TPTP Problem File: NUM925+6.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925+6 : TPTP v9.0.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 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_l192 [Bla11]
% Status : Theorem
% Rating : 0.18 v9.0.0, 0.19 v8.2.0, 0.22 v7.4.0, 0.10 v7.3.0, 0.17 v7.2.0, 0.14 v7.1.0, 0.17 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.21 v6.2.0, 0.28 v6.1.0, 0.43 v6.0.0, 0.35 v5.5.0, 0.48 v5.4.0, 0.50 v5.3.0
% Syntax : Number of formulae : 660 ( 228 unt; 0 def)
% Number of atoms : 1567 ( 334 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 1012 ( 105 ~; 27 |; 69 &)
% ( 146 <=>; 665 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 41 ( 40 usr; 0 prp; 1-2 aty)
% Number of functors : 29 ( 29 usr; 15 con; 0-4 aty)
% Number of variables : 1326 (1310 !; 16 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:18:40
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (45)
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_a] :
( one(X_a)
=> ti(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_arg1,axiom,
! [B_1_1,X_a] :
( monoid_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [B_1_1,X_a] :
( monoid_add(X_a)
=> ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_arg1,axiom,
! [B_1_1,X_a] :
( ab_semigroup_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [B_1_1,X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_arg1,axiom,
! [B_1_1,X_a] :
( cancel_semigroup_add(X_a)
=> plus_plus(X_a,ti(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
! [B_1_1,X_a] :
( cancel_semigroup_add(X_a)
=> ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)) = plus_plus(X_a,B_1_1) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_res,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_res,axiom,
! [X_a] :
( zero(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_If_res,axiom,
! [X_a] : ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a) ).
fof(tsy_c_Int_OBit0_res,hypothesis,
ti(fun(int,int),bit0) = bit0 ).
fof(tsy_c_Int_OBit1_res,hypothesis,
ti(fun(int,int),bit1) = bit1 ).
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_Int_Osucc_res,axiom,
ti(fun(int,int),succ) = succ ).
fof(tsy_c_Nat_Osemiring__1__class_Oof__nat_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(nat,X_a),semiring_1_of_nat(X_a)) = semiring_1_of_nat(X_a) ) ).
fof(tsy_c_NthRoot_Osqrt_res,axiom,
ti(fun(real,real),sqrt) = sqrt ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg1,axiom,
! [B_1_1,X_a] :
( power(X_a)
=> power_power(X_a,ti(X_a,B_1_1)) = power_power(X_a,B_1_1) ) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [B_1_1,X_a] :
( power(X_a)
=> ti(fun(nat,X_a),power_power(X_a,B_1_1)) = power_power(X_a,B_1_1) ) ).
fof(tsy_c_fFalse_res,axiom,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_hAPP_0_arg1,axiom,
! [X_a,B_1_1,B_2,X_b] :
( ord(X_b)
=> hAPP(X_a,X_b,ti(fun(X_a,X_b),B_1_1),B_2) = hAPP(X_a,X_b,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_0_arg2,axiom,
! [X_a,B_1_1,B_2,X_b] :
( ord(X_b)
=> hAPP(X_a,X_b,B_1_1,ti(X_a,B_2)) = hAPP(X_a,X_b,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_0_res,axiom,
! [X_a,B_1_1,B_2,X_b] :
( ord(X_b)
=> ti(X_b,hAPP(X_a,X_b,B_1_1,B_2)) = hAPP(X_a,X_b,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_1_arg1,axiom,
! [B_1_1,B_2,X_a] :
( power(X_a)
=> hAPP(nat,X_a,ti(fun(nat,X_a),B_1_1),B_2) = hAPP(nat,X_a,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_1_arg2,axiom,
! [B_1_1,B_2,X_a] :
( power(X_a)
=> hAPP(nat,X_a,B_1_1,ti(nat,B_2)) = hAPP(nat,X_a,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_1_res,axiom,
! [B_1_1,B_2,X_a] :
( power(X_a)
=> ti(X_a,hAPP(nat,X_a,B_1_1,B_2)) = hAPP(nat,X_a,B_1_1,B_2) ) ).
fof(tsy_c_hAPP_2_arg1,axiom,
! [X_a,B_1_1,B_2] : hAPP(bool,fun(X_a,fun(X_a,X_a)),ti(fun(bool,fun(X_a,fun(X_a,X_a))),B_1_1),B_2) = hAPP(bool,fun(X_a,fun(X_a,X_a)),B_1_1,B_2) ).
fof(tsy_c_hAPP_2_arg2,axiom,
! [X_a,B_1_1,B_2] : hAPP(bool,fun(X_a,fun(X_a,X_a)),B_1_1,ti(bool,B_2)) = hAPP(bool,fun(X_a,fun(X_a,X_a)),B_1_1,B_2) ).
fof(tsy_c_hAPP_2_res,axiom,
! [X_a,B_1_1,B_2] : ti(fun(X_a,fun(X_a,X_a)),hAPP(bool,fun(X_a,fun(X_a,X_a)),B_1_1,B_2)) = hAPP(bool,fun(X_a,fun(X_a,X_a)),B_1_1,B_2) ).
fof(tsy_c_hAPP_3_arg1,axiom,
! [X_a,B_1_1,B_2] : hAPP(X_a,fun(X_a,X_a),ti(fun(X_a,fun(X_a,X_a)),B_1_1),B_2) = hAPP(X_a,fun(X_a,X_a),B_1_1,B_2) ).
fof(tsy_c_hAPP_3_arg2,axiom,
! [X_a,B_1_1,B_2] : hAPP(X_a,fun(X_a,X_a),B_1_1,ti(X_a,B_2)) = hAPP(X_a,fun(X_a,X_a),B_1_1,B_2) ).
fof(tsy_c_hAPP_3_res,axiom,
! [X_a,B_1_1,B_2] : ti(fun(X_a,X_a),hAPP(X_a,fun(X_a,X_a),B_1_1,B_2)) = hAPP(X_a,fun(X_a,X_a),B_1_1,B_2) ).
fof(tsy_c_hAPP_4_arg1,axiom,
! [X_a,B_1_1,B_2] : hAPP(X_a,X_a,ti(fun(X_a,X_a),B_1_1),B_2) = hAPP(X_a,X_a,B_1_1,B_2) ).
fof(tsy_c_hAPP_4_arg2,axiom,
! [X_a,B_1_1,B_2] : hAPP(X_a,X_a,B_1_1,ti(X_a,B_2)) = hAPP(X_a,X_a,B_1_1,B_2) ).
fof(tsy_c_hAPP_4_res,axiom,
! [X_a,B_1_1,B_2] : ti(X_a,hAPP(X_a,X_a,B_1_1,B_2)) = hAPP(X_a,X_a,B_1_1,B_2) ).
fof(tsy_c_hBOOL_arg1,axiom,
! [B_1_1] :
( hBOOL(ti(bool,B_1_1))
<=> hBOOL(B_1_1) ) ).
fof(tsy_v_m1_____res,axiom,
ti(int,m1) = m1 ).
fof(tsy_v_n_____res,hypothesis,
ti(nat,n) = n ).
fof(tsy_v_t_____res,axiom,
ti(int,t) = t ).
fof(tsy_v_tn_____res,axiom,
ti(nat,tn) = tn ).
%----Relevant facts (496)
fof(fact_0_n1pos,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).
fof(fact_1_t1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) ).
fof(fact_2_sum__power2__eq__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(X_a)
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_3_one__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,power_power(X_a,one_one(X_a)),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a) ) ).
fof(fact_4_zero__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a) ) ).
fof(fact_5_zero__eq__power2,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [A_2] :
( hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_6_add__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W_1] : hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,W_1)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,hAPP(int,int,bit1,pls)),W_1)) ) ).
fof(fact_7_add__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1] : hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V_1),hAPP(int,int,bit1,pls))) ) ).
fof(fact_8_one__add__one__is__two,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).
fof(fact_9_semiring__one__add__one__is__two,axiom,
! [X_a] :
( number_semiring(X_a)
=> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).
fof(fact_10_quartic__square__square,axiom,
! [X] : hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,int,power_power(int,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_11_power__0__left__number__of,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [W_1] :
( ( number_number_of(nat,W_1) = zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W_1)) = one_one(X_a) )
& ( number_number_of(nat,W_1) != zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W_1)) = zero_zero(X_a) ) ) ) ).
fof(fact_12_semiring__norm_I110_J,axiom,
! [X_a] :
( number_ring(X_a)
=> one_one(X_a) = number_number_of(X_a,hAPP(int,int,bit1,pls)) ) ).
fof(fact_13_numeral__1__eq__1,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).
fof(fact_14_n0,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),n)) ).
fof(fact_15_zless__linear,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y))
| X = Y
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) ) ).
fof(fact_16_less__number__of__int__code,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,K_2)),number_number_of(int,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),L_1)) ) ).
fof(fact_17_plus__numeral__code_I9_J,axiom,
! [V_1,W_1] : hAPP(int,int,plus_plus(int,number_number_of(int,V_1)),number_number_of(int,W_1)) = number_number_of(int,hAPP(int,int,plus_plus(int,V_1),W_1)) ).
fof(fact_18_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ) ).
fof(fact_19_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
fof(fact_20_zpower__int,axiom,
! [M_1,N] : hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M_1)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M_1),N)) ).
fof(fact_21_int__power,axiom,
! [M_1,N] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M_1),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M_1)),N) ).
fof(fact_22_zadd__int__left,axiom,
! [M_1,N,Z_1] : hAPP(int,int,plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(int,int,plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),N)),Z_1)) = hAPP(int,int,plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,plus_plus(nat,M_1),N))),Z_1) ).
fof(fact_23_zadd__int,axiom,
! [M_1,N] : hAPP(int,int,plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,plus_plus(nat,M_1),N)) ).
fof(fact_24_int__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
fof(fact_25_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
fof(fact_26_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
fof(fact_27_int__eq__0__conv,axiom,
! [Na] :
( hAPP(nat,int,semiring_1_of_nat(int),Na) = zero_zero(int)
<=> Na = zero_zero(nat) ) ).
fof(fact_28_int__0,axiom,
hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).
fof(fact_29_nat__1__add__1,axiom,
hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) = number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ).
fof(fact_30_less__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_31_rel__simps_I17_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K_2)),hAPP(int,int,bit1,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),L_1)) ) ).
fof(fact_32_rel__simps_I2_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),pls)) ).
fof(fact_33_less__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_34_rel__simps_I14_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K_2)),hAPP(int,int,bit0,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),L_1)) ) ).
fof(fact_35_zadd__strict__right__mono,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,I_2),K)),hAPP(int,int,plus_plus(int,J_2),K))) ) ).
fof(fact_36_add__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V_1)),number_number_of(nat,V_2)) = number_number_of(nat,V_2) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
=> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V_1)),number_number_of(nat,V_2)) = number_number_of(nat,V_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
=> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V_1)),number_number_of(nat,V_2)) = number_number_of(nat,hAPP(int,int,plus_plus(int,V_1),V_2)) ) ) ) ) ).
fof(fact_37_one__is__num__one,axiom,
one_one(int) = number_number_of(int,hAPP(int,int,bit1,pls)) ).
fof(fact_38_nat__numeral__1__eq__1,axiom,
number_number_of(nat,hAPP(int,int,bit1,pls)) = one_one(nat) ).
fof(fact_39_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,hAPP(int,int,bit1,pls)) ).
fof(fact_40_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [Xa,Ya] :
( number_number_of(X_a,Xa) = number_number_of(X_a,Ya)
<=> Xa = Ya ) ) ).
fof(fact_41_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [Wa,Xa] :
( number_number_of(X_a,Wa) = ti(X_a,Xa)
<=> ti(X_a,Xa) = number_number_of(X_a,Wa) ) ) ).
fof(fact_42_rel__simps_I51_J,axiom,
! [K_2,L_1] :
( hAPP(int,int,bit1,K_2) = hAPP(int,int,bit1,L_1)
<=> K_2 = L_1 ) ).
fof(fact_43_rel__simps_I48_J,axiom,
! [K_2,L_1] :
( hAPP(int,int,bit0,K_2) = hAPP(int,int,bit0,L_1)
<=> K_2 = L_1 ) ).
fof(fact_44_even__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_45_zadd__assoc,axiom,
! [Z1,Z2,Z3] : hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,Z1),Z2)),Z3) = hAPP(int,int,plus_plus(int,Z1),hAPP(int,int,plus_plus(int,Z2),Z3)) ).
fof(fact_46_zadd__left__commute,axiom,
! [X,Y,Z_1] : hAPP(int,int,plus_plus(int,X),hAPP(int,int,plus_plus(int,Y),Z_1)) = hAPP(int,int,plus_plus(int,Y),hAPP(int,int,plus_plus(int,X),Z_1)) ).
fof(fact_47_zadd__commute,axiom,
! [Z_1,W_1] : hAPP(int,int,plus_plus(int,Z_1),W_1) = hAPP(int,int,plus_plus(int,W_1),Z_1) ).
fof(fact_48_int__int__eq,axiom,
! [Ma,Na] :
( hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(nat,int,semiring_1_of_nat(int),Na)
<=> Ma = Na ) ).
fof(fact_49_less__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),pls)) ) ) ).
fof(fact_50_less__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Ya)) ) ) ).
fof(fact_51_rel__simps_I12_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K_2)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),pls)) ) ).
fof(fact_52_less__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_53_rel__simps_I16_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K_2)),hAPP(int,int,bit0,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),L_1)) ) ).
fof(fact_54_rel__simps_I10_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K_2)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),pls)) ) ).
fof(fact_55_rel__simps_I4_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit0,K_2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K_2)) ) ).
fof(fact_56_bin__less__0__simps_I4_J,axiom,
! [Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,Wa)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).
fof(fact_57_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),zero_zero(int))) ).
fof(fact_58_bin__less__0__simps_I3_J,axiom,
! [Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,Wa)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).
fof(fact_59_int__0__less__1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))) ).
fof(fact_60_zless__add1__eq,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,plus_plus(int,Z_2),one_one(int))))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_2))
| Wa = Z_2 ) ) ).
fof(fact_61_int__less__0__conv,axiom,
! [K] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K)),zero_zero(int))) ).
fof(fact_62_less__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),hAPP(int,int,bit1,pls))) ) ) ).
fof(fact_63_less__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),Ya)) ) ) ).
fof(fact_64_odd__less__0,axiom,
! [Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_2)),Z_2)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),zero_zero(int))) ) ).
fof(fact_65_double__eq__0__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_66_rel__simps_I46_J,axiom,
! [K] : hAPP(int,int,bit1,K) != pls ).
fof(fact_67_rel__simps_I39_J,axiom,
! [L] : pls != hAPP(int,int,bit1,L) ).
fof(fact_68_rel__simps_I50_J,axiom,
! [K,L] : hAPP(int,int,bit1,K) != hAPP(int,int,bit0,L) ).
fof(fact_69_rel__simps_I49_J,axiom,
! [K,L] : hAPP(int,int,bit0,K) != hAPP(int,int,bit1,L) ).
fof(fact_70_rel__simps_I44_J,axiom,
! [K_2] :
( hAPP(int,int,bit0,K_2) = pls
<=> K_2 = pls ) ).
fof(fact_71_rel__simps_I38_J,axiom,
! [L_1] :
( pls = hAPP(int,int,bit0,L_1)
<=> pls = L_1 ) ).
fof(fact_72_Bit0__Pls,axiom,
hAPP(int,int,bit0,pls) = pls ).
fof(fact_73_Pls__def,axiom,
pls = zero_zero(int) ).
fof(fact_74_int__0__neq__1,axiom,
zero_zero(int) != one_one(int) ).
fof(fact_75_add__Pls__right,axiom,
! [K] : hAPP(int,int,plus_plus(int,K),pls) = K ).
fof(fact_76_add__Pls,axiom,
! [K] : hAPP(int,int,plus_plus(int,pls),K) = K ).
fof(fact_77_add__Bit0__Bit0,axiom,
! [K,L] : hAPP(int,int,plus_plus(int,hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L)) = hAPP(int,int,bit0,hAPP(int,int,plus_plus(int,K),L)) ).
fof(fact_78_Bit0__def,axiom,
! [K] : hAPP(int,int,bit0,K) = hAPP(int,int,plus_plus(int,K),K) ).
fof(fact_79_zadd__0__right,axiom,
! [Z_1] : hAPP(int,int,plus_plus(int,Z_1),zero_zero(int)) = Z_1 ).
fof(fact_80_zadd__0,axiom,
! [Z_1] : hAPP(int,int,plus_plus(int,zero_zero(int)),Z_1) = Z_1 ).
fof(fact_81_semiring__numeral__0__eq__0,axiom,
! [X_a] :
( number_semiring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_82_number__of__Pls,axiom,
! [X_a] :
( number_ring(X_a)
=> number_number_of(X_a,pls) = zero_zero(X_a) ) ).
fof(fact_83_semiring__norm_I112_J,axiom,
! [X_a] :
( number_ring(X_a)
=> zero_zero(X_a) = number_number_of(X_a,pls) ) ).
fof(fact_84_add__numeral__0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,pls)),A_1) = ti(X_a,A_1) ) ).
fof(fact_85_add__numeral__0__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,A_1),number_number_of(X_a,pls)) = ti(X_a,A_1) ) ).
fof(fact_86_power__eq__0__iff__number__of,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [A_2,Wa] :
( hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,Wa)) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& number_number_of(nat,Wa) != zero_zero(nat) ) ) ) ).
fof(fact_87_add__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W_1,Z_1] : hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,W_1)),Z_1)) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,hAPP(int,int,plus_plus(int,V_1),W_1))),Z_1) ) ).
fof(fact_88_add__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W_1] : hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),number_number_of(X_a,W_1)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V_1),W_1)) ) ).
fof(fact_89_number__of__add,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1,W_1] : number_number_of(X_a,hAPP(int,int,plus_plus(int,V_1),W_1)) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),number_number_of(X_a,W_1)) ) ).
fof(fact_90_add__Bit1__Bit0,axiom,
! [K,L] : hAPP(int,int,plus_plus(int,hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L)) = hAPP(int,int,bit1,hAPP(int,int,plus_plus(int,K),L)) ).
fof(fact_91_add__Bit0__Bit1,axiom,
! [K,L] : hAPP(int,int,plus_plus(int,hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L)) = hAPP(int,int,bit1,hAPP(int,int,plus_plus(int,K),L)) ).
fof(fact_92_Bit1__def,axiom,
! [K] : hAPP(int,int,bit1,K) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K)),K) ).
fof(fact_93_odd__nonzero,axiom,
! [Z_1] : hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_1)),Z_1) != zero_zero(int) ).
fof(fact_94_number__of__int,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [N] : number_number_of(X_a,hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).
fof(fact_95_zero__less__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
<=> ti(X_a,A_2) != zero_zero(X_a) ) ) ).
fof(fact_96_power2__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(X_a))) ) ).
fof(fact_97_sum__power2__gt__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
<=> ( ti(X_a,Xa) != zero_zero(X_a)
| ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).
fof(fact_98_not__sum__power2__lt__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a))) ) ).
fof(fact_99_number__of__Bit0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W_1] : number_number_of(X_a,hAPP(int,int,bit0,W_1)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),number_number_of(X_a,W_1))),number_number_of(X_a,W_1)) ) ).
fof(fact_100_number__of__Bit1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W_1] : number_number_of(X_a,hAPP(int,int,bit1,W_1)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,W_1))),number_number_of(X_a,W_1)) ) ).
fof(fact_101_semiring__numeral__1__eq__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> number_number_of(X_a,hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).
fof(fact_102_mn,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),m1),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).
fof(fact_103_of__nat__less__two__power,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) ) ).
fof(fact_104_transfer__int__nat__numerals_I3_J,axiom,
number_number_of(int,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_105_transfer__int__nat__numerals_I4_J,axiom,
number_number_of(int,hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) = hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).
fof(fact_106_realpow__two__sum__zero__iff,axiom,
! [Xa,Ya] :
( hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,Xa),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,power_power(real,Ya),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real)
<=> ( Xa = zero_zero(real)
& Ya = zero_zero(real) ) ) ).
fof(fact_107_of__nat__0__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).
fof(fact_108_one__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ).
fof(fact_109_power__0__left,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [N] :
( ( N = zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = one_one(X_a) )
& ( N != zero_zero(nat)
=> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = zero_zero(X_a) ) ) ) ).
fof(fact_110_power__strict__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ) ).
fof(fact_111_zero__less__two,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)))) ) ).
fof(fact_112_int__gr__induct,axiom,
! [P_1,K_2,I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),I))
=> ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,K_2),one_one(int))))
=> ( ! [I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),I_1))
=> ( hBOOL(hAPP(int,bool,P_1,I_1))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,I_1),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).
fof(fact_113_tn0,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),tn)) ).
fof(fact_114_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_115_real__zero__not__eq__one,axiom,
zero_zero(real) != one_one(real) ).
fof(fact_116_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).
fof(fact_117_not__add__less1,axiom,
! [I_2,J_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),J_2)),I_2)) ).
fof(fact_118_not__add__less2,axiom,
! [J_2,I_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,J_2),I_2)),I_2)) ).
fof(fact_119_number__of__is__id,axiom,
! [K] : number_number_of(int,K) = K ).
fof(fact_120_nat__neq__iff,axiom,
! [Ma,Na] :
( Ma != Na
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ) ).
fof(fact_121_nat__add__commute,axiom,
! [M_1,N] : hAPP(nat,nat,plus_plus(nat,M_1),N) = hAPP(nat,nat,plus_plus(nat,N),M_1) ).
fof(fact_122_nat__add__left__commute,axiom,
! [X,Y,Z_1] : hAPP(nat,nat,plus_plus(nat,X),hAPP(nat,nat,plus_plus(nat,Y),Z_1)) = hAPP(nat,nat,plus_plus(nat,Y),hAPP(nat,nat,plus_plus(nat,X),Z_1)) ).
fof(fact_123_nat__add__assoc,axiom,
! [M_1,N,K] : hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,M_1),N)),K) = hAPP(nat,nat,plus_plus(nat,M_1),hAPP(nat,nat,plus_plus(nat,N),K)) ).
fof(fact_124_nat__add__left__cancel,axiom,
! [K_2,Ma,Na] :
( hAPP(nat,nat,plus_plus(nat,K_2),Ma) = hAPP(nat,nat,plus_plus(nat,K_2),Na)
<=> Ma = Na ) ).
fof(fact_125_nat__add__right__cancel,axiom,
! [Ma,K_2,Na] :
( hAPP(nat,nat,plus_plus(nat,Ma),K_2) = hAPP(nat,nat,plus_plus(nat,Na),K_2)
<=> Ma = Na ) ).
fof(fact_126_nat__add__left__cancel__less,axiom,
! [K_2,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,K_2),Ma)),hAPP(nat,nat,plus_plus(nat,K_2),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).
fof(fact_127_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
fof(fact_128_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).
fof(fact_129_less__not__refl2,axiom,
! [N,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M_1))
=> M_1 != N ) ).
fof(fact_130_less__not__refl3,axiom,
! [S,T_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S),T_3))
=> S != T_3 ) ).
fof(fact_131_trans__less__add1,axiom,
! [M_1,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),hAPP(nat,nat,plus_plus(nat,J_2),M_1))) ) ).
fof(fact_132_trans__less__add2,axiom,
! [M_1,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),hAPP(nat,nat,plus_plus(nat,M_1),J_2))) ) ).
fof(fact_133_add__less__mono1,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),hAPP(nat,nat,plus_plus(nat,J_2),K))) ) ).
fof(fact_134_add__less__mono,axiom,
! [K,L,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),hAPP(nat,nat,plus_plus(nat,J_2),L))) ) ) ).
fof(fact_135_less__add__eq__less,axiom,
! [M_1,N,K,L] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K),L))
=> ( hAPP(nat,nat,plus_plus(nat,M_1),L) = hAPP(nat,nat,plus_plus(nat,K),N)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ).
fof(fact_136_add__lessD1,axiom,
! [I_2,J_2,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),J_2)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),K)) ) ).
fof(fact_137_nat__less__cases,axiom,
! [P_1,Ma,Na] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> ( ( Ma = Na
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) ) ) ) ).
fof(fact_138_gr0I,axiom,
! [N] :
( N != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_139_gr__implies__not0,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
=> N != zero_zero(nat) ) ).
fof(fact_140_nat__power__less__imp__less,axiom,
! [M_1,N,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,I_2),M_1)),hAPP(nat,nat,power_power(nat,I_2),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ).
fof(fact_141_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_142_nat__zero__less__power__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa))
| Na = zero_zero(nat) ) ) ).
fof(fact_143_add__gr__0,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,Ma),Na)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).
fof(fact_144_neq0__conv,axiom,
! [Na] :
( Na != zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
fof(fact_145_not__less0,axiom,
! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
fof(fact_146_zero__less__power__nat__eq,axiom,
! [Xa,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na)))
<=> ( Na = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).
fof(fact_147_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ).
fof(fact_148_zero__less__power__nat__eq__number__of,axiom,
! [Xa,Wa] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),number_number_of(nat,Wa))))
<=> ( number_number_of(nat,Wa) = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).
fof(fact_149_zless__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).
fof(fact_150_less__nat__number__of,axiom,
! [Va,V] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,Va)),number_number_of(nat,V)))
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V)) )
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V)) ) ) ).
fof(fact_151_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_152_add__eq__self__zero,axiom,
! [M_1,N] :
( hAPP(nat,nat,plus_plus(nat,M_1),N) = M_1
=> N = zero_zero(nat) ) ).
fof(fact_153_add__is__0,axiom,
! [Ma,Na] :
( hAPP(nat,nat,plus_plus(nat,Ma),Na) = zero_zero(nat)
<=> ( Ma = zero_zero(nat)
& Na = zero_zero(nat) ) ) ).
fof(fact_154_Nat_Oadd__0__right,axiom,
! [M_1] : hAPP(nat,nat,plus_plus(nat,M_1),zero_zero(nat)) = M_1 ).
fof(fact_155_plus__nat_Oadd__0,axiom,
! [N] : hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),N) = N ).
fof(fact_156_power__one__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat)) = ti(X_a,A_1) ) ).
fof(fact_157_of__nat__eq__iff,axiom,
! [X_a] :
( semiring_char_0(X_a)
=> ! [Ma,Na] :
( hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma) = hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)
<=> Ma = Na ) ) ).
fof(fact_158_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Xa,Ya] :
( hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya)
<=> Xa = Ya ) ).
fof(fact_159_int__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya)) )
& ( ~ hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya)) ) ) ).
fof(fact_160_less__0__number__of,axiom,
! [Va] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,Va)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)) ) ).
fof(fact_161_zero__less__int__conv,axiom,
! [Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
fof(fact_162_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_163_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_164_field__power__not__zero,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [N,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(nat,X_a,power_power(X_a,A_1),N) != zero_zero(X_a) ) ) ).
fof(fact_165_power__one,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [N] : hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N) = one_one(X_a) ) ).
fof(fact_166_of__nat__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).
fof(fact_167_less__imp__of__nat__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ) ).
fof(fact_168_of__nat__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ).
fof(fact_169_of__nat__add,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,plus_plus(nat,M_1),N)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ).
fof(fact_170_of__nat__1,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a) ) ).
fof(fact_171_of__nat__power,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,power_power(nat,M_1),N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),N) ) ).
fof(fact_172_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).
fof(fact_173_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP(int,int,plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,plus_plus(nat,X),Y)) ).
fof(fact_174_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N] : hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),X)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,X),N)) ).
fof(fact_175_pos__add__strict,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [B,C_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1))) ) ) ) ).
fof(fact_176_not__one__less__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_177_zero__less__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_178_zero__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_179_less__add__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a)))) ) ).
fof(fact_180_power__inject__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_2))
=> ( hAPP(nat,X_a,power_power(X_a,A_2),Ma) = hAPP(nat,X_a,power_power(X_a,A_2),Na)
<=> Ma = Na ) ) ) ).
fof(fact_181_power__strict__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Xa,Ya,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,B_1),Xa)),hAPP(nat,X_a,power_power(X_a,B_1),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ) ) ).
fof(fact_182_power__less__imp__less__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M_1)),hAPP(nat,X_a,power_power(X_a,A_1),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ) ).
fof(fact_183_power__strict__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,A_1),N_2))) ) ) ) ).
fof(fact_184_power__eq__0__iff,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [A_2,Na] :
( hAPP(nat,X_a,power_power(X_a,A_2),Na) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
& Na != zero_zero(nat) ) ) ) ).
fof(fact_185_of__nat__less__0__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),zero_zero(X_a))) ) ).
fof(fact_186_power__0,axiom,
! [X_a] :
( power(X_a)
=> ! [A_1] : hAPP(nat,X_a,power_power(X_a,A_1),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_187_of__nat__0,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a) ) ).
fof(fact_188_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) ).
fof(fact_189_pos2,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_190_zero__less__imp__eq__int,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ? [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
& K = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ) ).
fof(fact_191_less__imp__add__positive,axiom,
! [I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
& hAPP(nat,nat,plus_plus(nat,I_2),K_1) = J_2 ) ) ).
fof(fact_192_exp__eq__1,axiom,
! [Xa,Na] :
( hAPP(nat,nat,power_power(nat,Xa),Na) = one_one(nat)
<=> ( Xa = one_one(nat)
| Na = zero_zero(nat) ) ) ).
fof(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,power_power(X_a,X),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_194_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_195_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_196_zero__reorient,axiom,
! [X_a] :
( zero(X_a)
=> ! [Xa] :
( zero_zero(X_a) = ti(X_a,Xa)
<=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).
fof(fact_197_add__right__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B,A_1,C_1] :
( hAPP(X_a,X_a,plus_plus(X_a,B),A_1) = hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1)
=> ti(X_a,B) = ti(X_a,C_1) ) ) ).
fof(fact_198_add__imp__eq,axiom,
! [X_a] :
( cancel146912293up_add(X_a)
=> ! [A_1,B,C_1] :
( hAPP(X_a,X_a,plus_plus(X_a,A_1),B) = hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)
=> ti(X_a,B) = ti(X_a,C_1) ) ) ).
fof(fact_199_add__left__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_1,B,C_1] :
( hAPP(X_a,X_a,plus_plus(X_a,A_1),B) = hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)
=> ti(X_a,B) = ti(X_a,C_1) ) ) ).
fof(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C_1,D] : hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),hAPP(X_a,X_a,plus_plus(X_a,C_1),D)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),D)) ) ).
fof(fact_201_add__right__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B_1,A_2,C] :
( hAPP(X_a,X_a,plus_plus(X_a,B_1),A_2) = hAPP(X_a,X_a,plus_plus(X_a,C),A_2)
<=> ti(X_a,B_1) = ti(X_a,C) ) ) ).
fof(fact_202_add__left__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_2,B_1,C] :
( hAPP(X_a,X_a,plus_plus(X_a,A_2),B_1) = hAPP(X_a,X_a,plus_plus(X_a,A_2),C)
<=> ti(X_a,B_1) = ti(X_a,C) ) ) ).
fof(fact_203_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C_1] : hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C_1) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),B) ) ).
fof(fact_204_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ! [A_1,B,C_1] : hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C_1) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C_1)) ) ).
fof(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C_1] : hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C_1) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C_1)) ) ).
fof(fact_206_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C_1,D] : hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,C_1),D)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),D) ) ).
fof(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C_1,D] : hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,C_1),D)) = hAPP(X_a,X_a,plus_plus(X_a,C_1),hAPP(X_a,X_a,plus_plus(X_a,A_1),D)) ) ).
fof(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C_1] : hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1) = hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1) ) ).
fof(fact_209_one__reorient,axiom,
! [X_a] :
( one(X_a)
=> ! [Xa] :
( one_one(X_a) = ti(X_a,Xa)
<=> ti(X_a,Xa) = one_one(X_a) ) ) ).
fof(fact_210_add__0__iff,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [B_1,A_2] :
( ti(X_a,B_1) = hAPP(X_a,X_a,plus_plus(X_a,B_1),A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_211_add_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_213_add__0__right,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_214_double__zero__sym,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( zero_zero(X_a) = hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_215_add__0,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_217_add__0__left,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_218_add__less__imp__less__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_219_add__less__imp__less__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),C_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_220_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),D))) ) ) ) ).
fof(fact_221_add__strict__left__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B))) ) ) ).
fof(fact_222_add__strict__right__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),C_1))) ) ) ).
fof(fact_223_add__less__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_2,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_2)),hAPP(X_a,X_a,plus_plus(X_a,C),B_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_1)) ) ) ).
fof(fact_224_add__less__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_2),C)),hAPP(X_a,X_a,plus_plus(X_a,B_1),C)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_1)) ) ) ).
fof(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,power_power(X_a,X),one_one(nat)) = ti(X_a,X) ) ).
fof(fact_226_nat__power__eq__0__iff,axiom,
! [Ma,Na] :
( hAPP(nat,nat,power_power(nat,Ma),Na) = zero_zero(nat)
<=> ( Na != zero_zero(nat)
& Ma = zero_zero(nat) ) ) ).
fof(fact_227_add__neg__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_228_add__pos__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B))) ) ) ) ).
fof(fact_229_realpow__pos__nth__unique,axiom,
! [A_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
& hAPP(nat,real,power_power(real,X_1),N) = A_1
& ! [Y_1] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
& hAPP(nat,real,power_power(real,Y_1),N) = A_1 )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_230_realpow__pos__nth,axiom,
! [A_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [R] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
& hAPP(nat,real,power_power(real,R),N) = A_1 ) ) ) ).
fof(fact_231_tpos,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),t)) ).
fof(fact_232_nat__number__of__add__1,axiom,
! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V_1)),one_one(nat)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V_1)),one_one(nat)) = number_number_of(nat,hAPP(int,int,succ,V_1)) ) ) ).
fof(fact_233_nat__1__add__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> hAPP(nat,nat,plus_plus(nat,one_one(nat)),number_number_of(nat,V_1)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
=> hAPP(nat,nat,plus_plus(nat,one_one(nat)),number_number_of(nat,V_1)) = number_number_of(nat,hAPP(int,int,succ,V_1)) ) ) ).
fof(fact_234_add__le__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,C,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_2),C)),hAPP(X_a,X_a,plus_plus(X_a,B_1),C)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_1)) ) ) ).
fof(fact_235_add__le__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_2,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_2)),hAPP(X_a,X_a,plus_plus(X_a,C),B_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_1)) ) ) ).
fof(fact_236_add__right__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),C_1))) ) ) ).
fof(fact_237_add__left__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B))) ) ) ).
fof(fact_238_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),D))) ) ) ) ).
fof(fact_239_add__le__imp__le__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),C_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_240_add__le__imp__le__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_241_power__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,A_1),N_2))) ) ) ) ).
fof(fact_242_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_243_rel__simps_I34_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K_2)),hAPP(int,int,bit1,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),L_1)) ) ).
fof(fact_244_rel__simps_I19_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),pls)) ).
fof(fact_245_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_246_rel__simps_I31_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K_2)),hAPP(int,int,bit0,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),L_1)) ) ).
fof(fact_247_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).
fof(fact_248_zle__refl,axiom,
! [W_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),W_1)) ).
fof(fact_249_zle__linear,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1)) ) ).
fof(fact_250_zle__trans,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_2),K))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K)) ) ) ).
fof(fact_251_zle__antisym,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1))
=> Z_1 = W_1 ) ) ).
fof(fact_252_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ) ).
fof(fact_253_of__nat__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Ma,Na] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).
fof(fact_254_zless__le,axiom,
! [Z_2,Wa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),Wa))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),Wa))
& Z_2 != Wa ) ) ).
fof(fact_255_zadd__left__mono,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,K),I_2)),hAPP(int,int,plus_plus(int,K),J_2))) ) ).
fof(fact_256_less__eq__number__of__int__code,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,K_2)),number_number_of(int,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),L_1)) ) ).
fof(fact_257_le__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)) ) ) ).
fof(fact_258_le__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),pls)) ) ) ).
fof(fact_259_power__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ) ) ).
fof(fact_260_power__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [Xa,Ya,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,B_1),Xa)),hAPP(nat,X_a,power_power(X_a,B_1),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ) ) ).
fof(fact_261_power__le__imp__le__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M_1)),hAPP(nat,X_a,power_power(X_a,A_1),N)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ) ) ).
fof(fact_262_le__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),number_number_of(X_a,Ya)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),Ya)) ) ) ).
fof(fact_263_le__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),hAPP(int,int,bit1,pls))) ) ) ).
fof(fact_264_zero__le__double__add__iff__zero__le__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_265_double__add__le__zero__iff__single__add__le__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_266_add__nonneg__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B))) ) ) ) ).
fof(fact_267_add__nonneg__eq__0__iff,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [Ya,Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ya))
=> ( hAPP(X_a,X_a,plus_plus(X_a,Xa),Ya) = zero_zero(X_a)
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_268_add__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1))) ) ) ) ).
fof(fact_269_add__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1))) ) ) ) ).
fof(fact_270_add__nonpos__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_271_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),D))) ) ) ) ).
fof(fact_272_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B),D))) ) ) ) ).
fof(fact_273_not__one__le__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_274_zero__le__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_275_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [Va,Wa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,Wa)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Wa)),number_number_of(X_a,Va))) ) ) ).
fof(fact_276_zero__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_277_power__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B),N))) ) ) ) ).
fof(fact_278_one__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N))) ) ) ).
fof(fact_279_zero__le__imp__of__nat,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1))) ) ).
fof(fact_280_of__nat__0__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ).
fof(fact_281_rel__simps_I22_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit1,K_2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K_2)) ) ).
fof(fact_282_rel__simps_I32_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K_2)),hAPP(int,int,bit1,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),L_1)) ) ).
fof(fact_283_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_284_rel__simps_I21_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit0,K_2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K_2)) ) ).
fof(fact_285_rel__simps_I27_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K_2)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),pls)) ) ).
fof(fact_286_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))) ).
fof(fact_287_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,X),Y))) ) ) ).
fof(fact_288_zadd__zless__mono,axiom,
! [Z_3,Z_1,W_2,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_3),Z_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,W_2),Z_3)),hAPP(int,int,plus_plus(int,W_1),Z_1))) ) ) ).
fof(fact_289_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z_1))) ).
fof(fact_290_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(hAPP(int,bool,P_1,X_1)) )
<=> ? [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).
fof(fact_291_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(hAPP(int,bool,P_1,X_1)) )
<=> ! [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).
fof(fact_292_zero__zle__int,axiom,
! [N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N))) ).
fof(fact_293_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,X),N))) ) ).
fof(fact_294_zle__iff__zadd,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_2))
<=> ? [N_1] : Z_2 = hAPP(int,int,plus_plus(int,Wa),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ) ).
fof(fact_295_semiring__1__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,V_1)) = number_number_of(X_a,hAPP(int,int,succ,V_1)) ) ) ).
fof(fact_296_semiring__number__of__add__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,succ,V_1)) ) ) ).
fof(fact_297_add__pos__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B))) ) ) ) ).
fof(fact_298_add__nonneg__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B))) ) ) ) ).
fof(fact_299_add__strict__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1))) ) ) ) ).
fof(fact_300_add__strict__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C_1))) ) ) ) ).
fof(fact_301_add__neg__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_302_add__nonpos__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_303_power__less__imp__less__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B),N)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_304_eq__0__number__of,axiom,
! [Va] :
( zero_zero(nat) = number_number_of(nat,Va)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).
fof(fact_305_eq__number__of__0,axiom,
! [Va] :
( number_number_of(nat,Va) = zero_zero(nat)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).
fof(fact_306_rel__simps_I5_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit1,K_2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K_2)) ) ).
fof(fact_307_rel__simps_I29_J,axiom,
! [K_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K_2)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),pls)) ) ).
fof(fact_308_rel__simps_I15_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K_2)),hAPP(int,int,bit1,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),L_1)) ) ).
fof(fact_309_less__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_310_rel__simps_I33_J,axiom,
! [K_2,L_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K_2)),hAPP(int,int,bit0,L_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_2),L_1)) ) ).
fof(fact_311_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_312_int__one__le__iff__zero__less,axiom,
! [Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_2))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_2)) ) ).
fof(fact_313_int__le__0__conv,axiom,
! [Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),zero_zero(int)))
<=> Na = zero_zero(nat) ) ).
fof(fact_314_succ__Pls,axiom,
hAPP(int,int,succ,pls) = hAPP(int,int,bit1,pls) ).
fof(fact_315_succ__Bit0,axiom,
! [K] : hAPP(int,int,succ,hAPP(int,int,bit0,K)) = hAPP(int,int,bit1,K) ).
fof(fact_316_succ__Bit1,axiom,
! [K] : hAPP(int,int,succ,hAPP(int,int,bit1,K)) = hAPP(int,int,bit0,hAPP(int,int,succ,K)) ).
fof(fact_317_zle__add1__eq__le,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,plus_plus(int,Z_2),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_2)) ) ).
fof(fact_318_add1__zle__eq,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,Wa),one_one(int))),Z_2))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_2)) ) ).
fof(fact_319_zless__imp__add1__zle,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,W_1),one_one(int))),Z_1)) ) ).
fof(fact_320_succ__def,axiom,
! [K] : hAPP(int,int,succ,K) = hAPP(int,int,plus_plus(int,K),one_one(int)) ).
fof(fact_321_power__eq__imp__eq__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N,B] :
( hAPP(nat,X_a,power_power(X_a,A_1),N) = hAPP(nat,X_a,power_power(X_a,B),N)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ) ).
fof(fact_322_semiring__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_2,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2))
=> hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V_1)),number_number_of(X_a,V_2)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V_1),V_2)) ) ) ) ).
fof(fact_323_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).
fof(fact_324_le__imp__0__less,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,one_one(int)),Z_1))) ) ).
fof(fact_325_number__of__succ,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [K] : number_number_of(X_a,hAPP(int,int,succ,K)) = hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,K)) ) ).
fof(fact_326_add__Bit1__Bit1,axiom,
! [K,L] : hAPP(int,int,plus_plus(int,hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L)) = hAPP(int,int,bit0,hAPP(int,int,plus_plus(int,K),hAPP(int,int,succ,L))) ).
fof(fact_327_power__strict__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B),N))) ) ) ) ) ).
fof(fact_328_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_329_power2__ge__self,axiom,
! [X] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),hAPP(nat,int,power_power(int,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_330_zero__le__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_331_power2__le__imp__le,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ) ).
fof(fact_332_power2__eq__imp__eq,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ) ).
fof(fact_333_of__nat__number__of__lemma,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = number_number_of(X_a,V_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = zero_zero(X_a) ) ) ) ).
fof(fact_334_power2__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).
fof(fact_335_sum__power2__ge__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_336_sum__power2__le__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a)))
<=> ( ti(X_a,Xa) = zero_zero(X_a)
& ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).
fof(fact_337_int__pos__lt__two__imp__zero__or__one,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),number_number_of(int,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
=> ( X = zero_zero(int)
| X = one_one(int) ) ) ) ).
fof(fact_338_number__of1,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,N)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,hAPP(int,int,bit0,N))))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,hAPP(int,int,bit1,N)))) ) ) ).
fof(fact_339_number__of2,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,pls))) ).
fof(fact_340_zero__le__imp__eq__int,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ? [N_1] : K = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).
fof(fact_341_nonneg__int__cases,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ~ ! [N_1] : K != hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).
fof(fact_342_le0,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
fof(fact_343_less__eq__nat_Osimps_I1_J,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
fof(fact_344_le__0__eq,axiom,
! [Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat)))
<=> Na = zero_zero(nat) ) ).
fof(fact_345_less__or__eq__imp__le,axiom,
! [M_1,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
| M_1 = N )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_346_le__neq__implies__less,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> ( M_1 != N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ).
fof(fact_347_less__imp__le__nat,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_348_le__eq__less__or__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
| Ma = Na ) ) ).
fof(fact_349_nat__less__le,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
& Ma != Na ) ) ).
fof(fact_350_le__refl,axiom,
! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).
fof(fact_351_nat__le__linear,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M_1)) ) ).
fof(fact_352_eq__imp__le,axiom,
! [M_1,N] :
( M_1 = N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_353_le__trans,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K)) ) ) ).
fof(fact_354_le__antisym,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M_1))
=> M_1 = N ) ) ).
fof(fact_355_less__eq__real__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_356_real__less__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_357_le__add2,axiom,
! [N,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,plus_plus(nat,M_1),N))) ).
fof(fact_358_le__add1,axiom,
! [N,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,plus_plus(nat,N),M_1))) ).
fof(fact_359_le__iff__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
<=> ? [K_1] : Na = hAPP(nat,nat,plus_plus(nat,Ma),K_1) ) ).
fof(fact_360_nat__add__left__cancel__le,axiom,
! [K_2,Ma,Na] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,K_2),Ma)),hAPP(nat,nat,plus_plus(nat,K_2),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).
fof(fact_361_trans__le__add1,axiom,
! [M_1,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),hAPP(nat,nat,plus_plus(nat,J_2),M_1))) ) ).
fof(fact_362_trans__le__add2,axiom,
! [M_1,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),hAPP(nat,nat,plus_plus(nat,M_1),J_2))) ) ).
fof(fact_363_add__le__mono1,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),hAPP(nat,nat,plus_plus(nat,J_2),K))) ) ).
fof(fact_364_add__le__mono,axiom,
! [K,L,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),hAPP(nat,nat,plus_plus(nat,J_2),L))) ) ) ).
fof(fact_365_add__leD2,axiom,
! [M_1,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M_1),K)),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).
fof(fact_366_add__leD1,axiom,
! [M_1,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M_1),K)),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_367_add__leE,axiom,
! [M_1,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M_1),K)),N))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).
fof(fact_368_real__add__left__mono,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,Z_1),X)),hAPP(real,real,plus_plus(real,Z_1),Y))) ) ).
fof(fact_369_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ).
fof(fact_370_zle__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).
fof(fact_371_le__nat__number__of,axiom,
! [Va,V] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,Va)),number_number_of(nat,V)))
<=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),V))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ) ).
fof(fact_372_two__realpow__ge__one,axiom,
! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,power_power(real,number_number_of(real,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) ).
fof(fact_373_conj__le__cong,axiom,
! [P_2,P_1,Xa] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
& hBOOL(P_2) ) ) ) ).
fof(fact_374_imp__le__cong,axiom,
! [P_2,P_1,Xa] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
=> hBOOL(P_2) ) ) ) ).
fof(fact_375_ex__least__nat__less,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,P_1,Na))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),Na))
& ! [I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_1))
=> ~ hBOOL(hAPP(nat,bool,P_1,I_1)) )
& hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,plus_plus(nat,K_1),one_one(nat)))) ) ) ) ).
fof(fact_376_nonneg__eq__int,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> ~ ! [M] : Z_1 != hAPP(nat,int,semiring_1_of_nat(int),M) ) ).
fof(fact_377_mono__nat__linear__lb,axiom,
! [Ma,K_2,F] :
( ! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,M)),hAPP(nat,nat,F,N_1))) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,F,Ma)),K_2)),hAPP(nat,nat,F,hAPP(nat,nat,plus_plus(nat,Ma),K_2)))) ) ).
fof(fact_378_real__le__refl,axiom,
! [W_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W_1),W_1)) ).
fof(fact_379_real__le__linear,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W_1))
| hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W_1),Z_1)) ) ).
fof(fact_380_real__le__trans,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_2),J_2))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_2),K))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_2),K)) ) ) ).
fof(fact_381_real__le__antisym,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W_1),Z_1))
=> Z_1 = W_1 ) ) ).
fof(fact_382_int__ge__induct,axiom,
! [P_1,K_2,I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),I))
=> ( hBOOL(hAPP(int,bool,P_1,K_2))
=> ( ! [I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_2),I_1))
=> ( hBOOL(hAPP(int,bool,P_1,I_1))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,I_1),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).
fof(fact_383_ex__least__nat__le,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,P_1,Na))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),Na))
& ! [I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_1))
=> ~ hBOOL(hAPP(nat,bool,P_1,I_1)) )
& hBOOL(hAPP(nat,bool,P_1,K_1)) ) ) ) ).
fof(fact_384_less__mono__imp__le__mono,axiom,
! [I,J,F] :
( ! [I_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,I_1)),hAPP(nat,nat,F,J_1))) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I)),hAPP(nat,nat,F,J))) ) ) ).
fof(fact_385_order__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X)) ) ).
fof(fact_386_linorder__le__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_387_le__funE,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [Xa,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))) ) ) ).
fof(fact_388_xt1_I6_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z_1,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),X)) ) ) ) ).
fof(fact_389_xt1_I5_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_390_order__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z_1)) ) ) ) ).
fof(fact_391_order__antisym,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_392_xt1_I4_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1)) ) ) ) ).
fof(fact_393_ord__le__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( B = C_1
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1)) ) ) ) ).
fof(fact_394_xt1_I3_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1)) ) ) ) ).
fof(fact_395_ord__eq__le__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_1,B] :
( A_1 = B
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1)) ) ) ) ).
fof(fact_396_order__antisym__conv,axiom,
! [X_a] :
( order(X_a)
=> ! [Ya,Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
<=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_397_le__funD,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [Xa,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))) ) ) ).
fof(fact_398_order__eq__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( X = Y
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_399_order__eq__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [Xa,Ya] :
( ti(X_a,Xa) = ti(X_a,Ya)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ) ).
fof(fact_400_linorder__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_401_le__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
<=> ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_402_linorder__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_403_order__less__asym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_404_xt1_I10_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z_1,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).
fof(fact_405_order__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).
fof(fact_406_xt1_I2_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1)) ) ) ) ).
fof(fact_407_ord__less__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( B = C_1
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1)) ) ) ) ).
fof(fact_408_xt1_I1_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1)) ) ) ) ).
fof(fact_409_ord__eq__less__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_1,B] :
( A_1 = B
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1)) ) ) ) ).
fof(fact_410_xt1_I9_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_411_order__less__asym_H,axiom,
! [X_a] :
( preorder(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ).
fof(fact_412_order__less__imp__triv,axiom,
! [X_a] :
( preorder(X_a)
=> ! [P_1,Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
=> hBOOL(P_1) ) ) ) ).
fof(fact_413_order__less__imp__not__eq2,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,Y) != ti(X_a,X) ) ) ).
fof(fact_414_order__less__imp__not__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_415_order__less__imp__not__less,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_416_order__less__not__sym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_417_less__imp__neq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_418_linorder__neqE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_419_linorder__antisym__conv3,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Ya,Xa] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_420_linorder__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_421_not__less__iff__gr__or__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
| ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_422_linorder__neq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( ti(X_a,Xa) != ti(X_a,Ya)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) ) ) ) ).
fof(fact_423_order__less__irrefl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),X)) ) ).
fof(fact_424_linorder__not__less,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ).
fof(fact_425_linorder__not__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) ) ) ).
fof(fact_426_linorder__le__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_427_order__less__le,axiom,
! [X_a] :
( order(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
& ti(X_a,Xa) != ti(X_a,Ya) ) ) ) ).
fof(fact_428_less__le__not__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
& ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ) ).
fof(fact_429_order__le__less,axiom,
! [X_a] :
( order(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
| ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_430_leI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_431_not__leE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_432_linorder__antisym__conv1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
<=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_433_order__neq__le__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_434_xt1_I12_J,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_435_leD,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_436_order__less__imp__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_437_linorder__antisym__conv2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
<=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).
fof(fact_438_order__le__imp__less__or__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_439_order__le__neq__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_440_xt1_I11_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_441_order__less__le__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).
fof(fact_442_xt1_I7_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z_1,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).
fof(fact_443_order__le__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).
fof(fact_444_xt1_I8_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z_1,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).
fof(fact_445_real__lbound__gt__zero,axiom,
! [D2,D1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
=> ? [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D1))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D2)) ) ) ) ).
fof(fact_446_dense__le,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z_1,Y] :
( ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)) ) ) ).
fof(fact_447_less__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G))
<=> ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
& ~ hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) ) ) ) ).
fof(fact_448_le__funI,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1)))
=> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) ) ) ).
fof(fact_449_pinf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z] :
! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_3)) ) ) ).
fof(fact_450_dense__le__bounded,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ! [W] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z_1)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)) ) ) ) ).
fof(fact_451_xt5,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C,F,B_1,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),hAPP(X_a,X_b,F,B_1)))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_2))) ) ) ) ) ).
fof(fact_452_minf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z] :
! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_3)) ) ) ).
fof(fact_453_pinf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z] :
! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_3),X_1)) ) ) ).
fof(fact_454_xt7,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C,F,B_1,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,B_1)))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_2))) ) ) ) ) ).
fof(fact_455_order__less__le__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,C,A_2,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B_1)),C))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_2)),C)) ) ) ) ) ).
fof(fact_456_minf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z] :
! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_3),X_1)) ) ) ).
fof(fact_457_xt6,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [C,F,B_1,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B_1)),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),B_1))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_2)) ) ) ) ) ).
fof(fact_458_order__le__less__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,C,A_2,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,B_1)),C))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_2)),C)) ) ) ) ) ).
fof(fact_459_order__less__le__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [C,A_2,F,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,B_1)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_1),C))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,C))) ) ) ) ) ).
fof(fact_460_xt4,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C,F,B_1,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,B_1)),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),B_1))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_2)) ) ) ) ) ).
fof(fact_461_order__le__less__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [C,A_2,F,B_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),hAPP(X_b,X_a,F,B_1)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_1),C))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,C))) ) ) ) ) ).
fof(fact_462_real__sqrt__sum__squares__ge__zero,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,power_power(real,Y),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_463_zadd__power2,axiom,
! [A_1,B] : hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A_1),B)),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),number_number_of(int,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,power_power(int,B),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_464_mult__Bit1,axiom,
! [K,L] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit1,K)),L) = hAPP(int,int,plus_plus(int,hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),L))),L) ).
fof(fact_465_mult__left__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_466_mult__right__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_467_mult__less__imp__less__left,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_468_mult__left__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_469_mult__less__imp__less__right,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_470_mult__right__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [A_1,C_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_471_mult__le__less__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_472_mult__less__le__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_473_mult__strict__mono_H,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_474_mult__strict__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_475_mult__le__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_2)) ) ) ) ).
fof(fact_476_mult__le__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_1)) ) ) ) ).
fof(fact_477_sum__squares__gt__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Xa,Ya] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya))))
<=> ( ti(X_a,Xa) != zero_zero(X_a)
| ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).
fof(fact_478_not__sum__squares__lt__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))),zero_zero(X_a))) ) ).
fof(fact_479_power__less__power__Suc,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,power_power(X_a,A_1),N)))) ) ) ).
fof(fact_480_power__gt1__lemma,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,power_power(X_a,A_1),N)))) ) ) ).
fof(fact_481_real__sqrt__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,X))) ) ).
fof(fact_482_real__sqrt__not__eq__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,sqrt,X) != zero_zero(real) ) ).
fof(fact_483_real__sqrt__lt__0__iff,axiom,
! [Xa] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) ) ).
fof(fact_484_real__sqrt__gt__0__iff,axiom,
! [Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,Ya)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) ) ).
fof(fact_485_real__sqrt__lt__1__iff,axiom,
! [Xa] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) ) ).
fof(fact_486_real__sqrt__gt__1__iff,axiom,
! [Ya] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,Ya)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Ya)) ) ).
fof(fact_487_mult__strict__left__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C_1,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).
fof(fact_488_mult__strict__right__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C_1,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).
fof(fact_489_comm__mult__strict__left__mono,axiom,
! [X_a] :
( linord893533164strict(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).
fof(fact_490_mult__strict__left__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).
fof(fact_491_mult__strict__right__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).
fof(fact_492_mult__neg__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_493_mult__neg__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_494_mult__less__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),A_2)) ) ) ) ).
fof(fact_495_zero__less__mult__pos2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).
%----Arities (114)
fof(arity_fun___Orderings_Opreorder,axiom,
! [T_2,T_1] :
( preorder(T_1)
=> preorder(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oorder,axiom,
! [T_2,T_1] :
( order(T_1)
=> order(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oord,axiom,
! [T_2,T_1] :
( ord(T_1)
=> ord(fun(T_2,T_1)) ) ).
fof(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(int) ).
fof(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
fof(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
fof(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
fof(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
fof(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(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___Groups_Omonoid__add,axiom,
monoid_add(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
fof(arity_Int_Oint___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___Orderings_Oord,axiom,
ord(int) ).
fof(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
fof(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
fof(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring,axiom,
linordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
fof(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
fof(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(nat) ).
fof(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
fof(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
fof(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
fof(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
fof(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
fof(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
fof(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
fof(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
fof(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
fof(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp,axiom,
semiri456707255roduct(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(real) ).
fof(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring,axiom,
linordered_semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
fof(arity_RealDef_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
fof(arity_RealDef_Oreal___Nat_Osemiring__char__0,axiom,
semiring_char_0(real) ).
fof(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
fof(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(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___Groups_Omonoid__add,axiom,
monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
semiring_0(real) ).
fof(arity_RealDef_Oreal___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___Orderings_Oord,axiom,
ord(real) ).
fof(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
fof(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
fof(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
fof(arity_RealDef_Oreal___Groups_Oone,axiom,
one(real) ).
%----Helper facts (4)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
fof(help_If_1_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fTrue),X),Y) = ti(X_a,X) ).
fof(help_If_2_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fFalse),X),Y) = ti(X_a,Y) ).
fof(help_If_3_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(int) ).
%------------------------------------------------------------------------------