1,M] : hBOOL(hAPP(int,bool,zcong(K_1,K_1),M)) ).
fof(fact_883_zcong__sym,axiom,
! [A_2,B,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),Ma))
<=> hBOOL(hAPP(int,bool,zcong(B,A_2),Ma)) ) ).
fof(fact_884_order__le__neq__implies__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).
fof(fact_885_IntPrimes_Ozcong__zero,axiom,
! [A_2,B] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),zero_zero(int)))
<=> ti(int,A_2) = ti(int,B) ) ).
fof(fact_886_zcong__1,axiom,
! [A_1,B_1] : hBOOL(hAPP(int,bool,zcong(A_1,B_1),one_one(int))) ).
fof(fact_887_zcong__zmult__self,axiom,
! [A_1,M,B_1] : hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,M),times_times(int,B_1,M)),M)) ).
fof(fact_888_zcong__scalar,axiom,
! [K_1,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,K_1),times_times(int,B_1,K_1)),M)) ) ).
fof(fact_889_zcong__scalar2,axiom,
! [K_1,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> hBOOL(hAPP(int,bool,zcong(times_times(int,K_1,A_1),times_times(int,K_1,B_1)),M)) ) ).
fof(fact_890_zcong__zmult,axiom,
! [C,D_1,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(C,D_1),M))
=> hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,C),times_times(int,B_1,D_1)),M)) ) ) ).
fof(fact_891_zcong__zadd,axiom,
! [C,D_1,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(C,D_1),M))
=> hBOOL(hAPP(int,bool,zcong(plus_plus(int,A_1,C),plus_plus(int,B_1,D_1)),M)) ) ) ).
fof(fact_892_zcong__zdiff,axiom,
! [C,D_1,A_1,B_1,M] :
( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(C,D_1),M))
=> hBOOL(hAPP(int,bool,zcong(minus_minus(int,A_1,C),minus_minus(int,B_1,D_1)),M)) ) ) ).
fof(fact_893_zcong__not,axiom,
! [B_1,M,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),A_1))
=> ~ hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) ) ) ) ) ).
fof(fact_894_xzgcda__linear__aux1,axiom,
! [A_1,R_1,B_1,M,C,D_1,N] : plus_plus(int,times_times(int,minus_minus(int,A_1,times_times(int,R_1,B_1)),M),times_times(int,minus_minus(int,C,times_times(int,R_1,D_1)),N)) = minus_minus(int,plus_plus(int,times_times(int,A_1,M),times_times(int,C,N)),times_times(int,R_1,plus_plus(int,times_times(int,B_1,M),times_times(int,D_1,N)))) ).
fof(fact_895_zcong__iff__lin,axiom,
! [A_2,B,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),Ma))
<=> ? [K_2] : ti(int,B) = plus_plus(int,A_2,times_times(int,Ma,K_2)) ) ).
fof(fact_896_zcong__def,axiom,
! [A_2,B,Ma] :
( hBOOL(hAPP(int,bool,zcong(A_2,B),Ma))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),minus_minus(int,A_2,B))) ) ).
fof(fact_897_norR__mem__unique__aux,axiom,
! [A_1,B_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),minus_minus(int,B_1,one_one(int))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B_1)) ) ).
fof(fact_898_zcong__zless__0,axiom,
! [M,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),M))
=> ti(int,A_1) = zero_zero(int) ) ) ) ).
fof(fact_899_zcong__zless__imp__eq,axiom,
! [B_1,M,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),M))
=> ( hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))
=> ti(int,A_1) = ti(int,B_1) ) ) ) ) ) ).
fof(fact_900_zprime__zdvd__zmult,axiom,
! [N,P_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),times_times(int,M,N)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),M))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)) ) ) ) ) ).
fof(fact_901_zprime__def,axiom,
! [P_3] :
( hBOOL(hAPP(int,bool,zprime,P_3))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_3))
& ! [M_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),P_3)) )
=> ( ti(int,M_1) = one_one(int)
| ti(int,M_1) = ti(int,P_3) ) ) ) ) ).
fof(fact_902_zcong__square,axiom,
! [A_1,P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,A_1),one_one(int)),P_2))
=> ( hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_2))
| hBOOL(hAPP(int,bool,zcong(A_1,minus_minus(int,P_2,one_one(int))),P_2)) ) ) ) ) ).
fof(fact_903_zdiff__int__split,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,Xa,Ya))))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ya),Xa))
=> hBOOL(hAPP(int,bool,P_1,minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),Xa),hAPP(nat,int,semiring_1_of_nat(int),Ya)))) )
& ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya))
=> hBOOL(hAPP(int,bool,P_1,zero_zero(int))) ) ) ) ).
fof(fact_904_prime__g__5,axiom,
! [P_2] :
( hBOOL(hAPP(int,bool,zprime,P_2))
=> ( ti(int,P_2) != number_number_of(int,bit0(bit1(pls)))
=> ( ti(int,P_2) != number_number_of(int,bit1(bit1(pls)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_2)) ) ) ) ).
fof(fact_905_conj__le__cong,axiom,
! [P_4,P_1,Xa] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
& hBOOL(P_4) ) ) ) ).
fof(fact_906_imp__le__cong,axiom,
! [P_4,P_1,Xa] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_4) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> hBOOL(P_4) ) ) ) ).
fof(fact_907_zdvd__mono,axiom,
! [Ma,Ta,K] :
( ti(int,K) != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K,Ma)),times_times(int,K,Ta))) ) ) ).
fof(fact_908_unity__coeff__ex,axiom,
! [X_a] :
( ( dvd(X_a)
& semiring_0(X_a) )
=> ! [P_1,L_1] :
( ? [X_1] : hBOOL(hAPP(X_a,bool,P_1,times_times(X_a,L_1,X_1)))
<=> ? [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_1),plus_plus(X_a,X_1,zero_zero(X_a))))
& hBOOL(hAPP(X_a,bool,P_1,X_1)) ) ) ) ).
fof(fact_909_inf__period_I3_J,axiom,
! [X_a] :
( ( comm_ring(X_a)
& dvd(X_a) )
=> ! [Ta,D,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_2))
=> ! [X_1,K_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,X_1,Ta)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,minus_minus(X_a,X_1,times_times(X_a,K_2,D_2)),Ta))) ) ) ) ).
fof(fact_910_inf__period_I4_J,axiom,
! [X_a] :
( ( comm_ring(X_a)
& dvd(X_a) )
=> ! [Ta,D,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_2))
=> ! [X_1,K_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,X_1,Ta)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,minus_minus(X_a,X_1,times_times(X_a,K_2,D_2)),Ta))) ) ) ) ).
fof(fact_911_number__of2,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,pls))) ).
fof(fact_912_inv__not__p__minus__1__aux,axiom,
! [A_2,P_3] :
( hBOOL(hAPP(int,bool,zcong(times_times(int,A_2,minus_minus(int,P_3,one_one(int))),one_one(int)),P_3))
<=> hBOOL(hAPP(int,bool,zcong(A_2,minus_minus(int,P_3,one_one(int))),P_3)) ) ).
fof(fact_913_zcong__zpower__zmult,axiom,
! [Z,X,Y,P_2] :
( hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),Y),one_one(int)),P_2))
=> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z)),one_one(int)),P_2)) ) ).
fof(fact_914_number__of1,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,N)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(N))))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit1(N)))) ) ) ).
fof(fact_915__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An_,axiom,
~ ! [S,W_1] :
~ ( ti(int,W_1) = minus_minus(int,y,times_times(int,S,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,W_1))),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).
fof(fact_916__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An_,axiom,
~ ! [R,V] :
~ ( ti(int,V) = minus_minus(int,x,times_times(int,R,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,V))),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).
fof(fact_917_dvd__imp__le__int,axiom,
! [D_1,I_2] :
( ti(int,I_2) != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),I_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,D_1)),abs_abs(int,I_2))) ) ) ).
fof(fact_918_nat__abs__mult__distrib,axiom,
! [W,Z] : nat_1(abs_abs(int,times_times(int,W,Z))) = times_times(nat,nat_1(abs_abs(int,W)),nat_1(abs_abs(int,Z))) ).
fof(fact_919_dvd__if__abs__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [L,K_1] :
( abs_abs(X_a,L) = abs_abs(X_a,K_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),K_1)) ) ) ).
fof(fact_920_abs__dvd__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Ma,K] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),abs_abs(X_a,Ma)),K))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) ) ) ).
fof(fact_921_dvd__abs__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Ma,K] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),abs_abs(X_a,K)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) ) ) ).
fof(fact_922_zdvd__antisym__abs,axiom,
! [A_1,B_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B_1),A_1))
=> abs_abs(int,A_1) = abs_abs(int,B_1) ) ) ).
fof(fact_923_zdvd1__eq,axiom,
! [Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),one_one(int)))
<=> abs_abs(int,Xa) = one_one(int) ) ).
fof(fact_924_abs__mult__pos,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> times_times(X_a,abs_abs(X_a,Y),X) = abs_abs(X_a,times_times(X_a,Y,X)) ) ) ).
fof(fact_925_abs__eq__mult,axiom,
! [X_a] :
( ordered_ring_abs(X_a)
=> ! [B_1,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a))) ) )
=> abs_abs(X_a,times_times(X_a,A_1,B_1)) = times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)) ) ) ).
fof(fact_926_zero__le__power__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,abs_abs(X_a,A_1)),N))) ) ).
fof(fact_927_abs__diff__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1,C,D_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,plus_plus(X_a,A_1,B_1),plus_plus(X_a,C,D_1)))),plus_plus(X_a,abs_abs(X_a,minus_minus(X_a,A_1,C)),abs_abs(X_a,minus_minus(X_a,B_1,D_1))))) ) ).
fof(fact_928_abs__triangle__ineq4,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,A_1,B_1))),plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))) ) ).
fof(fact_929_abs__minus__one,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> abs_abs(X_a,number_number_of(X_a,min)) = one_one(X_a) ) ).
fof(fact_930_abs__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> abs_abs(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_931_abs__eq__0,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( abs_abs(X_a,A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_932_abs__add__abs,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : abs_abs(X_a,plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))) = plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)) ) ).
fof(fact_933_abs__one,axiom,
! [X_a] :
( linordered_idom(X_a)
=> abs_abs(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(fact_934_power__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N] : abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N)) = hAPP(nat,X_a,power_power(X_a,abs_abs(X_a,A_1)),N) ) ).
fof(fact_935_abs__of__nat,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [N] : abs_abs(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).
fof(fact_936_abs__int__eq,axiom,
! [M] : abs_abs(int,hAPP(nat,int,semiring_1_of_nat(int),M)) = hAPP(nat,int,semiring_1_of_nat(int),M) ).
fof(fact_937_abs__idempotent,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : abs_abs(X_a,abs_abs(X_a,A_1)) = abs_abs(X_a,A_1) ) ).
fof(fact_938_abs__zmult__eq__1,axiom,
! [M,N] :
( abs_abs(int,times_times(int,M,N)) = one_one(int)
=> abs_abs(int,M) = one_one(int) ) ).
fof(fact_939_abs__triangle__ineq2__sym,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),abs_abs(X_a,minus_minus(X_a,B_1,A_1)))) ) ).
fof(fact_940_abs__triangle__ineq2,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),abs_abs(X_a,minus_minus(X_a,A_1,B_1)))) ) ).
fof(fact_941_abs__triangle__ineq3,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))),abs_abs(X_a,minus_minus(X_a,A_1,B_1)))) ) ).
fof(fact_942_abs__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,plus_plus(X_a,A_1,B_1))),plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))) ) ).
fof(fact_943_abs__mult__less,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [B_1,D_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,A_1)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,B_1)),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),times_times(X_a,C,D_1))) ) ) ) ).
fof(fact_944_abs__of__pos,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> abs_abs(X_a,A_1) = ti(X_a,A_1) ) ) ).
fof(fact_945_zero__less__abs__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),abs_abs(X_a,A_2)))
<=> ti(X_a,A_2) != zero_zero(X_a) ) ) ).
fof(fact_946_abs__not__less__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,A_1)),zero_zero(X_a))) ) ).
fof(fact_947_abs__of__nonneg,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> abs_abs(X_a,A_1) = ti(X_a,A_1) ) ) ).
fof(fact_948_abs__le__zero__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,A_2)),zero_zero(X_a)))
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_949_abs__ge__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),abs_abs(X_a,A_1))) ) ).
fof(fact_950_abs__minus__commute,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] : abs_abs(X_a,minus_minus(X_a,A_1,B_1)) = abs_abs(X_a,minus_minus(X_a,B_1,A_1)) ) ).
fof(fact_951_abs__mult,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,B_1] : abs_abs(X_a,times_times(X_a,A_1,B_1)) = times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)) ) ).
fof(fact_952_abs__mult__self,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,A_1)) = times_times(X_a,A_1,A_1) ) ).
fof(fact_953_abs__ge__self,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),abs_abs(X_a,A_1))) ) ).
fof(fact_954_abs__le__D1,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,A_1)),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) ) ) ).
fof(fact_955_zero__le__zpower__abs,axiom,
! [X,N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,abs_abs(int,X)),N))) ).
fof(fact_956_zabs__less__one__iff,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,Z_1)),one_one(int)))
<=> ti(int,Z_1) = zero_zero(int) ) ).
fof(fact_957_abs__eq__1__iff,axiom,
! [Z_1] :
( abs_abs(int,Z_1) = one_one(int)
<=> ( ti(int,Z_1) = one_one(int)
| ti(int,Z_1) = number_number_of(int,min) ) ) ).
fof(fact_958_abs__power__minus__one,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [N] : abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),N)) = one_one(X_a) ) ).
fof(fact_959_abs__power3__distrib,axiom,
! [X] : abs_abs(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit1(bit1(pls))))) = hAPP(nat,int,power_power(int,abs_abs(int,X)),number_number_of(nat,bit1(bit1(pls)))) ).
fof(fact_960_zero__less__zpower__abs__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,abs_abs(int,Xa)),Na)))
<=> ( ti(int,Xa) != zero_zero(int)
| Na = zero_zero(nat) ) ) ).
fof(fact_961_zdvd__mult__cancel1,axiom,
! [Na,Ma] :
( ti(int,Ma) != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,Ma,Na)),Ma))
<=> abs_abs(int,Na) = one_one(int) ) ) ).
fof(fact_962_int__dvd__iff,axiom,
! [Ma,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z_1))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),nat_1(abs_abs(int,Z_1)))) ) ).
fof(fact_963_dvd__int__iff,axiom,
! [Z_1,Ma] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(abs_abs(int,Z_1))),Ma)) ) ).
fof(fact_964_abs__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_965_power2__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,abs_abs(X_a,A_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_966_abs__power2__distrib,axiom,
! [A_1] : abs_abs(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,abs_abs(int,A_1)),number_number_of(nat,bit0(bit1(pls)))) ).
fof(fact_967_power2__eq__iff__abs__eq,axiom,
! [A_2,B] :
( hAPP(nat,int,power_power(int,A_2),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))
<=> abs_abs(int,A_2) = abs_abs(int,B) ) ).
fof(fact_968_power2__eq1__iff,axiom,
! [A_1] :
( hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))) = one_one(int)
=> abs_abs(int,A_1) = one_one(int) ) ).
fof(fact_969_incr__lemma,axiom,
! [Z,X,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),plus_plus(int,X,times_times(int,plus_plus(int,abs_abs(int,minus_minus(int,X,Z)),one_one(int)),D_1)))) ) ).
fof(fact_970_decr__lemma,axiom,
! [X,Z,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,X,times_times(int,plus_plus(int,abs_abs(int,minus_minus(int,X,Z)),one_one(int)),D_1))),Z)) ) ).
fof(fact_971_best__division__abs,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ? [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,minus_minus(int,Y,times_times(int,N_1,X))))),X)) ) ).
fof(fact_972_decr__mult__lemma,axiom,
! [K,P_1,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> ( ! [X_1] :
( hBOOL(hAPP(int,bool,P_1,X_1))
=> hBOOL(hAPP(int,bool,P_1,minus_minus(int,X_1,D))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_1] :
( hBOOL(hAPP(int,bool,P_1,X_1))
=> hBOOL(hAPP(int,bool,P_1,minus_minus(int,X_1,times_times(int,K,D)))) ) ) ) ) ).
fof(fact_973_abs__add__one__not__less__self,axiom,
! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,abs_abs(real,X),one_one(real))),X)) ).
fof(fact_974_abs__add__one__gt__zero,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,one_one(real),abs_abs(real,X)))) ).
fof(fact_975_less__one__imp__sqr__less__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,X)),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls))))),one_one(real))) ) ).
fof(fact_976_ex__least__nat__less,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,P_1,Na))
=> ? [K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),Na))
& ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2))
=> ~ hBOOL(hAPP(nat,bool,P_1,I)) )
& hBOOL(hAPP(nat,bool,P_1,plus_plus(nat,K_2,one_one(nat)))) ) ) ) ).
fof(fact_977_incr__mult__lemma,axiom,
! [K,P_1,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> ( ! [X_1] :
( hBOOL(hAPP(int,bool,P_1,X_1))
=> hBOOL(hAPP(int,bool,P_1,plus_plus(int,X_1,D))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_1] :
( hBOOL(hAPP(int,bool,P_1,X_1))
=> hBOOL(hAPP(int,bool,P_1,plus_plus(int,X_1,times_times(int,K,D)))) ) ) ) ) ).
fof(fact_978_nat__less__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)),Na)) ) ) ).
fof(fact_979_nat__less__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na))) ) ) ).
fof(fact_980_nat__mult__eq__cancel__disj,axiom,
! [K,Ma,Na] :
( times_times(nat,K,Ma) = times_times(nat,K,Na)
<=> ( K = zero_zero(nat)
| Ma = Na ) ) ).
fof(fact_981_left__add__mult__distrib,axiom,
! [I_2,U_1,J_1,K_1] : plus_plus(nat,times_times(nat,I_2,U_1),plus_plus(nat,times_times(nat,J_1,U_1),K_1)) = plus_plus(nat,times_times(nat,plus_plus(nat,I_2,J_1),U_1),K_1) ).
fof(fact_982_nat__mult__less__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).
fof(fact_983_nat__mult__eq__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( times_times(nat,K,Ma) = times_times(nat,K,Na)
<=> Ma = Na ) ) ).
fof(fact_984_nat__mult__dvd__cancel__disj,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> ( K = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).
fof(fact_985_nat__mult__dvd__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).
fof(fact_986_nat__mult__le__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).
fof(fact_987_nat__le__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)),Na)) ) ) ).
fof(fact_988_nat__diff__add__eq1,axiom,
! [U_1,M,N,J_1,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_2))
=> minus_minus(nat,plus_plus(nat,times_times(nat,I_2,U_1),M),plus_plus(nat,times_times(nat,J_1,U_1),N)) = minus_minus(nat,plus_plus(nat,times_times(nat,minus_minus(nat,I_2,J_1),U_1),M),N) ) ).
fof(fact_989_nat__eq__add__iff1,axiom,
! [U,Ma,Na,J,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1))
=> ( plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na)
<=> plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma) = Na ) ) ).
fof(fact_990_nat__le__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na))) ) ) ).
fof(fact_991_nat__diff__add__eq2,axiom,
! [U_1,M,N,I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1))
=> minus_minus(nat,plus_plus(nat,times_times(nat,I_2,U_1),M),plus_plus(nat,times_times(nat,J_1,U_1),N)) = minus_minus(nat,M,plus_plus(nat,times_times(nat,minus_minus(nat,J_1,I_2),U_1),N)) ) ).
fof(fact_992_nat__eq__add__iff2,axiom,
! [U,Ma,Na,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na)
<=> Ma = plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na) ) ) ).
fof(fact_993_nat0__intermed__int__val,axiom,
! [K,F,Na] :
( ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Na))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,hAPP(nat,int,F,plus_plus(nat,I,one_one(nat))),hAPP(nat,int,F,I)))),one_one(int))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na)))
=> ? [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na))
& hAPP(nat,int,F,I) = ti(int,K) ) ) ) ) ).
fof(fact_994_int__val__lemma,axiom,
! [K,F,Na] :
( ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Na))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,hAPP(nat,int,F,plus_plus(nat,I,one_one(nat))),hAPP(nat,int,F,I)))),one_one(int))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na)))
=> ? [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na))
& hAPP(nat,int,F,I) = ti(int,K) ) ) ) ) ).
%----Arities (154)
fof(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(int) ).
fof(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
fof(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(int) ).
fof(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
fof(arity_Int_Oint___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
fof(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
fof(arity_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(int) ).
fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
fof(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
fof(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
fof(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
fof(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
fof(arity_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
fof(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
fof(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
fof(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
fof(arity_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
fof(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
fof(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
fof(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
fof(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
fof(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
fof(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
fof(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
fof(arity_Int_Oint___Rings_Oidom,axiom,
idom(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
fof(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
fof(arity_Int_Oint___Rings_Odvd,axiom,
dvd(int) ).
fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring,axiom,
linordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring(nat) ).
fof(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(nat) ).
fof(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
fof(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
fof(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
fof(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
fof(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
fof(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
fof(arity_Nat_Onat___Rings_Odvd,axiom,
dvd(nat) ).
fof(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
fof(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp,axiom,
semiri456707255roduct(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(real) ).
fof(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring,axiom,
linordered_semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__semiring,axiom,
ordered_semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(real) ).
fof(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring,axiom,
comm_semiring(real) ).
fof(arity_RealDef_Oreal___Nat_Osemiring__char__0,axiom,
semiring_char_0(real) ).
fof(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__group__add,axiom,
ab_group_add(real) ).
fof(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__ring,axiom,
ordered_ring(real) ).
fof(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
fof(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
monoid_mult(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__ring__1,axiom,
comm_ring_1(real) ).
fof(arity_RealDef_Oreal___Groups_Omonoid__add,axiom,
monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
semiring_0(real) ).
fof(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
group_add(real) ).
fof(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__ring,axiom,
comm_ring(real) ).
fof(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
fof(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
fof(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring,axiom,
semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__1,axiom,
ring_1(real) ).
fof(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
fof(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
fof(arity_RealDef_Oreal___Rings_Oring,axiom,
ring(real) ).
fof(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
fof(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
fof(arity_RealDef_Oreal___Groups_Oone,axiom,
one(real) ).
fof(arity_RealDef_Oreal___Rings_Odvd,axiom,
dvd(real) ).
%----Helper facts (4)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
fof(help_If_1_1_T,axiom,
! [X_a,X,Y] : if(X_a,fTrue,X,Y) = ti(X_a,X) ).
fof(help_If_2_1_T,axiom,
! [X_a,X,Y] : if(X_a,fFalse,X,Y) = ti(X_a,Y) ).
fof(help_If_3_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int) ).
%------------------------------------------------------------------------------