TPTP Problem File: NUM924+6.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM924+6 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_500_fofpt_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.31 v8.2.0, 0.25 v8.1.0, 0.28 v7.5.0, 0.34 v7.4.0, 0.14 v7.3.0, 0.00 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.25 v6.2.0, 0.28 v6.1.0, 0.43 v6.0.0, 0.35 v5.5.0, 0.48 v5.4.0, 0.54 v5.3.0
% Syntax : Number of formulae : 650 ( 224 unt; 0 def)
% Number of atoms : 1447 ( 420 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 890 ( 93 ~; 34 |; 73 &)
% ( 135 <=>; 555 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 41 ( 40 usr; 0 prp; 1-3 aty)
% Number of functors : 22 ( 22 usr; 10 con; 0-3 aty)
% Number of variables : 1280 (1276 !; 4 ?)
% SPC : FOF_CAX_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:11:40
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (65)
fof(tsy_c_Groups_Ominus__class_Ominus_0_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( ab_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] :
( ab_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] :
( ab_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_0_res,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> ti(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(tsy_c_Groups_Oone__class_Oone_1_res,axiom,
! [X_a] :
( power(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] :
( semiring(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] :
( semiring(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] :
( semiring(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_group_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_group_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_group_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] :
( 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_1_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_1_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_2_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_2_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_2_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_3_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_3_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_3_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_4_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_4_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_4_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_Ozero__class_Ozero_0_res,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_1_res,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_2_res,axiom,
! [X_a] :
( mult_zero(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_3_res,axiom,
! [X_a] :
( linord219039673up_add(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_IntPrimes_Ozcong_arg1,axiom,
! [B_1_1,B_2_1,B_3] :
( zcong(ti(int,B_1_1),B_2_1,B_3)
<=> zcong(B_1_1,B_2_1,B_3) ) ).
fof(tsy_c_IntPrimes_Ozcong_arg2,axiom,
! [B_1_1,B_2_1,B_3] :
( zcong(B_1_1,ti(int,B_2_1),B_3)
<=> zcong(B_1_1,B_2_1,B_3) ) ).
fof(tsy_c_IntPrimes_Ozcong_arg3,axiom,
! [B_1_1,B_2_1,B_3] :
( zcong(B_1_1,B_2_1,ti(int,B_3))
<=> zcong(B_1_1,B_2_1,B_3) ) ).
fof(tsy_c_IntPrimes_Ozprime_arg1,axiom,
! [B_1_1] :
( zprime(ti(int,B_1_1))
<=> zprime(B_1_1) ) ).
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_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_Orderings_Oord__class_Oless_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ( ord_less(X_a,ti(X_a,B_1_1),B_2_1)
<=> ord_less(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ( ord_less(X_a,B_1_1,ti(X_a,B_2_1))
<=> ord_less(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ( ord_less_eq(X_a,ti(X_a,B_1_1),B_2_1)
<=> ord_less_eq(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( order(X_a)
=> ( ord_less_eq(X_a,B_1_1,ti(X_a,B_2_1))
<=> ord_less_eq(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> power_power(X_a,ti(X_a,B_1_1),B_2_1) = power_power(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> power_power(X_a,B_1_1,ti(nat,B_2_1)) = power_power(X_a,B_1_1,B_2_1) ) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [B_1_1,B_2_1,X_a] :
( power(X_a)
=> ti(X_a,power_power(X_a,B_1_1,B_2_1)) = power_power(X_a,B_1_1,B_2_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,B_2_1] :
( quadRes(ti(int,B_1_1),B_2_1)
<=> quadRes(B_1_1,B_2_1) ) ).
fof(tsy_c_Residues_OQuadRes_arg2,axiom,
! [B_1_1,B_2_1] :
( quadRes(B_1_1,ti(int,B_2_1))
<=> quadRes(B_1_1,B_2_1) ) ).
fof(tsy_c_Rings_Odvd__class_Odvd_arg1,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> ( dvd_dvd(X_a,ti(X_a,B_1_1),B_2_1)
<=> dvd_dvd(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_Rings_Odvd__class_Odvd_arg2,axiom,
! [B_1_1,B_2_1,X_a] :
( dvd(X_a)
=> ( dvd_dvd(X_a,B_1_1,ti(X_a,B_2_1))
<=> dvd_dvd(X_a,B_1_1,B_2_1) ) ) ).
fof(tsy_c_TwoSquares__Mirabelle__iprzenvtji_Ois__sum2sq_arg1,axiom,
! [B_1_1] :
( twoSqu420862341sum2sq(ti(int,B_1_1))
<=> twoSqu420862341sum2sq(B_1_1) ) ).
fof(tsy_v_m_res,axiom,
ti(int,m) = m ).
fof(tsy_v_s1_____res,axiom,
ti(int,s1) = s1 ).
fof(tsy_v_s_____res,hypothesis,
ti(int,s) = s ).
fof(tsy_v_t_____res,axiom,
ti(int,t) = t ).
%----Relevant facts (497)
fof(fact_0__096t_A_060_A0_096,axiom,
ord_less(int,t,zero_zero(int)) ).
fof(fact_1_calculation_I1_J,axiom,
ord_less(int,t,one_one(int)) ).
fof(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less(int,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t),times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),zero_zero(int))) ).
fof(fact_3_t,axiom,
plus_plus(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_4_calculation_I2_J,axiom,
( t = zero_zero(int)
=> plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = zero_zero(int) ) ).
fof(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq(int,one_one(int),t) ).
fof(fact_6_p0,axiom,
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_7_not__sum__power2__lt__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ~ ord_less(X_a,plus_plus(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a)) ) ).
fof(fact_8_sum__power2__gt__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_2,Y_2] :
( ord_less(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls))))))
<=> ( ti(X_a,X_2) != zero_zero(X_a)
| ti(X_a,Y_2) != zero_zero(X_a) ) ) ) ).
fof(fact_9_sum__power2__eq__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_2,Y_2] :
( plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a)
<=> ( ti(X_a,X_2) = zero_zero(X_a)
& ti(X_a,Y_2) = zero_zero(X_a) ) ) ) ).
fof(fact_10_power2__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ~ ord_less(X_a,power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))),zero_zero(X_a)) ) ).
fof(fact_11_zero__less__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( ord_less(X_a,zero_zero(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_12_one__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a) ) ).
fof(fact_13_zero__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> power_power(X_a,zero_zero(X_a),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a) ) ).
fof(fact_14_zero__eq__power2,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [A_2] :
( 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_15_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_16_add__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] : plus_plus(X_a,number_number_of(X_a,V),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V,bit1(pls))) ) ).
fof(fact_17_t__l__p,axiom,
ord_less(int,t,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_18__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_1] : plus_plus(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_19_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_20_qf1pt,axiom,
twoSqu420862341sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t)) ).
fof(fact_21_zle__refl,axiom,
! [W] : ord_less_eq(int,W,W) ).
fof(fact_22_number__of__is__id,axiom,
! [K] : number_number_of(int,K) = ti(int,K) ).
fof(fact_23_zmult__commute,axiom,
! [Z,W] : times_times(int,Z,W) = times_times(int,W,Z) ).
fof(fact_24_zle__linear,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
| ord_less_eq(int,W,Z) ) ).
fof(fact_25_times__numeral__code_I5_J,axiom,
! [V,W] : times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W)) ).
fof(fact_26_less__eq__number__of__int__code,axiom,
! [K_1,L_1] :
( ord_less_eq(int,number_number_of(int,K_1),number_number_of(int,L_1))
<=> ord_less_eq(int,K_1,L_1) ) ).
fof(fact_27_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2,Y_2] :
( ord_less_eq(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2))
<=> ord_less_eq(int,X_2,Y_2) ) ) ).
fof(fact_28_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_29_zle__trans,axiom,
! [K,I,J] :
( ord_less_eq(int,I,J)
=> ( ord_less_eq(int,J,K)
=> ord_less_eq(int,I,K) ) ) ).
fof(fact_30_zle__antisym,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
=> ( ord_less_eq(int,W,Z)
=> ti(int,Z) = ti(int,W) ) ) ).
fof(fact_31_zpower__zadd__distrib,axiom,
! [X,Y,Z] : power_power(int,X,plus_plus(nat,Y,Z)) = times_times(int,power_power(int,X,Y),power_power(int,X,Z)) ).
fof(fact_32_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit1(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_33_rel__simps_I34_J,axiom,
! [K_1,L_1] :
( ord_less_eq(int,bit1(K_1),bit1(L_1))
<=> ord_less_eq(int,K_1,L_1) ) ).
fof(fact_34_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
fof(fact_35_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit0(K1),bit0(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_36_rel__simps_I31_J,axiom,
! [K_1,L_1] :
( ord_less_eq(int,bit0(K_1),bit0(L_1))
<=> ord_less_eq(int,K_1,L_1) ) ).
fof(fact_37_zless__le,axiom,
! [Z_1,W_1] :
( ord_less(int,Z_1,W_1)
<=> ( ord_less_eq(int,Z_1,W_1)
& ti(int,Z_1) != ti(int,W_1) ) ) ).
fof(fact_38_zadd__left__mono,axiom,
! [K,I,J] :
( ord_less_eq(int,I,J)
=> ord_less_eq(int,plus_plus(int,K,I),plus_plus(int,K,J)) ) ).
fof(fact_39_eq__number__of__0,axiom,
! [V_3] :
( number_number_of(nat,V_3) = zero_zero(nat)
<=> ord_less_eq(int,V_3,pls) ) ).
fof(fact_40_eq__0__number__of,axiom,
! [V_3] :
( zero_zero(nat) = number_number_of(nat,V_3)
<=> ord_less_eq(int,V_3,pls) ) ).
fof(fact_41_semiring__mult__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1,V] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V_1)
=> times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,times_times(int,V,V_1)) ) ) ) ).
fof(fact_42_mult__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,Z] : times_times(X_a,number_number_of(X_a,V),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V,W)),Z) ) ).
fof(fact_43_arith__simps_I32_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V,W)) ) ).
fof(fact_44_number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : number_number_of(X_a,times_times(int,V,W)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) ) ).
fof(fact_45_sum__squares__le__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_2,Y_2] :
( ord_less_eq(X_a,plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)),zero_zero(X_a))
<=> ( ti(X_a,X_2) = zero_zero(X_a)
& ti(X_a,Y_2) = zero_zero(X_a) ) ) ) ).
fof(fact_46_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : 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_47_le__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2] :
( ord_less_eq(X_a,number_number_of(X_a,X_2),zero_zero(X_a))
<=> ord_less_eq(int,X_2,pls) ) ) ).
fof(fact_48_le__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_2] :
( ord_less_eq(X_a,zero_zero(X_a),number_number_of(X_a,Y_2))
<=> ord_less_eq(int,pls,Y_2) ) ) ).
fof(fact_49_less__0__number__of,axiom,
! [V_3] :
( ord_less(nat,zero_zero(nat),number_number_of(nat,V_3))
<=> ord_less(int,pls,V_3) ) ).
fof(fact_50_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [V_3,W_1] :
( ord_less_eq(X_a,number_number_of(X_a,V_3),number_number_of(X_a,W_1))
<=> ~ ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_3)) ) ) ).
fof(fact_51_rel__simps_I22_J,axiom,
! [K_1] :
( ord_less_eq(int,pls,bit1(K_1))
<=> ord_less_eq(int,pls,K_1) ) ).
fof(fact_52_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_53_rel__simps_I32_J,axiom,
! [K_1,L_1] :
( ord_less_eq(int,bit0(K_1),bit1(L_1))
<=> ord_less_eq(int,K_1,L_1) ) ).
fof(fact_54_rel__simps_I27_J,axiom,
! [K_1] :
( ord_less_eq(int,bit0(K_1),pls)
<=> ord_less_eq(int,K_1,pls) ) ).
fof(fact_55_rel__simps_I21_J,axiom,
! [K_1] :
( ord_less_eq(int,pls,bit0(K_1))
<=> ord_less_eq(int,pls,K_1) ) ).
fof(fact_56_zadd__zless__mono,axiom,
! [Z_2,Z,W_2,W] :
( ord_less(int,W_2,W)
=> ( ord_less_eq(int,Z_2,Z)
=> ord_less(int,plus_plus(int,W_2,Z_2),plus_plus(int,W,Z)) ) ) ).
fof(fact_57_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
fof(fact_58_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
fof(fact_59_le__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2] :
( ord_less_eq(X_a,number_number_of(X_a,X_2),one_one(X_a))
<=> ord_less_eq(int,X_2,bit1(pls)) ) ) ).
fof(fact_60_le__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_2] :
( ord_less_eq(X_a,one_one(X_a),number_number_of(X_a,Y_2))
<=> ord_less_eq(int,bit1(pls),Y_2) ) ) ).
fof(fact_61_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
fof(fact_62_mult__Pls,axiom,
! [W] : times_times(int,pls,W) = pls ).
fof(fact_63_mult__Bit0,axiom,
! [K,L] : times_times(int,bit0(K),L) = bit0(times_times(int,K,L)) ).
fof(fact_64_less__number__of__int__code,axiom,
! [K_1,L_1] :
( ord_less(int,number_number_of(int,K_1),number_number_of(int,L_1))
<=> ord_less(int,K_1,L_1) ) ).
fof(fact_65_zmult__1__right,axiom,
! [Z] : times_times(int,Z,one_one(int)) = ti(int,Z) ).
fof(fact_66_zmult__1,axiom,
! [Z] : times_times(int,one_one(int),Z) = ti(int,Z) ).
fof(fact_67_plus__numeral__code_I9_J,axiom,
! [V,W] : plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W)) ).
fof(fact_68_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_69_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_70_rel__simps_I29_J,axiom,
! [K_1] :
( ord_less_eq(int,bit1(K_1),pls)
<=> ord_less(int,K_1,pls) ) ).
fof(fact_71_rel__simps_I5_J,axiom,
! [K_1] :
( ord_less(int,pls,bit1(K_1))
<=> ord_less_eq(int,pls,K_1) ) ).
fof(fact_72_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_73_rel__simps_I33_J,axiom,
! [K_1,L_1] :
( ord_less_eq(int,bit1(K_1),bit0(L_1))
<=> ord_less(int,K_1,L_1) ) ).
fof(fact_74_less__int__code_I14_J,axiom,
! [K1,K2] :
( ord_less(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_75_rel__simps_I15_J,axiom,
! [K_1,L_1] :
( ord_less(int,bit0(K_1),bit1(L_1))
<=> ord_less_eq(int,K_1,L_1) ) ).
fof(fact_76_less__nat__number__of,axiom,
! [V_3,V_2] :
( ord_less(nat,number_number_of(nat,V_3),number_number_of(nat,V_2))
<=> ( ( ord_less(int,V_3,V_2)
=> ord_less(int,pls,V_2) )
& ord_less(int,V_3,V_2) ) ) ).
fof(fact_77_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( ord_less_eq(int,one_one(int),Z_1)
<=> ord_less(int,zero_zero(int),Z_1) ) ).
fof(fact_78_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
fof(fact_79_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
fof(fact_80_zless__imp__add1__zle,axiom,
! [W,Z] :
( ord_less(int,W,Z)
=> ord_less_eq(int,plus_plus(int,W,one_one(int)),Z) ) ).
fof(fact_81_add1__zle__eq,axiom,
! [W_1,Z_1] :
( ord_less_eq(int,plus_plus(int,W_1,one_one(int)),Z_1)
<=> ord_less(int,W_1,Z_1) ) ).
fof(fact_82_zle__add1__eq__le,axiom,
! [W_1,Z_1] :
( ord_less(int,W_1,plus_plus(int,Z_1,one_one(int)))
<=> ord_less_eq(int,W_1,Z_1) ) ).
fof(fact_83_semiring__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1,V] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V_1)
=> plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,plus_plus(int,V,V_1)) ) ) ) ).
fof(fact_84_add__nat__number__of,axiom,
! [V_1,V] :
( ( ord_less(int,V,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V_1) )
& ( ~ ord_less(int,V,pls)
=> ( ( ord_less(int,V_1,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V) )
& ( ~ ord_less(int,V_1,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,plus_plus(int,V,V_1)) ) ) ) ) ).
fof(fact_85_le__imp__0__less,axiom,
! [Z] :
( ord_less_eq(int,zero_zero(int),Z)
=> ord_less(int,zero_zero(int),plus_plus(int,one_one(int),Z)) ) ).
fof(fact_86_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [X_2,Y_2] :
( number_number_of(X_a,X_2) = number_number_of(X_a,Y_2)
<=> ti(int,X_2) = ti(int,Y_2) ) ) ).
fof(fact_87_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [W_1,X_2] :
( number_number_of(X_a,W_1) = ti(X_a,X_2)
<=> ti(X_a,X_2) = number_number_of(X_a,W_1) ) ) ).
fof(fact_88_rel__simps_I51_J,axiom,
! [K_1,L_1] :
( bit1(K_1) = bit1(L_1)
<=> ti(int,K_1) = ti(int,L_1) ) ).
fof(fact_89_rel__simps_I48_J,axiom,
! [K_1,L_1] :
( bit0(K_1) = bit0(L_1)
<=> ti(int,K_1) = ti(int,L_1) ) ).
fof(fact_90_zless__linear,axiom,
! [X,Y] :
( ord_less(int,X,Y)
| ti(int,X) = ti(int,Y)
| ord_less(int,Y,X) ) ).
fof(fact_91_sum__squares__eq__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_2,Y_2] :
( plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)) = zero_zero(X_a)
<=> ( ti(X_a,X_2) = zero_zero(X_a)
& ti(X_a,Y_2) = zero_zero(X_a) ) ) ) ).
fof(fact_92_left__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [A_1,B,V] : times_times(X_b,plus_plus(X_b,A_1,B),number_number_of(X_b,V)) = plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V))) ) ).
fof(fact_93_right__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [V,B,C] : times_times(X_b,number_number_of(X_b,V),plus_plus(X_b,B,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C)) ) ).
fof(fact_94_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_95_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_96_zadd__commute,axiom,
! [Z,W] : plus_plus(int,Z,W) = plus_plus(int,W,Z) ).
fof(fact_97_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
fof(fact_98_zmult__zless__mono2,axiom,
! [K,I,J] :
( ord_less(int,I,J)
=> ( ord_less(int,zero_zero(int),K)
=> ord_less(int,times_times(int,K,I),times_times(int,K,J)) ) ) ).
fof(fact_99_power2__eq__imp__eq,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))) = power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls))))
=> ( ord_less_eq(X_a,zero_zero(X_a),X)
=> ( ord_less_eq(X_a,zero_zero(X_a),Y)
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ) ).
fof(fact_100_power2__le__imp__le,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( ord_less_eq(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(X_a,zero_zero(X_a),Y)
=> ord_less_eq(X_a,X,Y) ) ) ) ).
fof(fact_101_zero__le__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls))))) ) ).
fof(fact_102_power2__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( ord_less(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(X_a,zero_zero(X_a),Y)
=> ord_less(X_a,X,Y) ) ) ) ).
fof(fact_103_sum__power2__le__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_2,Y_2] :
( ord_less_eq(X_a,plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a))
<=> ( ti(X_a,X_2) = zero_zero(X_a)
& ti(X_a,Y_2) = zero_zero(X_a) ) ) ) ).
fof(fact_104_sum__power2__ge__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ord_less_eq(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls)))))) ) ).
fof(fact_105_sum__squares__gt__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_2,Y_2] :
( ord_less(X_a,zero_zero(X_a),plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)))
<=> ( ti(X_a,X_2) != zero_zero(X_a)
| ti(X_a,Y_2) != zero_zero(X_a) ) ) ) ).
fof(fact_106_not__sum__squares__lt__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ~ 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_107_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_108_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_109_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
fof(fact_110_mult__Bit1,axiom,
! [K,L] : times_times(int,bit1(K),L) = plus_plus(int,bit0(times_times(int,K,L)),L) ).
fof(fact_111_pos__zmult__eq__1__iff,axiom,
! [N_1,Ma] :
( ord_less(int,zero_zero(int),Ma)
=> ( times_times(int,Ma,N_1) = one_one(int)
<=> ( ti(int,Ma) = one_one(int)
& ti(int,N_1) = one_one(int) ) ) ) ).
fof(fact_112_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_113_rel__simps_I46_J,axiom,
! [K] : bit1(K) != pls ).
fof(fact_114_rel__simps_I39_J,axiom,
! [L] : pls != bit1(L) ).
fof(fact_115_rel__simps_I50_J,axiom,
! [K,L] : bit1(K) != bit0(L) ).
fof(fact_116_rel__simps_I49_J,axiom,
! [K,L] : bit0(K) != bit1(L) ).
fof(fact_117_rel__simps_I44_J,axiom,
! [K_1] :
( bit0(K_1) = pls
<=> ti(int,K_1) = pls ) ).
fof(fact_118_rel__simps_I38_J,axiom,
! [L_1] :
( pls = bit0(L_1)
<=> pls = ti(int,L_1) ) ).
fof(fact_119_Bit0__Pls,axiom,
bit0(pls) = pls ).
fof(fact_120_Pls__def,axiom,
pls = zero_zero(int) ).
fof(fact_121_less__int__code_I16_J,axiom,
! [K1,K2] :
( ord_less(int,bit1(K1),bit1(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_122_rel__simps_I17_J,axiom,
! [K_1,L_1] :
( ord_less(int,bit1(K_1),bit1(L_1))
<=> ord_less(int,K_1,L_1) ) ).
fof(fact_123_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
fof(fact_124_less__int__code_I13_J,axiom,
! [K1,K2] :
( ord_less(int,bit0(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_125_rel__simps_I14_J,axiom,
! [K_1,L_1] :
( ord_less(int,bit0(K_1),bit0(L_1))
<=> ord_less(int,K_1,L_1) ) ).
fof(fact_126_int__0__neq__1,axiom,
zero_zero(int) != one_one(int) ).
fof(fact_127_add__Pls__right,axiom,
! [K] : plus_plus(int,K,pls) = ti(int,K) ).
fof(fact_128_add__Pls,axiom,
! [K] : plus_plus(int,pls,K) = ti(int,K) ).
fof(fact_129_add__Bit0__Bit0,axiom,
! [K,L] : plus_plus(int,bit0(K),bit0(L)) = bit0(plus_plus(int,K,L)) ).
fof(fact_130_Bit0__def,axiom,
! [K] : bit0(K) = plus_plus(int,K,K) ).
fof(fact_131_zadd__0__right,axiom,
! [Z] : plus_plus(int,Z,zero_zero(int)) = ti(int,Z) ).
fof(fact_132_zadd__0,axiom,
! [Z] : plus_plus(int,zero_zero(int),Z) = ti(int,Z) ).
fof(fact_133_zadd__strict__right__mono,axiom,
! [K,I,J] :
( ord_less(int,I,J)
=> ord_less(int,plus_plus(int,I,K),plus_plus(int,J,K)) ) ).
fof(fact_134_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_135_power3__eq__cube,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : 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_136_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_137_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_138_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_139_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_140_power2__eq__square,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,A_1,A_1) ) ).
fof(fact_141_even__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( ord_less(X_a,plus_plus(X_a,A_2,A_2),zero_zero(X_a))
<=> ord_less(X_a,A_2,zero_zero(X_a)) ) ) ).
fof(fact_142_semiring__numeral__0__eq__0,axiom,
! [X_a] :
( number_semiring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_143_number__of__Pls,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_144_semiring__norm_I112_J,axiom,
! [X_a] :
( number_ring(X_a)
=> zero_zero(X_a) = number_number_of(X_a,pls) ) ).
fof(fact_145_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_146_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_147_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2,Y_2] :
( ord_less(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2))
<=> ord_less(int,X_2,Y_2) ) ) ).
fof(fact_148_add__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,Z] : plus_plus(X_a,number_number_of(X_a,V),plus_plus(X_a,number_number_of(X_a,W),Z)) = plus_plus(X_a,number_number_of(X_a,plus_plus(int,V,W)),Z) ) ).
fof(fact_149_add__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,V,W)) ) ).
fof(fact_150_number__of__add,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : number_number_of(X_a,plus_plus(int,V,W)) = plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) ) ).
fof(fact_151_rel__simps_I12_J,axiom,
! [K_1] :
( ord_less(int,bit1(K_1),pls)
<=> ord_less(int,K_1,pls) ) ).
fof(fact_152_less__int__code_I15_J,axiom,
! [K1,K2] :
( ord_less(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_153_rel__simps_I16_J,axiom,
! [K_1,L_1] :
( ord_less(int,bit1(K_1),bit0(L_1))
<=> ord_less(int,K_1,L_1) ) ).
fof(fact_154_bin__less__0__simps_I4_J,axiom,
! [W_1] :
( ord_less(int,bit1(W_1),zero_zero(int))
<=> ord_less(int,W_1,zero_zero(int)) ) ).
fof(fact_155_rel__simps_I10_J,axiom,
! [K_1] :
( ord_less(int,bit0(K_1),pls)
<=> ord_less(int,K_1,pls) ) ).
fof(fact_156_rel__simps_I4_J,axiom,
! [K_1] :
( ord_less(int,pls,bit0(K_1))
<=> ord_less(int,pls,K_1) ) ).
fof(fact_157_bin__less__0__simps_I1_J,axiom,
~ ord_less(int,pls,zero_zero(int)) ).
fof(fact_158_bin__less__0__simps_I3_J,axiom,
! [W_1] :
( ord_less(int,bit0(W_1),zero_zero(int))
<=> ord_less(int,W_1,zero_zero(int)) ) ).
fof(fact_159_add__Bit1__Bit0,axiom,
! [K,L] : plus_plus(int,bit1(K),bit0(L)) = bit1(plus_plus(int,K,L)) ).
fof(fact_160_add__Bit0__Bit1,axiom,
! [K,L] : plus_plus(int,bit0(K),bit1(L)) = bit1(plus_plus(int,K,L)) ).
fof(fact_161_int__0__less__1,axiom,
ord_less(int,zero_zero(int),one_one(int)) ).
fof(fact_162_Bit1__def,axiom,
! [K] : bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K) ).
fof(fact_163_odd__nonzero,axiom,
! [Z] : plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int) ).
fof(fact_164_zless__add1__eq,axiom,
! [W_1,Z_1] :
( ord_less(int,W_1,plus_plus(int,Z_1,one_one(int)))
<=> ( ord_less(int,W_1,Z_1)
| ti(int,W_1) = ti(int,Z_1) ) ) ).
fof(fact_165_power2__eq__square__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [W] : 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_166_power2__sum,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [X,Y] : 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,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),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_167_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_168_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_169_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_170_numeral__1__eq__1,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,bit1(pls)) = one_one(X_a) ) ).
fof(fact_171_semiring__norm_I110_J,axiom,
! [X_a] :
( number_ring(X_a)
=> one_one(X_a) = number_number_of(X_a,bit1(pls)) ) ).
fof(fact_172_odd__less__0,axiom,
! [Z_1] :
( ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1),zero_zero(int))
<=> ord_less(int,Z_1,zero_zero(int)) ) ).
fof(fact_173_less__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2] :
( ord_less(X_a,number_number_of(X_a,X_2),zero_zero(X_a))
<=> ord_less(int,X_2,pls) ) ) ).
fof(fact_174_less__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_2] :
( ord_less(X_a,zero_zero(X_a),number_number_of(X_a,Y_2))
<=> ord_less(int,pls,Y_2) ) ) ).
fof(fact_175_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_176_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_177_less__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2] :
( ord_less(X_a,number_number_of(X_a,X_2),one_one(X_a))
<=> ord_less(int,X_2,bit1(pls)) ) ) ).
fof(fact_178_less__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_2] :
( ord_less(X_a,one_one(X_a),number_number_of(X_a,Y_2))
<=> ord_less(int,bit1(pls),Y_2) ) ) ).
fof(fact_179__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int))) ).
fof(fact_180_zadd__power3,axiom,
! [A_1,B] : power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(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))),power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) ).
fof(fact_181_zadd__power2,axiom,
! [A_1,B] : power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,plus_plus(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)),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_182_int__pos__lt__two__imp__zero__or__one,axiom,
! [X] :
( ord_less_eq(int,zero_zero(int),X)
=> ( 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_183_s0p,axiom,
( ord_less_eq(int,zero_zero(int),s)
& ord_less(int,s,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& zcong(s1,s,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ) ).
fof(fact_184_cube__square,axiom,
! [A_1] : times_times(int,A_1,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))) = power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))) ).
fof(fact_185_power2__ge__self,axiom,
! [X] : ord_less_eq(int,X,power_power(int,X,number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_186_self__quotient__aux1,axiom,
! [R,Q,A_1] :
( ord_less(int,zero_zero(int),A_1)
=> ( ti(int,A_1) = plus_plus(int,R,times_times(int,A_1,Q))
=> ( ord_less(int,R,A_1)
=> ord_less_eq(int,one_one(int),Q) ) ) ) ).
fof(fact_187_self__quotient__aux2,axiom,
! [R,Q,A_1] :
( ord_less(int,zero_zero(int),A_1)
=> ( ti(int,A_1) = plus_plus(int,R,times_times(int,A_1,Q))
=> ( ord_less_eq(int,zero_zero(int),R)
=> ord_less_eq(int,Q,one_one(int)) ) ) ) ).
fof(fact_188_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,bit0(bit1(pls)))) ).
fof(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N] : power_power(X_a,X,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(X_a,power_power(X_a,X,N),power_power(X_a,X,N)) ) ).
fof(fact_190__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong(power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),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_191__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] :
( ord_less_eq(int,zero_zero(int),X_1)
& ord_less(int,X_1,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& zcong(s1,X_1,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& ! [Y_1] :
( ( ord_less_eq(int,zero_zero(int),Y_1)
& ord_less(int,Y_1,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& 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_192__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] :
~ ( ord_less_eq(int,zero_zero(int),S)
& ord_less(int,S,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& zcong(s1,S,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ) ).
fof(fact_193_s1,axiom,
zcong(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_194_zero__less__power__nat__eq,axiom,
! [X_2,N_1] :
( ord_less(nat,zero_zero(nat),power_power(nat,X_2,N_1))
<=> ( N_1 = zero_zero(nat)
| ord_less(nat,zero_zero(nat),X_2) ) ) ).
fof(fact_195_zprime__zdvd__power,axiom,
! [A_1,N,P] :
( zprime(P)
=> ( dvd_dvd(int,P,power_power(int,A_1,N))
=> dvd_dvd(int,P,A_1) ) ) ).
fof(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P,Q] : power_power(X_a,power_power(X_a,X,P),Q) = power_power(X_a,X,times_times(nat,P,Q)) ) ).
fof(fact_197_zprime__power__zdvd__cancel__right,axiom,
! [N,A_1,B,P] :
( zprime(P)
=> ( ~ dvd_dvd(int,P,B)
=> ( dvd_dvd(int,power_power(int,P,N),times_times(int,A_1,B))
=> dvd_dvd(int,power_power(int,P,N),A_1) ) ) ) ).
fof(fact_198_zprime__power__zdvd__cancel__left,axiom,
! [N,B,A_1,P] :
( zprime(P)
=> ( ~ dvd_dvd(int,P,A_1)
=> ( dvd_dvd(int,power_power(int,P,N),times_times(int,A_1,B))
=> dvd_dvd(int,power_power(int,P,N),B) ) ) ) ).
fof(fact_199_zpower__zpower,axiom,
! [X,Y,Z] : power_power(int,power_power(int,X,Y),Z) = power_power(int,X,times_times(nat,Y,Z)) ).
fof(fact_200_zdvd__not__zless,axiom,
! [N,M] :
( ord_less(int,zero_zero(int),M)
=> ( ord_less(int,M,N)
=> ~ dvd_dvd(int,N,M) ) ) ).
fof(fact_201_zdvd__antisym__nonneg,axiom,
! [N,M] :
( ord_less_eq(int,zero_zero(int),M)
=> ( ord_less_eq(int,zero_zero(int),N)
=> ( dvd_dvd(int,M,N)
=> ( dvd_dvd(int,N,M)
=> ti(int,M) = ti(int,N) ) ) ) ) ).
fof(fact_202_zdvd__mult__cancel,axiom,
! [K,M,N] :
( dvd_dvd(int,times_times(int,K,M),times_times(int,K,N))
=> ( ti(int,K) != zero_zero(int)
=> dvd_dvd(int,M,N) ) ) ).
fof(fact_203_zdvd__reduce,axiom,
! [K_1,N_1,Ma] :
( dvd_dvd(int,K_1,plus_plus(int,N_1,times_times(int,K_1,Ma)))
<=> dvd_dvd(int,K_1,N_1) ) ).
fof(fact_204_zdvd__period,axiom,
! [C_1,X_2,Ta,A_2,D_1] :
( dvd_dvd(int,A_2,D_1)
=> ( dvd_dvd(int,A_2,plus_plus(int,X_2,Ta))
<=> dvd_dvd(int,A_2,plus_plus(int,plus_plus(int,X_2,times_times(int,C_1,D_1)),Ta)) ) ) ).
fof(fact_205_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
fof(fact_206_zdvd__imp__le,axiom,
! [Z,N] :
( dvd_dvd(int,Z,N)
=> ( ord_less(int,zero_zero(int),N)
=> ord_less_eq(int,Z,N) ) ) ).
fof(fact_207_is__mult__sum2sq,axiom,
! [Y,X] :
( twoSqu420862341sum2sq(X)
=> ( twoSqu420862341sum2sq(Y)
=> twoSqu420862341sum2sq(times_times(int,X,Y)) ) ) ).
fof(fact_208_le__nat__number__of,axiom,
! [V_3,V_2] :
( ord_less_eq(nat,number_number_of(nat,V_3),number_number_of(nat,V_2))
<=> ( ~ ord_less_eq(int,V_3,V_2)
=> ord_less_eq(int,V_3,pls) ) ) ).
fof(fact_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : times_times(X_a,A_1,B) = times_times(X_a,B,A_1) ) ).
fof(fact_217_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_218_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : plus_plus(X_a,A_1,plus_plus(X_a,C,D)) = plus_plus(X_a,C,plus_plus(X_a,A_1,D)) ) ).
fof(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : plus_plus(X_a,A_1,plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),D) ) ).
fof(fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,A_1,plus_plus(X_a,B,C)) ) ).
fof(fact_221_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B) ) ).
fof(fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D] : plus_plus(X_a,plus_plus(X_a,A_1,B),plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D)) ) ).
fof(fact_223_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : power_power(X_a,X,one_one(nat)) = ti(X_a,X) ) ).
fof(fact_224_zero__less__power__nat__eq__number__of,axiom,
! [X_2,W_1] :
( ord_less(nat,zero_zero(nat),power_power(nat,X_2,number_number_of(nat,W_1)))
<=> ( number_number_of(nat,W_1) = zero_zero(nat)
| ord_less(nat,zero_zero(nat),X_2) ) ) ).
fof(fact_225_nat__mult__2__right,axiom,
! [Z] : times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z) ).
fof(fact_226_nat__mult__2,axiom,
! [Z] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
fof(fact_227_mult__nat__number__of,axiom,
! [V_1,V] :
( ( ord_less(int,V,pls)
=> times_times(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = zero_zero(nat) )
& ( ~ ord_less(int,V,pls)
=> times_times(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,times_times(int,V,V_1)) ) ) ).
fof(fact_228_nat__number__of__mult__left,axiom,
! [V_1,K,V] :
( ( ord_less(int,V,pls)
=> times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V_1),K)) = zero_zero(nat) )
& ( ~ ord_less(int,V,pls)
=> times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V_1),K)) = times_times(nat,number_number_of(nat,times_times(int,V,V_1)),K) ) ) ).
fof(fact_229_power__even__eq,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N] : power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = power_power(X_a,power_power(X_a,A_1,N),number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_230_even__power__le__0__imp__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,K] :
( ord_less_eq(X_a,power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),K)),zero_zero(X_a))
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_231_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_232_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_233_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_234_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_235_add__0__iff,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [B_1,A_2] :
( ti(X_a,B_1) = plus_plus(X_a,B_1,A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_236_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_237_crossproduct__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C_1,D_1,A_2,B_1] :
( ( ti(X_a,A_2) != ti(X_a,B_1)
& ti(X_a,C_1) != ti(X_a,D_1) )
<=> plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B_1,D_1)) != plus_plus(X_a,times_times(X_a,A_2,D_1),times_times(X_a,B_1,C_1)) ) ) ).
fof(fact_238_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : times_times(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B,C)) ) ).
fof(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,B] : plus_plus(X_a,times_times(X_a,A_1,M),times_times(X_a,B,M)) = times_times(X_a,plus_plus(X_a,A_1,B),M) ) ).
fof(fact_240_crossproduct__eq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [W_1,Y_2,X_2,Z_1] :
( plus_plus(X_a,times_times(X_a,W_1,Y_2),times_times(X_a,X_2,Z_1)) = plus_plus(X_a,times_times(X_a,W_1,Z_1),times_times(X_a,X_2,Y_2))
<=> ( ti(X_a,W_1) = ti(X_a,X_2)
| ti(X_a,Y_2) = ti(X_a,Z_1) ) ) ) ).
fof(fact_241_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_242_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_243_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Q] : power_power(X_a,times_times(X_a,X,Y),Q) = times_times(X_a,power_power(X_a,X,Q),power_power(X_a,Y,Q)) ) ).
fof(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P,Q] : times_times(X_a,power_power(X_a,X,P),power_power(X_a,X,Q)) = power_power(X_a,X,plus_plus(nat,P,Q)) ) ).
fof(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : power_power(X_a,X,zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_246_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
fof(fact_247_zero__le__even__power_H,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N] : ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))) ) ).
fof(fact_248_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D,A_1,B,R] :
( ti(X_a,R) != zero_zero(X_a)
=> ( ( ti(X_a,A_1) = ti(X_a,B)
& ti(X_a,C) != ti(X_a,D) )
=> plus_plus(X_a,A_1,times_times(X_a,R,C)) != plus_plus(X_a,B,times_times(X_a,R,D)) ) ) ) ).
fof(fact_249_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_250_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_251_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_252_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,W_1] :
( power_power(X_a,A_2,number_number_of(nat,W_1)) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& number_number_of(nat,W_1) != zero_zero(nat) ) ) ) ).
fof(fact_253_pos__zmult__pos,axiom,
! [B,A_1] :
( ord_less(int,zero_zero(int),A_1)
=> ( ord_less(int,zero_zero(int),times_times(int,A_1,B))
=> ord_less(int,zero_zero(int),B) ) ) ).
fof(fact_254_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq(int,zero_zero(int),one_one(int)) ).
fof(fact_255_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ord_less_eq(int,zero_zero(int),times_times(int,X,Y)) ) ) ).
fof(fact_256_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ord_less_eq(int,zero_zero(int),plus_plus(int,X,Y)) ) ) ).
fof(fact_257_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,X] :
( ord_less_eq(int,zero_zero(int),X)
=> ord_less_eq(int,zero_zero(int),power_power(int,X,N)) ) ).
fof(fact_258_power__0__left__number__of,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [W] :
( ( number_number_of(nat,W) = zero_zero(nat)
=> 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)
=> power_power(X_a,zero_zero(X_a),number_number_of(nat,W)) = zero_zero(X_a) ) ) ) ).
fof(fact_259_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,bit1(bit1(pls)))) ).
fof(fact_260_q__pos__lemma,axiom,
! [B_2,Q_1,R_1] :
( ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,B_2,Q_1),R_1))
=> ( ord_less(int,R_1,B_2)
=> ( ord_less(int,zero_zero(int),B_2)
=> ord_less_eq(int,zero_zero(int),Q_1) ) ) ) ).
fof(fact_261_q__neg__lemma,axiom,
! [B_2,Q_1,R_1] :
( ord_less(int,plus_plus(int,times_times(int,B_2,Q_1),R_1),zero_zero(int))
=> ( ord_less_eq(int,zero_zero(int),R_1)
=> ( ord_less(int,zero_zero(int),B_2)
=> ord_less_eq(int,Q_1,zero_zero(int)) ) ) ) ).
fof(fact_262_unique__quotient__lemma,axiom,
! [B,Q_1,R_1,Q,R] :
( ord_less_eq(int,plus_plus(int,times_times(int,B,Q_1),R_1),plus_plus(int,times_times(int,B,Q),R))
=> ( ord_less_eq(int,zero_zero(int),R_1)
=> ( ord_less(int,R_1,B)
=> ( ord_less(int,R,B)
=> ord_less_eq(int,Q_1,Q) ) ) ) ) ).
fof(fact_263_zdiv__mono2__lemma,axiom,
! [B,Q,R,B_2,Q_1,R_1] :
( plus_plus(int,times_times(int,B,Q),R) = plus_plus(int,times_times(int,B_2,Q_1),R_1)
=> ( ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,B_2,Q_1),R_1))
=> ( ord_less(int,R_1,B_2)
=> ( ord_less_eq(int,zero_zero(int),R)
=> ( ord_less(int,zero_zero(int),B_2)
=> ( ord_less_eq(int,B_2,B)
=> ord_less_eq(int,Q,Q_1) ) ) ) ) ) ) ).
fof(fact_264_unique__quotient__lemma__neg,axiom,
! [B,Q_1,R_1,Q,R] :
( ord_less_eq(int,plus_plus(int,times_times(int,B,Q_1),R_1),plus_plus(int,times_times(int,B,Q),R))
=> ( ord_less_eq(int,R,zero_zero(int))
=> ( ord_less(int,B,R)
=> ( ord_less(int,B,R_1)
=> ord_less_eq(int,Q,Q_1) ) ) ) ) ).
fof(fact_265_zdiv__mono2__neg__lemma,axiom,
! [B,Q,R,B_2,Q_1,R_1] :
( plus_plus(int,times_times(int,B,Q),R) = plus_plus(int,times_times(int,B_2,Q_1),R_1)
=> ( ord_less(int,plus_plus(int,times_times(int,B_2,Q_1),R_1),zero_zero(int))
=> ( ord_less(int,R,B)
=> ( ord_less_eq(int,zero_zero(int),R_1)
=> ( ord_less(int,zero_zero(int),B_2)
=> ( ord_less_eq(int,B_2,B)
=> ord_less_eq(int,Q_1,Q) ) ) ) ) ) ) ).
fof(fact_266_quartic__square__square,axiom,
! [X] : power_power(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X,number_number_of(nat,bit0(bit0(bit1(pls))))) ).
fof(fact_267_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : times_times(X_a,X,X) = power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_268_s,axiom,
zcong(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_269_Euler_Oaux____1,axiom,
! [Y,X,P] :
( ~ zcong(X,zero_zero(int),P)
=> ( zcong(power_power(int,Y,number_number_of(nat,bit0(bit1(pls)))),X,P)
=> ~ dvd_dvd(int,P,Y) ) ) ).
fof(fact_270_zprime__def,axiom,
! [P_1] :
( zprime(P_1)
<=> ( ord_less(int,one_one(int),P_1)
& ! [M_1] :
( ( ord_less_eq(int,zero_zero(int),M_1)
& dvd_dvd(int,M_1,P_1) )
=> ( ti(int,M_1) = one_one(int)
| ti(int,M_1) = ti(int,P_1) ) ) ) ) ).
fof(fact_271_prime__g__5,axiom,
! [P] :
( zprime(P)
=> ( ti(int,P) != number_number_of(int,bit0(bit1(pls)))
=> ( ti(int,P) != number_number_of(int,bit1(bit1(pls)))
=> ord_less_eq(int,number_number_of(int,bit1(bit0(bit1(pls)))),P) ) ) ) ).
fof(fact_272_pos2,axiom,
ord_less(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))) ).
fof(fact_273__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1] : ~ zcong(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_274__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_275_Bit1__Min,axiom,
bit1(min) = min ).
fof(fact_276_rel__simps_I43_J,axiom,
! [L_1] :
( min = bit1(L_1)
<=> min = ti(int,L_1) ) ).
fof(fact_277_rel__simps_I47_J,axiom,
! [K_1] :
( bit1(K_1) = min
<=> ti(int,K_1) = min ) ).
fof(fact_278_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_279_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_280_rel__simps_I42_J,axiom,
! [L] : min != bit0(L) ).
fof(fact_281_rel__simps_I45_J,axiom,
! [K] : bit0(K) != min ).
fof(fact_282_rel__simps_I7_J,axiom,
~ ord_less(int,min,min) ).
fof(fact_283_rel__simps_I24_J,axiom,
ord_less_eq(int,min,min) ).
fof(fact_284_rel__simps_I13_J,axiom,
! [K_1] :
( ord_less(int,bit1(K_1),min)
<=> ord_less(int,K_1,min) ) ).
fof(fact_285_rel__simps_I9_J,axiom,
! [K_1] :
( ord_less(int,min,bit1(K_1))
<=> ord_less(int,min,K_1) ) ).
fof(fact_286_rel__simps_I3_J,axiom,
~ ord_less(int,pls,min) ).
fof(fact_287_rel__simps_I6_J,axiom,
ord_less(int,min,pls) ).
fof(fact_288_rel__simps_I8_J,axiom,
! [K_1] :
( ord_less(int,min,bit0(K_1))
<=> ord_less(int,min,K_1) ) ).
fof(fact_289_bin__less__0__simps_I2_J,axiom,
ord_less(int,min,zero_zero(int)) ).
fof(fact_290_rel__simps_I26_J,axiom,
! [K_1] :
( ord_less_eq(int,min,bit1(K_1))
<=> ord_less_eq(int,min,K_1) ) ).
fof(fact_291_rel__simps_I30_J,axiom,
! [K_1] :
( ord_less_eq(int,bit1(K_1),min)
<=> ord_less_eq(int,K_1,min) ) ).
fof(fact_292_rel__simps_I23_J,axiom,
ord_less_eq(int,min,pls) ).
fof(fact_293_rel__simps_I20_J,axiom,
~ ord_less_eq(int,pls,min) ).
fof(fact_294_rel__simps_I28_J,axiom,
! [K_1] :
( ord_less_eq(int,bit0(K_1),min)
<=> ord_less_eq(int,K_1,min) ) ).
fof(fact_295_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
fof(fact_296_rel__simps_I11_J,axiom,
! [K_1] :
( ord_less(int,bit0(K_1),min)
<=> ord_less_eq(int,K_1,min) ) ).
fof(fact_297_rel__simps_I25_J,axiom,
! [K_1] :
( ord_less_eq(int,min,bit0(K_1))
<=> ord_less(int,min,K_1) ) ).
fof(fact_298_zmult__eq__1__iff,axiom,
! [Ma,N_1] :
( times_times(int,Ma,N_1) = one_one(int)
<=> ( ( ti(int,Ma) = one_one(int)
& ti(int,N_1) = one_one(int) )
| ( ti(int,Ma) = number_number_of(int,min)
& ti(int,N_1) = number_number_of(int,min) ) ) ) ).
fof(fact_299_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_300_zcong__sym,axiom,
! [A_2,B_1,Ma] :
( zcong(A_2,B_1,Ma)
<=> zcong(B_1,A_2,Ma) ) ).
fof(fact_301_zcong__refl,axiom,
! [K,M] : zcong(K,K,M) ).
fof(fact_302_zcong__trans,axiom,
! [C,A_1,B,M] :
( zcong(A_1,B,M)
=> ( zcong(B,C,M)
=> zcong(A_1,C,M) ) ) ).
fof(fact_303_order__le__neq__implies__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( ord_less_eq(X_a,X,Y)
=> ( ti(X_a,X) != ti(X_a,Y)
=> ord_less(X_a,X,Y) ) ) ) ).
fof(fact_304_Euler_Oaux2,axiom,
! [B,A_1,C] :
( ord_less(int,A_1,C)
=> ( ord_less(int,B,C)
=> ( ord_less_eq(int,A_1,B)
| ord_less_eq(int,B,A_1) ) ) ) ).
fof(fact_305_IntPrimes_Ozcong__zero,axiom,
! [A_2,B_1] :
( zcong(A_2,B_1,zero_zero(int))
<=> ti(int,A_2) = ti(int,B_1) ) ).
fof(fact_306_zcong__1,axiom,
! [A_1,B] : zcong(A_1,B,one_one(int)) ).
fof(fact_307_zcong__zmult__self,axiom,
! [A_1,M,B] : zcong(times_times(int,A_1,M),times_times(int,B,M),M) ).
fof(fact_308_zcong__scalar,axiom,
! [K,A_1,B,M] :
( zcong(A_1,B,M)
=> zcong(times_times(int,A_1,K),times_times(int,B,K),M) ) ).
fof(fact_309_zcong__scalar2,axiom,
! [K,A_1,B,M] :
( zcong(A_1,B,M)
=> zcong(times_times(int,K,A_1),times_times(int,K,B),M) ) ).
fof(fact_310_zcong__zmult,axiom,
! [C,D,A_1,B,M] :
( zcong(A_1,B,M)
=> ( zcong(C,D,M)
=> zcong(times_times(int,A_1,C),times_times(int,B,D),M) ) ) ).
fof(fact_311_zcong__zadd,axiom,
! [C,D,A_1,B,M] :
( zcong(A_1,B,M)
=> ( zcong(C,D,M)
=> zcong(plus_plus(int,A_1,C),plus_plus(int,B,D),M) ) ) ).
fof(fact_312_power__m1__even,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [N] : 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_313_zcong__not,axiom,
! [B,M,A_1] :
( ord_less(int,zero_zero(int),A_1)
=> ( ord_less(int,A_1,M)
=> ( ord_less(int,zero_zero(int),B)
=> ( ord_less(int,B,A_1)
=> ~ zcong(A_1,B,M) ) ) ) ) ).
fof(fact_314_zcong__iff__lin,axiom,
! [A_2,B_1,Ma] :
( zcong(A_2,B_1,Ma)
<=> ? [K_2] : ti(int,B_1) = plus_plus(int,A_2,times_times(int,Ma,K_2)) ) ).
fof(fact_315_four__x__squared,axiom,
! [X] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) = power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),number_number_of(nat,bit0(bit1(pls)))) ).
fof(fact_316_zcong__zless__0,axiom,
! [M,A_1] :
( ord_less_eq(int,zero_zero(int),A_1)
=> ( ord_less(int,A_1,M)
=> ( zcong(A_1,zero_zero(int),M)
=> ti(int,A_1) = zero_zero(int) ) ) ) ).
fof(fact_317_zcong__zless__imp__eq,axiom,
! [B,M,A_1] :
( ord_less_eq(int,zero_zero(int),A_1)
=> ( ord_less(int,A_1,M)
=> ( ord_less_eq(int,zero_zero(int),B)
=> ( ord_less(int,B,M)
=> ( zcong(A_1,B,M)
=> ti(int,A_1) = ti(int,B) ) ) ) ) ) ).
fof(fact_318_zcong__zpower__zmult,axiom,
! [Z,X,Y,P] :
( zcong(power_power(int,X,Y),one_one(int),P)
=> zcong(power_power(int,X,times_times(nat,Y,Z)),one_one(int),P) ) ).
fof(fact_319_zprime__zdvd__zmult,axiom,
! [N,P,M] :
( ord_less_eq(int,zero_zero(int),M)
=> ( zprime(P)
=> ( dvd_dvd(int,P,times_times(int,M,N))
=> ( dvd_dvd(int,P,M)
| dvd_dvd(int,P,N) ) ) ) ) ).
fof(fact_320__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
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_321__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),minus_minus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min))) ).
fof(fact_322_neg__one__power__eq__mod__m,axiom,
! [J,K,M] :
( ord_less(int,number_number_of(int,bit0(bit1(pls))),M)
=> ( zcong(power_power(int,number_number_of(int,min),J),power_power(int,number_number_of(int,min),K),M)
=> power_power(int,number_number_of(int,min),J) = power_power(int,number_number_of(int,min),K) ) ) ).
fof(fact_323_zcong__neg__1__impl__ne__1,axiom,
! [X,P] :
( ord_less(int,number_number_of(int,bit0(bit1(pls))),P)
=> ( zcong(X,number_number_of(int,min),P)
=> ~ zcong(X,one_one(int),P) ) ) ).
fof(fact_324__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
minus_minus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)) = plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) ).
fof(fact_325__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
( ~ 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_326_not__real__square__gt__zero,axiom,
! [X_2] :
( ~ ord_less(real,zero_zero(real),times_times(real,X_2,X_2))
<=> X_2 = zero_zero(real) ) ).
fof(fact_327_zcong__zdiff,axiom,
! [C,D,A_1,B,M] :
( zcong(A_1,B,M)
=> ( zcong(C,D,M)
=> zcong(minus_minus(int,A_1,C),minus_minus(int,B,D),M) ) ) ).
fof(fact_328_zdvd__zdiffD,axiom,
! [K,M,N] :
( dvd_dvd(int,K,minus_minus(int,M,N))
=> ( dvd_dvd(int,K,N)
=> dvd_dvd(int,K,M) ) ) ).
fof(fact_329_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_330_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_331_diff__bin__simps_I7_J,axiom,
! [K,L] : minus_minus(int,bit0(K),bit0(L)) = bit0(minus_minus(int,K,L)) ).
fof(fact_332_diff__bin__simps_I1_J,axiom,
! [K] : minus_minus(int,K,pls) = ti(int,K) ).
fof(fact_333_Int2_Oaux1,axiom,
! [A_1,B,C] :
( minus_minus(int,A_1,B) = ti(int,C)
=> ti(int,A_1) = plus_plus(int,C,B) ) ).
fof(fact_334_number__of__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : number_number_of(X_a,minus_minus(int,V,W)) = minus_minus(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) ) ).
fof(fact_335_right__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [V,B,C] : times_times(X_b,number_number_of(X_b,V),minus_minus(X_b,B,C)) = minus_minus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C)) ) ).
fof(fact_336_left__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [A_1,B,V] : times_times(X_b,minus_minus(X_b,A_1,B),number_number_of(X_b,V)) = minus_minus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V))) ) ).
fof(fact_337_diff__bin__simps_I9_J,axiom,
! [K,L] : minus_minus(int,bit1(K),bit0(L)) = bit1(minus_minus(int,K,L)) ).
fof(fact_338_diff__bin__simps_I10_J,axiom,
! [K,L] : minus_minus(int,bit1(K),bit1(L)) = bit0(minus_minus(int,K,L)) ).
fof(fact_339_diff__bin__simps_I3_J,axiom,
! [L] : minus_minus(int,pls,bit0(L)) = bit0(minus_minus(int,pls,L)) ).
fof(fact_340_less__bin__lemma,axiom,
! [K_1,L_1] :
( ord_less(int,K_1,L_1)
<=> ord_less(int,minus_minus(int,K_1,L_1),zero_zero(int)) ) ).
fof(fact_341_xzgcda__linear__aux1,axiom,
! [A_1,R,B,M,C,D,N] : plus_plus(int,times_times(int,minus_minus(int,A_1,times_times(int,R,B)),M),times_times(int,minus_minus(int,C,times_times(int,R,D)),N)) = minus_minus(int,plus_plus(int,times_times(int,A_1,M),times_times(int,C,N)),times_times(int,R,plus_plus(int,times_times(int,B,M),times_times(int,D,N)))) ).
fof(fact_342_zcong__def,axiom,
! [A_2,B_1,Ma] :
( zcong(A_2,B_1,Ma)
<=> dvd_dvd(int,Ma,minus_minus(int,A_2,B_1)) ) ).
fof(fact_343_add__number__of__diff1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,C] : plus_plus(X_a,number_number_of(X_a,V),minus_minus(X_a,number_number_of(X_a,W),C)) = minus_minus(X_a,number_number_of(X_a,plus_plus(int,V,W)),C) ) ).
fof(fact_344_Euler_Oaux1,axiom,
! [A_1,X] :
( ord_less(int,zero_zero(int),X)
=> ( ord_less(int,X,A_1)
=> ( ti(int,X) != minus_minus(int,A_1,one_one(int))
=> ord_less(int,X,minus_minus(int,A_1,one_one(int))) ) ) ) ).
fof(fact_345_zle__diff1__eq,axiom,
! [W_1,Z_1] :
( ord_less_eq(int,W_1,minus_minus(int,Z_1,one_one(int)))
<=> ord_less(int,W_1,Z_1) ) ).
fof(fact_346_diff__bin__simps_I4_J,axiom,
! [L] : minus_minus(int,pls,bit1(L)) = bit1(minus_minus(int,min,L)) ).
fof(fact_347_diff__bin__simps_I6_J,axiom,
! [L] : minus_minus(int,min,bit1(L)) = bit0(minus_minus(int,min,L)) ).
fof(fact_348_diff__bin__simps_I5_J,axiom,
! [L] : minus_minus(int,min,bit0(L)) = bit1(minus_minus(int,min,L)) ).
fof(fact_349_inv__not__p__minus__1__aux,axiom,
! [A_2,P_1] :
( zcong(times_times(int,A_2,minus_minus(int,P_1,one_one(int))),one_one(int),P_1)
<=> zcong(A_2,minus_minus(int,P_1,one_one(int)),P_1) ) ).
fof(fact_350_zcong__eq__trans,axiom,
! [D,C,A_1,B,M] :
( zcong(A_1,B,M)
=> ( ti(int,B) = ti(int,C)
=> ( zcong(C,D,M)
=> zcong(A_1,D,M) ) ) ) ).
fof(fact_351_zcong__square__zless,axiom,
! [A_1,P] :
( zprime(P)
=> ( ord_less(int,zero_zero(int),A_1)
=> ( ord_less(int,A_1,P)
=> ( zcong(times_times(int,A_1,A_1),one_one(int),P)
=> ( ti(int,A_1) = one_one(int)
| ti(int,A_1) = minus_minus(int,P,one_one(int)) ) ) ) ) ) ).
fof(fact_352_zcong__square,axiom,
! [A_1,P] :
( zprime(P)
=> ( ord_less(int,zero_zero(int),A_1)
=> ( zcong(times_times(int,A_1,A_1),one_one(int),P)
=> ( zcong(A_1,one_one(int),P)
| zcong(A_1,minus_minus(int,P,one_one(int)),P) ) ) ) ) ).
fof(fact_353_zspecial__product,axiom,
! [A_1,B] : times_times(int,plus_plus(int,A_1,B),minus_minus(int,A_1,B)) = minus_minus(int,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_354_zcong__id,axiom,
! [M] : zcong(M,zero_zero(int),M) ).
fof(fact_355_zcong__zmult__prop2,axiom,
! [C_1,D_1,A_2,B_1,Ma] :
( zcong(A_2,B_1,Ma)
=> ( zcong(C_1,times_times(int,D_1,A_2),Ma)
<=> zcong(C_1,times_times(int,D_1,B_1),Ma) ) ) ).
fof(fact_356_zcong__zmult__prop1,axiom,
! [C_1,D_1,A_2,B_1,Ma] :
( zcong(A_2,B_1,Ma)
=> ( zcong(C_1,times_times(int,A_2,D_1),Ma)
<=> zcong(C_1,times_times(int,B_1,D_1),Ma) ) ) ).
fof(fact_357_zcong__shift,axiom,
! [C,A_1,B,M] :
( zcong(A_1,B,M)
=> zcong(plus_plus(int,A_1,C),plus_plus(int,B,C),M) ) ).
fof(fact_358_zcong__zpower,axiom,
! [Z,X,Y,M] :
( zcong(X,Y,M)
=> zcong(power_power(int,X,Z),power_power(int,Y,Z),M) ) ).
fof(fact_359_power2__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X,Y] : 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,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),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_360_zdiff__power2,axiom,
! [A_1,B] : power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,minus_minus(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)),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_361_zdiff__power3,axiom,
! [A_1,B] : power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit1(bit1(pls)))) = minus_minus(int,plus_plus(int,minus_minus(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))),power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) ).
fof(fact_362_zcong__less__eq,axiom,
! [M,Y,X] :
( ord_less(int,zero_zero(int),X)
=> ( ord_less(int,zero_zero(int),Y)
=> ( ord_less(int,zero_zero(int),M)
=> ( zcong(X,Y,M)
=> ( ord_less(int,X,M)
=> ( ord_less(int,Y,M)
=> ti(int,X) = ti(int,Y) ) ) ) ) ) ) ).
fof(fact_363_zcong__not__zero,axiom,
! [M,X] :
( ord_less(int,zero_zero(int),X)
=> ( ord_less(int,X,M)
=> ~ zcong(X,zero_zero(int),M) ) ) ).
fof(fact_364_zdvd__bounds,axiom,
! [N,M] :
( dvd_dvd(int,N,M)
=> ( ord_less_eq(int,M,zero_zero(int))
| ord_less_eq(int,N,M) ) ) ).
fof(fact_365_zcong__zero__equiv__div,axiom,
! [A_2,Ma] :
( zcong(A_2,zero_zero(int),Ma)
<=> dvd_dvd(int,Ma,A_2) ) ).
fof(fact_366_zcong__eq__zdvd__prop,axiom,
! [X_2,P_1] :
( zcong(X_2,zero_zero(int),P_1)
<=> dvd_dvd(int,P_1,X_2) ) ).
fof(fact_367_zprime__zdvd__zmult__better,axiom,
! [M,N,P] :
( zprime(P)
=> ( dvd_dvd(int,P,times_times(int,M,N))
=> ( dvd_dvd(int,P,M)
| dvd_dvd(int,P,N) ) ) ) ).
fof(fact_368_Int2_Ozcong__zero,axiom,
! [M,X] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less(int,X,M)
=> ( zcong(X,zero_zero(int),M)
=> ti(int,X) = zero_zero(int) ) ) ) ).
fof(fact_369_zpower__zdvd__prop1,axiom,
! [P,Y,N] :
( ord_less(nat,zero_zero(nat),N)
=> ( dvd_dvd(int,P,Y)
=> dvd_dvd(int,P,power_power(int,Y,N)) ) ) ).
fof(fact_370_neg__one__power,axiom,
! [N] :
( power_power(int,number_number_of(int,min),N) = one_one(int)
| power_power(int,number_number_of(int,min),N) = number_number_of(int,min) ) ).
fof(fact_371_zcong__zmult__prop3,axiom,
! [Y,X,P] :
( zprime(P)
=> ( ~ zcong(X,zero_zero(int),P)
=> ( ~ zcong(Y,zero_zero(int),P)
=> ~ zcong(times_times(int,X,Y),zero_zero(int),P) ) ) ) ).
fof(fact_372_zcong__zprime__prod__zero__contra,axiom,
! [B,A_1,P] :
( zprime(P)
=> ( ord_less(int,zero_zero(int),A_1)
=> ( ( ~ zcong(A_1,zero_zero(int),P)
& ~ zcong(B,zero_zero(int),P) )
=> ~ zcong(times_times(int,A_1,B),zero_zero(int),P) ) ) ) ).
fof(fact_373_zcong__zprime__prod__zero,axiom,
! [B,A_1,P] :
( zprime(P)
=> ( ord_less(int,zero_zero(int),A_1)
=> ( zcong(times_times(int,A_1,B),zero_zero(int),P)
=> ( zcong(A_1,zero_zero(int),P)
| zcong(B,zero_zero(int),P) ) ) ) ) ).
fof(fact_374_zpower__zdvd__prop2,axiom,
! [Y,N,P] :
( zprime(P)
=> ( dvd_dvd(int,P,power_power(int,Y,N))
=> ( ord_less(nat,zero_zero(nat),N)
=> dvd_dvd(int,P,Y) ) ) ) ).
fof(fact_375_Legendre__1mod4,axiom,
! [M] :
( 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_376_one__not__neg__one__mod__m,axiom,
! [M] :
( ord_less(int,number_number_of(int,bit0(bit1(pls))),M)
=> ~ zcong(one_one(int),number_number_of(int,min),M) ) ).
fof(fact_377_Legendre__def,axiom,
! [A_1,P] :
( ( zcong(A_1,zero_zero(int),P)
=> legendre(A_1,P) = zero_zero(int) )
& ( ~ zcong(A_1,zero_zero(int),P)
=> ( ( quadRes(P,A_1)
=> legendre(A_1,P) = one_one(int) )
& ( ~ quadRes(P,A_1)
=> legendre(A_1,P) = number_number_of(int,min) ) ) ) ) ).
fof(fact_378_QuadRes__def,axiom,
! [Ma,X_2] :
( quadRes(Ma,X_2)
<=> ? [Y_1] : zcong(power_power(int,Y_1,number_number_of(nat,bit0(bit1(pls)))),X_2,Ma) ) ).
fof(fact_379_real__sum__squared__expand,axiom,
! [X,Y] : power_power(real,plus_plus(real,X,Y),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(real,plus_plus(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),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_380_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_381_power__eq__if,axiom,
! [P,M] :
( ( M = zero_zero(nat)
=> power_power(nat,P,M) = one_one(nat) )
& ( M != zero_zero(nat)
=> power_power(nat,P,M) = times_times(nat,P,power_power(nat,P,minus_minus(nat,M,one_one(nat)))) ) ) ).
fof(fact_382_diff__square,axiom,
! [X,Y] : minus_minus(nat,power_power(nat,X,number_number_of(nat,bit0(bit1(pls)))),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_383_realpow__two__sum__zero__iff,axiom,
! [X_2,Y_2] :
( plus_plus(real,power_power(real,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y_2,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(real)
<=> ( X_2 = zero_zero(real)
& Y_2 = zero_zero(real) ) ) ).
fof(fact_384_real__zero__not__eq__one,axiom,
zero_zero(real) != one_one(real) ).
fof(fact_385_real__le__eq__diff,axiom,
! [X_2,Y_2] :
( ord_less_eq(real,X_2,Y_2)
<=> ord_less_eq(real,minus_minus(real,X_2,Y_2),zero_zero(real)) ) ).
fof(fact_386_real__less__def,axiom,
! [X_2,Y_2] :
( ord_less(real,X_2,Y_2)
<=> ( ord_less_eq(real,X_2,Y_2)
& X_2 != Y_2 ) ) ).
fof(fact_387_less__eq__real__def,axiom,
! [X_2,Y_2] :
( ord_less_eq(real,X_2,Y_2)
<=> ( ord_less(real,X_2,Y_2)
| X_2 = Y_2 ) ) ).
fof(fact_388_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_389_real__mult__commute,axiom,
! [Z,W] : times_times(real,Z,W) = times_times(real,W,Z) ).
fof(fact_390_real__mult__1,axiom,
! [Z] : times_times(real,one_one(real),Z) = Z ).
fof(fact_391_real__add__left__mono,axiom,
! [Z,X,Y] :
( ord_less_eq(real,X,Y)
=> ord_less_eq(real,plus_plus(real,Z,X),plus_plus(real,Z,Y)) ) ).
fof(fact_392_add__diff__add,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,C,B,D] : minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D)) = plus_plus(X_a,minus_minus(X_a,A_1,B),minus_minus(X_a,C,D)) ) ).
fof(fact_393_real__mult__left__cancel,axiom,
! [A_2,B_1,C_1] :
( C_1 != zero_zero(real)
=> ( times_times(real,C_1,A_2) = times_times(real,C_1,B_1)
<=> A_2 = B_1 ) ) ).
fof(fact_394_real__mult__right__cancel,axiom,
! [A_2,B_1,C_1] :
( C_1 != zero_zero(real)
=> ( times_times(real,A_2,C_1) = times_times(real,B_1,C_1)
<=> A_2 = B_1 ) ) ).
fof(fact_395_nat__mult__eq__one,axiom,
! [N_1,Ma] :
( times_times(nat,N_1,Ma) = one_one(nat)
<=> ( N_1 = one_one(nat)
& Ma = one_one(nat) ) ) ).
fof(fact_396_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_397_nat__power__eq__0__iff,axiom,
! [Ma,N_1] :
( power_power(nat,Ma,N_1) = zero_zero(nat)
<=> ( N_1 != zero_zero(nat)
& Ma = zero_zero(nat) ) ) ).
fof(fact_398_mult__diff__mult,axiom,
! [X_a] :
( ring(X_a)
=> ! [X,Y,A_1,B] : minus_minus(X_a,times_times(X_a,X,Y),times_times(X_a,A_1,B)) = plus_plus(X_a,times_times(X_a,X,minus_minus(X_a,Y,B)),times_times(X_a,minus_minus(X_a,X,A_1),B)) ) ).
fof(fact_399_real__mult__le__cancel__iff1,axiom,
! [X_2,Y_2,Z_1] :
( ord_less(real,zero_zero(real),Z_1)
=> ( ord_less_eq(real,times_times(real,X_2,Z_1),times_times(real,Y_2,Z_1))
<=> ord_less_eq(real,X_2,Y_2) ) ) ).
fof(fact_400_real__mult__le__cancel__iff2,axiom,
! [X_2,Y_2,Z_1] :
( ord_less(real,zero_zero(real),Z_1)
=> ( ord_less_eq(real,times_times(real,Z_1,X_2),times_times(real,Z_1,Y_2))
<=> ord_less_eq(real,X_2,Y_2) ) ) ).
fof(fact_401_real__mult__less__mono2,axiom,
! [X,Y,Z] :
( ord_less(real,zero_zero(real),Z)
=> ( ord_less(real,X,Y)
=> ord_less(real,times_times(real,Z,X),times_times(real,Z,Y)) ) ) ).
fof(fact_402_real__mult__order,axiom,
! [Y,X] :
( ord_less(real,zero_zero(real),X)
=> ( ord_less(real,zero_zero(real),Y)
=> ord_less(real,zero_zero(real),times_times(real,X,Y)) ) ) ).
fof(fact_403_real__mult__less__iff1,axiom,
! [X_2,Y_2,Z_1] :
( ord_less(real,zero_zero(real),Z_1)
=> ( ord_less(real,times_times(real,X_2,Z_1),times_times(real,Y_2,Z_1))
<=> ord_less(real,X_2,Y_2) ) ) ).
fof(fact_404_real__two__squares__add__zero__iff,axiom,
! [X_2,Y_2] :
( plus_plus(real,times_times(real,X_2,X_2),times_times(real,Y_2,Y_2)) = zero_zero(real)
<=> ( X_2 = zero_zero(real)
& Y_2 = zero_zero(real) ) ) ).
fof(fact_405_exp__eq__1,axiom,
! [X_2,N_1] :
( power_power(nat,X_2,N_1) = one_one(nat)
<=> ( X_2 = one_one(nat)
| N_1 = zero_zero(nat) ) ) ).
fof(fact_406_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_407_realpow__minus__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [X,N] :
( ord_less(nat,zero_zero(nat),N)
=> times_times(X_a,power_power(X_a,X,minus_minus(nat,N,one_one(nat))),X) = power_power(X_a,X,N) ) ) ).
fof(fact_408_two__realpow__ge__one,axiom,
! [N] : ord_less_eq(real,one_one(real),power_power(real,number_number_of(real,bit0(bit1(pls))),N)) ).
fof(fact_409_realpow__num__eq__if,axiom,
! [X_a] :
( power(X_a)
=> ! [M,N] :
( ( N = zero_zero(nat)
=> power_power(X_a,M,N) = one_one(X_a) )
& ( N != zero_zero(nat)
=> power_power(X_a,M,N) = times_times(X_a,M,power_power(X_a,M,minus_minus(nat,N,one_one(nat)))) ) ) ) ).
fof(fact_410_power__strict__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B] :
( ord_less(X_a,A_1,B)
=> ( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less(nat,zero_zero(nat),N)
=> ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)) ) ) ) ) ).
fof(fact_411_real__le__refl,axiom,
! [W] : ord_less_eq(real,W,W) ).
fof(fact_412_real__le__linear,axiom,
! [Z,W] :
( ord_less_eq(real,Z,W)
| ord_less_eq(real,W,Z) ) ).
fof(fact_413_real__le__trans,axiom,
! [K,I,J] :
( ord_less_eq(real,I,J)
=> ( ord_less_eq(real,J,K)
=> ord_less_eq(real,I,K) ) ) ).
fof(fact_414_real__le__antisym,axiom,
! [Z,W] :
( ord_less_eq(real,Z,W)
=> ( ord_less_eq(real,W,Z)
=> Z = W ) ) ).
fof(fact_415_field__power__not__zero,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [N,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> power_power(X_a,A_1,N) != zero_zero(X_a) ) ) ).
fof(fact_416_power__mult__distrib,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1,B,N] : power_power(X_a,times_times(X_a,A_1,B),N) = times_times(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)) ) ).
fof(fact_417_power__commutes,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N] : times_times(X_a,power_power(X_a,A_1,N),A_1) = times_times(X_a,A_1,power_power(X_a,A_1,N)) ) ).
fof(fact_418_power__one,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [N] : power_power(X_a,one_one(X_a),N) = one_one(X_a) ) ).
fof(fact_419_dvd__power__same,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N,X,Y] :
( dvd_dvd(X_a,X,Y)
=> dvd_dvd(X_a,power_power(X_a,X,N),power_power(X_a,Y,N)) ) ) ).
fof(fact_420_power__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N] : power_power(X_a,A_1,times_times(nat,M,N)) = power_power(X_a,power_power(X_a,A_1,M),N) ) ).
fof(fact_421_power__one__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : power_power(X_a,A_1,one_one(nat)) = ti(X_a,A_1) ) ).
fof(fact_422_power__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B] :
( ord_less_eq(X_a,A_1,B)
=> ( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ord_less_eq(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)) ) ) ) ).
fof(fact_423_zero__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,N)) ) ) ).
fof(fact_424_zero__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less(X_a,zero_zero(X_a),A_1)
=> ord_less(X_a,zero_zero(X_a),power_power(X_a,A_1,N)) ) ) ).
fof(fact_425_one__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less_eq(X_a,one_one(X_a),A_1)
=> ord_less_eq(X_a,one_one(X_a),power_power(X_a,A_1,N)) ) ) ).
fof(fact_426_power__inject__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,N_1,A_2] :
( ord_less(X_a,one_one(X_a),A_2)
=> ( power_power(X_a,A_2,Ma) = power_power(X_a,A_2,N_1)
<=> Ma = N_1 ) ) ) ).
fof(fact_427_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,N_1] :
( power_power(X_a,A_2,N_1) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& N_1 != zero_zero(nat) ) ) ) ).
fof(fact_428_power__0,axiom,
! [X_a] :
( power(X_a)
=> ! [A_1] : power_power(X_a,A_1,zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_429_power__add,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N] : power_power(X_a,A_1,plus_plus(nat,M,N)) = times_times(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)) ) ).
fof(fact_430_power__le__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M,A_1,N,B] :
( dvd_dvd(X_a,power_power(X_a,A_1,N),B)
=> ( ord_less_eq(nat,M,N)
=> dvd_dvd(X_a,power_power(X_a,A_1,M),B) ) ) ) ).
fof(fact_431_dvd__power__le,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N,M,X,Y] :
( dvd_dvd(X_a,X,Y)
=> ( ord_less_eq(nat,N,M)
=> dvd_dvd(X_a,power_power(X_a,X,N),power_power(X_a,Y,M)) ) ) ) ).
fof(fact_432_le__imp__power__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,N] :
( ord_less_eq(nat,M,N)
=> dvd_dvd(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)) ) ) ).
fof(fact_433_nat__power__less__imp__less,axiom,
! [M,N,I] :
( ord_less(nat,zero_zero(nat),I)
=> ( ord_less(nat,power_power(nat,I,M),power_power(nat,I,N))
=> ord_less(nat,M,N) ) ) ).
fof(fact_434_nat__zero__less__power__iff,axiom,
! [X_2,N_1] :
( ord_less(nat,zero_zero(nat),power_power(nat,X_2,N_1))
<=> ( ord_less(nat,zero_zero(nat),X_2)
| N_1 = zero_zero(nat) ) ) ).
fof(fact_435_power__less__imp__less__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B] :
( ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N))
=> ( ord_less_eq(X_a,zero_zero(X_a),B)
=> ord_less(X_a,A_1,B) ) ) ) ).
fof(fact_436_power__less__power__Suc,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less(X_a,one_one(X_a),A_1)
=> ord_less(X_a,power_power(X_a,A_1,N),times_times(X_a,A_1,power_power(X_a,A_1,N))) ) ) ).
fof(fact_437_power__gt1__lemma,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less(X_a,one_one(X_a),A_1)
=> ord_less(X_a,one_one(X_a),times_times(X_a,A_1,power_power(X_a,A_1,N))) ) ) ).
fof(fact_438_power__0__left,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [N] :
( ( N = zero_zero(nat)
=> power_power(X_a,zero_zero(X_a),N) = one_one(X_a) )
& ( N != zero_zero(nat)
=> power_power(X_a,zero_zero(X_a),N) = zero_zero(X_a) ) ) ) ).
fof(fact_439_power__strict__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( ord_less(nat,N,N_2)
=> ( ord_less(X_a,one_one(X_a),A_1)
=> ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,A_1,N_2)) ) ) ) ).
fof(fact_440_power__less__imp__less__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N,A_1] :
( ord_less(X_a,one_one(X_a),A_1)
=> ( ord_less(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N))
=> ord_less(nat,M,N) ) ) ) ).
fof(fact_441_power__strict__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X_2,Y_2,B_1] :
( ord_less(X_a,one_one(X_a),B_1)
=> ( ord_less(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2))
<=> ord_less(nat,X_2,Y_2) ) ) ) ).
fof(fact_442_power__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( ord_less_eq(nat,N,N_2)
=> ( ord_less_eq(X_a,one_one(X_a),A_1)
=> ord_less_eq(X_a,power_power(X_a,A_1,N),power_power(X_a,A_1,N_2)) ) ) ) ).
fof(fact_443_power__Suc__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less(X_a,zero_zero(X_a),A_1)
=> ( ord_less(X_a,A_1,one_one(X_a))
=> ord_less(X_a,times_times(X_a,A_1,power_power(X_a,A_1,N)),power_power(X_a,A_1,N)) ) ) ) ).
fof(fact_444_power__strict__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( ord_less(nat,N,N_2)
=> ( ord_less(X_a,zero_zero(X_a),A_1)
=> ( ord_less(X_a,A_1,one_one(X_a))
=> ord_less(X_a,power_power(X_a,A_1,N_2),power_power(X_a,A_1,N)) ) ) ) ) ).
fof(fact_445_power__eq__imp__eq__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B] :
( power_power(X_a,A_1,N) = power_power(X_a,B,N)
=> ( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less_eq(X_a,zero_zero(X_a),B)
=> ( ord_less(nat,zero_zero(nat),N)
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ) ).
fof(fact_446_power__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( ord_less_eq(nat,N,N_2)
=> ( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less_eq(X_a,A_1,one_one(X_a))
=> ord_less_eq(X_a,power_power(X_a,A_1,N_2),power_power(X_a,A_1,N)) ) ) ) ) ).
fof(fact_447_power__le__imp__le__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N,A_1] :
( ord_less(X_a,one_one(X_a),A_1)
=> ( ord_less_eq(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N))
=> ord_less_eq(nat,M,N) ) ) ) ).
fof(fact_448_power__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X_2,Y_2,B_1] :
( ord_less(X_a,one_one(X_a),B_1)
=> ( ord_less_eq(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2))
<=> ord_less_eq(nat,X_2,Y_2) ) ) ) ).
fof(fact_449_one__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( ord_less(X_a,one_one(X_a),A_1)
=> ( ord_less(nat,zero_zero(nat),N)
=> ord_less(X_a,one_one(X_a),power_power(X_a,A_1,N)) ) ) ) ).
fof(fact_450_dvd__power,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N] :
( ( ord_less(nat,zero_zero(nat),N)
| ti(X_a,X) = one_one(X_a) )
=> dvd_dvd(X_a,X,power_power(X_a,X,N)) ) ) ).
fof(fact_451_convex__bound__lt,axiom,
! [X_a] :
( linord626643107strict(X_a)
=> ! [V,U,Y,X,A_1] :
( ord_less(X_a,X,A_1)
=> ( ord_less(X_a,Y,A_1)
=> ( ord_less_eq(X_a,zero_zero(X_a),U)
=> ( ord_less_eq(X_a,zero_zero(X_a),V)
=> ( plus_plus(X_a,U,V) = one_one(X_a)
=> ord_less(X_a,plus_plus(X_a,times_times(X_a,U,X),times_times(X_a,V,Y)),A_1) ) ) ) ) ) ) ).
fof(fact_452_realpow__pos__nth__unique,axiom,
! [A_1,N] :
( ord_less(nat,zero_zero(nat),N)
=> ( ord_less(real,zero_zero(real),A_1)
=> ? [X_1] :
( ord_less(real,zero_zero(real),X_1)
& power_power(real,X_1,N) = A_1
& ! [Y_1] :
( ( ord_less(real,zero_zero(real),Y_1)
& power_power(real,Y_1,N) = A_1 )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_453_dvd__0__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : dvd_dvd(X_a,A_1,zero_zero(X_a)) ) ).
fof(fact_454_divides__ge,axiom,
! [A_1,B] :
( dvd_dvd(nat,A_1,B)
=> ( B = zero_zero(nat)
| ord_less_eq(nat,A_1,B) ) ) ).
fof(fact_455_divides__exp,axiom,
! [N,X,Y] :
( dvd_dvd(nat,X,Y)
=> dvd_dvd(nat,power_power(nat,X,N),power_power(nat,Y,N)) ) ).
fof(fact_456_divides__mul__l,axiom,
! [C,A_1,B] :
( dvd_dvd(nat,A_1,B)
=> dvd_dvd(nat,times_times(nat,C,A_1),times_times(nat,C,B)) ) ).
fof(fact_457_divides__mul__r,axiom,
! [C,A_1,B] :
( dvd_dvd(nat,A_1,B)
=> dvd_dvd(nat,times_times(nat,A_1,C),times_times(nat,B,C)) ) ).
fof(fact_458_divides__add__revr,axiom,
! [B,D,A_1] :
( dvd_dvd(nat,D,A_1)
=> ( dvd_dvd(nat,D,plus_plus(nat,A_1,B))
=> dvd_dvd(nat,D,B) ) ) ).
fof(fact_459_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma,K_1,N_1] :
( dvd_dvd(nat,times_times(nat,Ma,K_1),times_times(nat,N_1,K_1))
<=> ( K_1 = zero_zero(nat)
| dvd_dvd(nat,Ma,N_1) ) ) ).
fof(fact_460_divides__rev,axiom,
! [A_1,N,B] :
( dvd_dvd(nat,power_power(nat,A_1,N),power_power(nat,B,N))
=> ( N != zero_zero(nat)
=> dvd_dvd(nat,A_1,B) ) ) ).
fof(fact_461_divides__exp2,axiom,
! [X,Y,N] :
( N != zero_zero(nat)
=> ( dvd_dvd(nat,power_power(nat,X,N),Y)
=> dvd_dvd(nat,X,Y) ) ) ).
fof(fact_462_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ ord_less(X_a,X,Y)
=> ord_less(X_a,Y,X) ) ) ) ).
fof(fact_463_dvd__trans,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( dvd_dvd(X_a,A_1,B)
=> ( dvd_dvd(X_a,B,C)
=> dvd_dvd(X_a,A_1,C) ) ) ) ).
fof(fact_464_dvd__refl,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : dvd_dvd(X_a,A_1,A_1) ) ).
fof(fact_465_divides__div__not,axiom,
! [X,Q,N,R] :
( X = plus_plus(nat,times_times(nat,Q,N),R)
=> ( ord_less(nat,zero_zero(nat),R)
=> ( ord_less(nat,R,N)
=> ~ dvd_dvd(nat,N,X) ) ) ) ).
fof(fact_466_power__dvd__imp__le,axiom,
! [I,M,N] :
( dvd_dvd(nat,power_power(nat,I,M),power_power(nat,I,N))
=> ( ord_less(nat,one_one(nat),I)
=> ord_less_eq(nat,M,N) ) ) ).
fof(fact_467_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_468_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_469_mult__eq__0__iff,axiom,
! [X_a] :
( ring_n68954251visors(X_a)
=> ! [A_2,B_1] :
( times_times(X_a,A_2,B_1) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
| ti(X_a,B_1) = zero_zero(X_a) ) ) ) ).
fof(fact_470_no__zero__divisors,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> times_times(X_a,A_1,B) != zero_zero(X_a) ) ) ) ).
fof(fact_471_divisors__zero,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [A_1,B] :
( times_times(X_a,A_1,B) = zero_zero(X_a)
=> ( ti(X_a,A_1) = zero_zero(X_a)
| ti(X_a,B) = zero_zero(X_a) ) ) ) ).
fof(fact_472_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_473_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_474_comm__semiring__class_Odistrib,axiom,
! [X_a] :
( comm_semiring(X_a)
=> ! [A_1,B,C] : times_times(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B,C)) ) ).
fof(fact_475_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E,B,C] : plus_plus(X_a,times_times(X_a,A_1,E),plus_plus(X_a,times_times(X_a,B,E),C)) = plus_plus(X_a,times_times(X_a,plus_plus(X_a,A_1,B),E),C) ) ).
fof(fact_476_dvd__0__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] :
( dvd_dvd(X_a,zero_zero(X_a),A_1)
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_477_dvd__mult__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] :
( dvd_dvd(X_a,times_times(X_a,A_1,B),C)
=> dvd_dvd(X_a,B,C) ) ) ).
fof(fact_478_dvd__mult__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] :
( dvd_dvd(X_a,times_times(X_a,A_1,B),C)
=> dvd_dvd(X_a,A_1,C) ) ) ).
fof(fact_479_dvdI,axiom,
! [X_a] :
( dvd(X_a)
=> ! [A_1,B,K] :
( A_1 = times_times(X_a,B,K)
=> dvd_dvd(X_a,B,A_1) ) ) ).
fof(fact_480_mult__dvd__mono,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,D,A_1,B] :
( dvd_dvd(X_a,A_1,B)
=> ( dvd_dvd(X_a,C,D)
=> dvd_dvd(X_a,times_times(X_a,A_1,C),times_times(X_a,B,D)) ) ) ) ).
fof(fact_481_dvd__mult,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [B,A_1,C] :
( dvd_dvd(X_a,A_1,C)
=> dvd_dvd(X_a,A_1,times_times(X_a,B,C)) ) ) ).
fof(fact_482_dvd__mult2,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( dvd_dvd(X_a,A_1,B)
=> dvd_dvd(X_a,A_1,times_times(X_a,B,C)) ) ) ).
fof(fact_483_dvd__triv__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : dvd_dvd(X_a,A_1,times_times(X_a,B,A_1)) ) ).
fof(fact_484_dvd__triv__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : dvd_dvd(X_a,A_1,times_times(X_a,A_1,B)) ) ).
fof(fact_485_dvd__add,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( dvd_dvd(X_a,A_1,B)
=> ( dvd_dvd(X_a,A_1,C)
=> dvd_dvd(X_a,A_1,plus_plus(X_a,B,C)) ) ) ) ).
fof(fact_486_one__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : dvd_dvd(X_a,one_one(X_a),A_1) ) ).
fof(fact_487_dvd__diff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [Z,X,Y] :
( dvd_dvd(X_a,X,Y)
=> ( dvd_dvd(X_a,X,Z)
=> dvd_dvd(X_a,X,minus_minus(X_a,Y,Z)) ) ) ) ).
fof(fact_488_divides__cases,axiom,
! [N,M] :
( dvd_dvd(nat,N,M)
=> ( M = zero_zero(nat)
| M = N
| ord_less_eq(nat,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N),M) ) ) ).
fof(fact_489_zero__le__square,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_1,A_1)) ) ).
fof(fact_490_zero__le__mult__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_1] :
( ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_2,B_1))
<=> ( ( ord_less_eq(X_a,zero_zero(X_a),A_2)
& ord_less_eq(X_a,zero_zero(X_a),B_1) )
| ( ord_less_eq(X_a,A_2,zero_zero(X_a))
& ord_less_eq(X_a,B_1,zero_zero(X_a)) ) ) ) ) ).
fof(fact_491_mult__le__0__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_1] :
( ord_less_eq(X_a,times_times(X_a,A_2,B_1),zero_zero(X_a))
<=> ( ( ord_less_eq(X_a,zero_zero(X_a),A_2)
& ord_less_eq(X_a,B_1,zero_zero(X_a)) )
| ( ord_less_eq(X_a,A_2,zero_zero(X_a))
& ord_less_eq(X_a,zero_zero(X_a),B_1) ) ) ) ) ).
fof(fact_492_mult__nonneg__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less_eq(X_a,zero_zero(X_a),B)
=> ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_1,B)) ) ) ) ).
fof(fact_493_mult__nonneg__nonpos,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less_eq(X_a,B,zero_zero(X_a))
=> ord_less_eq(X_a,times_times(X_a,A_1,B),zero_zero(X_a)) ) ) ) ).
fof(fact_494_mult__nonneg__nonpos2,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( ord_less_eq(X_a,zero_zero(X_a),A_1)
=> ( ord_less_eq(X_a,B,zero_zero(X_a))
=> ord_less_eq(X_a,times_times(X_a,B,A_1),zero_zero(X_a)) ) ) ) ).
fof(fact_495_mult__nonpos__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( ord_less_eq(X_a,A_1,zero_zero(X_a))
=> ( ord_less_eq(X_a,zero_zero(X_a),B)
=> ord_less_eq(X_a,times_times(X_a,A_1,B),zero_zero(X_a)) ) ) ) ).
fof(fact_496_mult__nonpos__nonpos,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [B,A_1] :
( ord_less_eq(X_a,A_1,zero_zero(X_a))
=> ( ord_less_eq(X_a,B,zero_zero(X_a))
=> ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_1,B)) ) ) ) ).
%----Arities (86)
fof(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_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_Olinordered__semidom,axiom,
linordered_semidom(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___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___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
fof(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(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___Rings_Oring,axiom,
ring(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(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___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___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___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___Int_Onumber,axiom,
number(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___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(real) ).
fof(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_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_Olinordered__semidom,axiom,
linordered_semidom(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(real) ).
fof(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___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___Rings_Osemiring__1,axiom,
semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
semiring_0(real) ).
fof(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(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___Rings_Oring,axiom,
ring(real) ).
fof(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
fof(arity_RealDef_Oreal___Rings_Odvd,axiom,
dvd(real) ).
%----Helper facts (1)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
%----Conjectures (1)
fof(conj_0,conjecture,
ord_less(int,plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)),zero_zero(int)) ).
%------------------------------------------------------------------------------