TPTP Problem File: NUM924+5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM924+5 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 100 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_100_fofpt_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.31 v8.2.0, 0.28 v8.1.0, 0.33 v7.5.0, 0.34 v7.4.0, 0.14 v7.3.0, 0.00 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.17 v6.2.0, 0.24 v6.1.0, 0.33 v6.0.0, 0.22 v5.5.0, 0.41 v5.4.0, 0.46 v5.3.0
% Syntax : Number of formulae : 155 ( 60 unt; 0 def)
% Number of atoms : 311 ( 95 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 168 ( 12 ~; 4 |; 26 &)
% ( 51 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 16 ( 16 usr; 6 con; 0-3 aty)
% Number of variables : 259 ( 259 !; 0 ?)
% SPC : FOF_CAX_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:00:53
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (39)
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_a] :
( semiring_1(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,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> plus_plus(X_b,ti(X_b,B_1),B_2) = plus_plus(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_arg2,axiom,
! [B_1,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> plus_plus(X_b,B_1,ti(X_b,B_2)) = plus_plus(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [B_1,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ti(X_b,plus_plus(X_b,B_1,B_2)) = plus_plus(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_arg1,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> plus_plus(X_a,ti(X_a,B_1),B_2) = plus_plus(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_arg2,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> plus_plus(X_a,B_1,ti(X_a,B_2)) = plus_plus(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> ti(X_a,plus_plus(X_a,B_1,B_2)) = plus_plus(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_arg1,axiom,
! [B_1,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> times_times(X_b,ti(X_b,B_1),B_2) = times_times(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_arg2,axiom,
! [B_1,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> times_times(X_b,B_1,ti(X_b,B_2)) = times_times(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
! [B_1,B_2,X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ti(X_b,times_times(X_b,B_1,B_2)) = times_times(X_b,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_arg1,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> times_times(X_a,ti(X_a,B_1),B_2) = times_times(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_arg2,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> times_times(X_a,B_1,ti(X_a,B_2)) = times_times(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> ti(X_a,times_times(X_a,B_1,B_2)) = times_times(X_a,B_1,B_2) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_0_res,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_1_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_IntPrimes_Ozprime_arg1,axiom,
! [B_1] :
( zprime(ti(int,B_1))
<=> zprime(B_1) ) ).
fof(tsy_c_Int_OBit0_arg1,hypothesis,
! [B_1] : bit0(ti(int,B_1)) = bit0(B_1) ).
fof(tsy_c_Int_OBit0_res,hypothesis,
! [B_1] : ti(int,bit0(B_1)) = bit0(B_1) ).
fof(tsy_c_Int_OBit1_arg1,hypothesis,
! [B_1] : bit1(ti(int,B_1)) = bit1(B_1) ).
fof(tsy_c_Int_OBit1_res,hypothesis,
! [B_1] : ti(int,bit1(B_1)) = bit1(B_1) ).
fof(tsy_c_Int_OPls_res,hypothesis,
ti(int,pls) = pls ).
fof(tsy_c_Int_Onumber__class_Onumber__of_arg1,axiom,
! [B_1,X_a] :
( number(X_a)
=> number_number_of(X_a,ti(int,B_1)) = number_number_of(X_a,B_1) ) ).
fof(tsy_c_Int_Onumber__class_Onumber__of_res,axiom,
! [B_1,X_a] :
( number(X_a)
=> ti(X_a,number_number_of(X_a,B_1)) = number_number_of(X_a,B_1) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_0_arg1,axiom,
! [B_1,B_2,X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ( ord_less(X_a,ti(X_a,B_1),B_2)
<=> ord_less(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_0_arg2,axiom,
! [B_1,B_2,X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ( ord_less(X_a,B_1,ti(X_a,B_2))
<=> ord_less(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_1_arg1,axiom,
! [B_1,B_2,X_a] :
( linordered_idom(X_a)
=> ( ord_less(X_a,ti(X_a,B_1),B_2)
<=> ord_less(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_1_arg2,axiom,
! [B_1,B_2,X_a] :
( linordered_idom(X_a)
=> ( ord_less(X_a,B_1,ti(X_a,B_2))
<=> ord_less(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_0_arg1,axiom,
! [B_1,B_2,X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ( ord_less_eq(X_a,ti(X_a,B_1),B_2)
<=> ord_less_eq(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_0_arg2,axiom,
! [B_1,B_2,X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ( ord_less_eq(X_a,B_1,ti(X_a,B_2))
<=> ord_less_eq(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_1_arg1,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> ( ord_less_eq(X_a,ti(X_a,B_1),B_2)
<=> ord_less_eq(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_1_arg2,axiom,
! [B_1,B_2,X_a] :
( linordered_ring(X_a)
=> ( ord_less_eq(X_a,B_1,ti(X_a,B_2))
<=> ord_less_eq(X_a,B_1,B_2) ) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg1,axiom,
! [B_1,B_2,X_a] :
( semiring_1(X_a)
=> power_power(X_a,ti(X_a,B_1),B_2) = power_power(X_a,B_1,B_2) ) ).
fof(tsy_c_Power_Opower__class_Opower_arg2,axiom,
! [B_1,B_2,X_a] :
( semiring_1(X_a)
=> power_power(X_a,B_1,ti(nat,B_2)) = power_power(X_a,B_1,B_2) ) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [B_1,B_2,X_a] :
( semiring_1(X_a)
=> ti(X_a,power_power(X_a,B_1,B_2)) = power_power(X_a,B_1,B_2) ) ).
fof(tsy_c_TwoSquares__Mirabelle__vsgmegnqdl_Ois__sum2sq_arg1,axiom,
! [B_1] :
( twoSqu33214720sum2sq(ti(int,B_1))
<=> twoSqu33214720sum2sq(B_1) ) ).
fof(tsy_v_m_res,axiom,
ti(int,m) = m ).
fof(tsy_v_s_____res,hypothesis,
ti(int,s) = s ).
fof(tsy_v_t_____res,axiom,
ti(int,t) = t ).
%----Relevant facts (98)
fof(fact_0__096t_A_060_A0_096,axiom,
ord_less(int,t,zero_zero(int)) ).
fof(fact_1_calculation_I1_J,axiom,
ord_less(int,t,one_one(int)) ).
fof(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less(int,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t),times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),zero_zero(int))) ).
fof(fact_3_t,axiom,
plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t) ).
fof(fact_4_calculation_I2_J,axiom,
( t = zero_zero(int)
=> plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = zero_zero(int) ) ).
fof(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq(int,one_one(int),t) ).
fof(fact_6_p0,axiom,
ord_less(int,zero_zero(int),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_7_not__sum__power2__lt__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ~ ord_less(X_a,plus_plus(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a)) ) ).
fof(fact_8_sum__power2__gt__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y_1] :
( ord_less(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))))
<=> ( ti(X_a,X_1) != zero_zero(X_a)
| ti(X_a,Y_1) != zero_zero(X_a) ) ) ) ).
fof(fact_9_sum__power2__eq__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y_1] :
( plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_10_power2__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ~ ord_less(X_a,power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))),zero_zero(X_a)) ) ).
fof(fact_11_zero__less__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( ord_less(X_a,zero_zero(X_a),power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls)))))
<=> ti(X_a,A_2) != zero_zero(X_a) ) ) ).
fof(fact_12_one__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a) ) ).
fof(fact_13_zero__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> power_power(X_a,zero_zero(X_a),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a) ) ).
fof(fact_14_zero__eq__power2,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [A_2] :
( power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_15_add__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,bit1(pls),W)) ) ).
fof(fact_16_add__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] : plus_plus(X_a,number_number_of(X_a,V),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V,bit1(pls))) ) ).
fof(fact_17_t__l__p,axiom,
ord_less(int,t,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_18__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_1] : plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T_1) ).
fof(fact_19_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_20_qf1pt,axiom,
twoSqu33214720sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t)) ).
fof(fact_21_zle__refl,axiom,
! [W] : ord_less_eq(int,W,W) ).
fof(fact_22_number__of__is__id,axiom,
! [K_1] : number_number_of(int,K_1) = K_1 ).
fof(fact_23_zmult__commute,axiom,
! [Z,W] : times_times(int,Z,W) = times_times(int,W,Z) ).
fof(fact_24_zle__linear,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
| ord_less_eq(int,W,Z) ) ).
fof(fact_25_times__numeral__code_I5_J,axiom,
! [V,W] : times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W)) ).
fof(fact_26_less__eq__number__of__int__code,axiom,
! [K,L] :
( ord_less_eq(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_27_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1,Y_1] :
( ord_less_eq(X_a,number_number_of(X_a,X_1),number_number_of(X_a,Y_1))
<=> ord_less_eq(int,X_1,Y_1) ) ) ).
fof(fact_28_zmult__assoc,axiom,
! [Z1,Z2,Z3] : times_times(int,times_times(int,Z1,Z2),Z3) = times_times(int,Z1,times_times(int,Z2,Z3)) ).
fof(fact_29_zle__trans,axiom,
! [K_1,I,J] :
( ord_less_eq(int,I,J)
=> ( ord_less_eq(int,J,K_1)
=> ord_less_eq(int,I,K_1) ) ) ).
fof(fact_30_zle__antisym,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
=> ( ord_less_eq(int,W,Z)
=> Z = W ) ) ).
fof(fact_31_zpower__zadd__distrib,axiom,
! [X,Y,Z] : power_power(int,X,plus_plus(nat,Y,Z)) = times_times(int,power_power(int,X,Y),power_power(int,X,Z)) ).
fof(fact_32_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit1(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_33_rel__simps_I34_J,axiom,
! [K,L] :
( ord_less_eq(int,bit1(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_34_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
fof(fact_35_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit0(K1),bit0(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_36_rel__simps_I31_J,axiom,
! [K,L] :
( ord_less_eq(int,bit0(K),bit0(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_37_zless__le,axiom,
! [Z_1,W_1] :
( ord_less(int,Z_1,W_1)
<=> ( ord_less_eq(int,Z_1,W_1)
& Z_1 != W_1 ) ) ).
fof(fact_38_zadd__left__mono,axiom,
! [K_1,I,J] :
( ord_less_eq(int,I,J)
=> ord_less_eq(int,plus_plus(int,K_1,I),plus_plus(int,K_1,J)) ) ).
fof(fact_39_eq__number__of__0,axiom,
! [V_3] :
( number_number_of(nat,V_3) = zero_zero(nat)
<=> ord_less_eq(int,V_3,pls) ) ).
fof(fact_40_eq__0__number__of,axiom,
! [V_3] :
( zero_zero(nat) = number_number_of(nat,V_3)
<=> ord_less_eq(int,V_3,pls) ) ).
fof(fact_41_semiring__mult__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1,V] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V_1)
=> times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,times_times(int,V,V_1)) ) ) ) ).
fof(fact_42_mult__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,Z] : times_times(X_a,number_number_of(X_a,V),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V,W)),Z) ) ).
fof(fact_43_arith__simps_I32_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V,W)) ) ).
fof(fact_44_number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : number_number_of(X_a,times_times(int,V,W)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) ) ).
fof(fact_45_sum__squares__le__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_1] :
( ord_less_eq(X_a,plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)),zero_zero(X_a))
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_46_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ord_less_eq(X_a,zero_zero(X_a),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y))) ) ).
fof(fact_47_le__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( ord_less_eq(X_a,number_number_of(X_a,X_1),zero_zero(X_a))
<=> ord_less_eq(int,X_1,pls) ) ) ).
fof(fact_48_le__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( ord_less_eq(X_a,zero_zero(X_a),number_number_of(X_a,Y_1))
<=> ord_less_eq(int,pls,Y_1) ) ) ).
fof(fact_49_less__0__number__of,axiom,
! [V_3] :
( ord_less(nat,zero_zero(nat),number_number_of(nat,V_3))
<=> ord_less(int,pls,V_3) ) ).
fof(fact_50_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [V_3,W_1] :
( ord_less_eq(X_a,number_number_of(X_a,V_3),number_number_of(X_a,W_1))
<=> ~ ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_3)) ) ) ).
fof(fact_51_rel__simps_I22_J,axiom,
! [K] :
( ord_less_eq(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
fof(fact_52_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_53_rel__simps_I32_J,axiom,
! [K,L] :
( ord_less_eq(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_54_rel__simps_I27_J,axiom,
! [K] :
( ord_less_eq(int,bit0(K),pls)
<=> ord_less_eq(int,K,pls) ) ).
fof(fact_55_rel__simps_I21_J,axiom,
! [K] :
( ord_less_eq(int,pls,bit0(K))
<=> ord_less_eq(int,pls,K) ) ).
fof(fact_56_zadd__zless__mono,axiom,
! [Z_2,Z,W_2,W] :
( ord_less(int,W_2,W)
=> ( ord_less_eq(int,Z_2,Z)
=> ord_less(int,plus_plus(int,W_2,Z_2),plus_plus(int,W,Z)) ) ) ).
fof(fact_57_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
fof(fact_58_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
fof(fact_59_le__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( ord_less_eq(X_a,number_number_of(X_a,X_1),one_one(X_a))
<=> ord_less_eq(int,X_1,bit1(pls)) ) ) ).
fof(fact_60_le__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( ord_less_eq(X_a,one_one(X_a),number_number_of(X_a,Y_1))
<=> ord_less_eq(int,bit1(pls),Y_1) ) ) ).
fof(fact_61_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
fof(fact_62_mult__Pls,axiom,
! [W] : times_times(int,pls,W) = pls ).
fof(fact_63_mult__Bit0,axiom,
! [K_1,L_1] : times_times(int,bit0(K_1),L_1) = bit0(times_times(int,K_1,L_1)) ).
fof(fact_64_less__number__of__int__code,axiom,
! [K,L] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less(int,K,L) ) ).
fof(fact_65_zmult__1__right,axiom,
! [Z] : times_times(int,Z,one_one(int)) = Z ).
fof(fact_66_zmult__1,axiom,
! [Z] : times_times(int,one_one(int),Z) = Z ).
fof(fact_67_plus__numeral__code_I9_J,axiom,
! [V,W] : plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W)) ).
fof(fact_68_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W)) ).
fof(fact_69_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times(int,W,plus_plus(int,Z1,Z2)) = plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2)) ).
fof(fact_70_rel__simps_I29_J,axiom,
! [K] :
( ord_less_eq(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
fof(fact_71_rel__simps_I5_J,axiom,
! [K] :
( ord_less(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
fof(fact_72_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_73_rel__simps_I33_J,axiom,
! [K,L] :
( ord_less_eq(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
fof(fact_74_less__int__code_I14_J,axiom,
! [K1,K2] :
( ord_less(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_75_rel__simps_I15_J,axiom,
! [K,L] :
( ord_less(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_76_less__nat__number__of,axiom,
! [V_3,V_2] :
( ord_less(nat,number_number_of(nat,V_3),number_number_of(nat,V_2))
<=> ( ( ord_less(int,V_3,V_2)
=> ord_less(int,pls,V_2) )
& ord_less(int,V_3,V_2) ) ) ).
fof(fact_77_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( ord_less_eq(int,one_one(int),Z_1)
<=> ord_less(int,zero_zero(int),Z_1) ) ).
fof(fact_78_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
fof(fact_79_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
fof(fact_80_zless__imp__add1__zle,axiom,
! [W,Z] :
( ord_less(int,W,Z)
=> ord_less_eq(int,plus_plus(int,W,one_one(int)),Z) ) ).
fof(fact_81_add1__zle__eq,axiom,
! [W_1,Z_1] :
( ord_less_eq(int,plus_plus(int,W_1,one_one(int)),Z_1)
<=> ord_less(int,W_1,Z_1) ) ).
fof(fact_82_zle__add1__eq__le,axiom,
! [W_1,Z_1] :
( ord_less(int,W_1,plus_plus(int,Z_1,one_one(int)))
<=> ord_less_eq(int,W_1,Z_1) ) ).
fof(fact_83_semiring__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_1,V] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V_1)
=> plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,plus_plus(int,V,V_1)) ) ) ) ).
fof(fact_84_add__nat__number__of,axiom,
! [V_1,V] :
( ( ord_less(int,V,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V_1) )
& ( ~ ord_less(int,V,pls)
=> ( ( ord_less(int,V_1,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V) )
& ( ~ ord_less(int,V_1,pls)
=> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,plus_plus(int,V,V_1)) ) ) ) ) ).
fof(fact_85_le__imp__0__less,axiom,
! [Z] :
( ord_less_eq(int,zero_zero(int),Z)
=> ord_less(int,zero_zero(int),plus_plus(int,one_one(int),Z)) ) ).
fof(fact_86_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [X_1,Y_1] :
( number_number_of(X_a,X_1) = number_number_of(X_a,Y_1)
<=> X_1 = Y_1 ) ) ).
fof(fact_87_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [W_1,X_1] :
( number_number_of(X_a,W_1) = ti(X_a,X_1)
<=> ti(X_a,X_1) = number_number_of(X_a,W_1) ) ) ).
fof(fact_88_rel__simps_I51_J,axiom,
! [K,L] :
( bit1(K) = bit1(L)
<=> K = L ) ).
fof(fact_89_rel__simps_I48_J,axiom,
! [K,L] :
( bit0(K) = bit0(L)
<=> K = L ) ).
fof(fact_90_zless__linear,axiom,
! [X,Y] :
( ord_less(int,X,Y)
| X = Y
| ord_less(int,Y,X) ) ).
fof(fact_91_sum__squares__eq__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_1] :
( plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_92_left__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [A_1,B,V] : times_times(X_b,plus_plus(X_b,A_1,B),number_number_of(X_b,V)) = plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V))) ) ).
fof(fact_93_right__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [V,B,C] : times_times(X_b,number_number_of(X_b,V),plus_plus(X_b,B,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C)) ) ).
fof(fact_94_zadd__assoc,axiom,
! [Z1,Z2,Z3] : plus_plus(int,plus_plus(int,Z1,Z2),Z3) = plus_plus(int,Z1,plus_plus(int,Z2,Z3)) ).
fof(fact_95_zadd__left__commute,axiom,
! [X,Y,Z] : plus_plus(int,X,plus_plus(int,Y,Z)) = plus_plus(int,Y,plus_plus(int,X,Z)) ).
fof(fact_96_zadd__commute,axiom,
! [Z,W] : plus_plus(int,Z,W) = plus_plus(int,W,Z) ).
fof(fact_97_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
%----Arities (16)
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___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
fof(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
fof(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
fof(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
%----Helper facts (1)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
%----Conjectures (1)
fof(conj_0,conjecture,
ord_less(int,plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)),zero_zero(int)) ).
%------------------------------------------------------------------------------