TPTP Problem File: NUM926+5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM926+5 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 258, 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_l258 [Bla11]
% Status : Theorem
% Rating : 0.14 v8.2.0, 0.11 v8.1.0, 0.17 v7.5.0, 0.19 v7.4.0, 0.23 v7.3.0, 0.21 v7.2.0, 0.17 v7.0.0, 0.13 v6.4.0, 0.19 v6.3.0, 0.21 v6.2.0, 0.24 v6.1.0, 0.20 v6.0.0, 0.22 v5.5.0, 0.37 v5.4.0, 0.43 v5.3.0
% Syntax : Number of formulae : 142 ( 52 unt; 0 def)
% Number of atoms : 261 ( 84 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 126 ( 7 ~; 3 |; 12 &)
% ( 40 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-3 aty)
% Number of variables : 266 ( 260 !; 6 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:14:19
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (27)
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_arg1,axiom,
! [B_1,B_2,X_a] :
( comm_semiring_1(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_arg2,axiom,
! [B_1,B_2,X_a] :
( comm_semiring_1(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_res,axiom,
! [B_1,B_2,X_a] :
( comm_semiring_1(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_arg1,axiom,
! [B_1,B_2,X_a] :
( monoid_mult(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_arg2,axiom,
! [B_1,B_2,X_a] :
( monoid_mult(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_res,axiom,
! [B_1,B_2,X_a] :
( monoid_mult(X_a)
=> ti(X_a,times_times(X_a,B_1,B_2)) = times_times(X_a,B_1,B_2) ) ).
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_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_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__eq_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_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_Power_Opower__class_Opower_arg1,axiom,
! [B_1,B_2,X_a] :
( monoid_mult(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] :
( monoid_mult(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] :
( monoid_mult(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,hypothesis,
ti(int,m) = m ).
fof(tsy_v_s_____res,axiom,
ti(int,s) = s ).
fof(tsy_v_t_____res,axiom,
ti(int,t) = t ).
%----Relevant facts (98)
fof(fact_0_tpos,axiom,
ord_less_eq(int,one_one(int),t) ).
fof(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( t = one_one(int)
=> ? [X,Y] : plus_plus(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),power_power(int,Y,number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) ) ).
fof(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06,axiom,
( ord_less(int,one_one(int),t)
=> ? [X,Y] : plus_plus(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),power_power(int,Y,number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) ) ).
fof(fact_3_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_4_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
fof(fact_5_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_6_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_7_zadd__power2,axiom,
! [A_1,B] : power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,plus_plus(int,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B)),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
fof(fact_8_zadd__power3,axiom,
! [A_1,B] : power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(int,power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) ).
fof(fact_9_power2__sum,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [X_1,Y_1] : power_power(X_a,plus_plus(X_a,X_1,Y_1),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(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))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X_1),Y_1)) ) ).
fof(fact_10_power2__eq__square__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [W] : power_power(X_b,number_number_of(X_b,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_b,number_number_of(X_b,W),number_number_of(X_b,W)) ) ).
fof(fact_11_cube__square,axiom,
! [A_1] : times_times(int,A_1,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))) = power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))) ).
fof(fact_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_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1] : times_times(X_a,X_1,X_1) = power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))) ) ).
fof(fact_14_power2__eq__square,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,A_1,A_1) ) ).
fof(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1,N] : power_power(X_a,X_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(X_a,power_power(X_a,X_1,N),power_power(X_a,X_1,N)) ) ).
fof(fact_16_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_17_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_18_one__add__one__is__two,axiom,
! [X_a] :
( number_ring(X_a)
=> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))) ) ).
fof(fact_19__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_20_zle__refl,axiom,
! [W] : ord_less_eq(int,W,W) ).
fof(fact_21_zle__linear,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
| ord_less_eq(int,W,Z) ) ).
fof(fact_22_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_23_zless__linear,axiom,
! [X_1,Y_1] :
( ord_less(int,X_1,Y_1)
| X_1 = Y_1
| ord_less(int,Y_1,X_1) ) ).
fof(fact_24_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_25_zle__antisym,axiom,
! [Z,W] :
( ord_less_eq(int,Z,W)
=> ( ord_less_eq(int,W,Z)
=> Z = W ) ) ).
fof(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1,P,Q] : power_power(X_a,power_power(X_a,X_1,P),Q) = power_power(X_a,X_1,times_times(nat,P,Q)) ) ).
fof(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1] : power_power(X_a,X_1,one_one(nat)) = ti(X_a,X_1) ) ).
fof(fact_28_zpower__zpower,axiom,
! [X_1,Y_1,Z] : power_power(int,power_power(int,X_1,Y_1),Z) = power_power(int,X_1,times_times(nat,Y_1,Z)) ).
fof(fact_29_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [V_2,W_1] :
( ord_less_eq(X_a,number_number_of(X_a,V_2),number_number_of(X_a,W_1))
<=> ~ ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_2)) ) ) ).
fof(fact_30_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2,Y_2] :
( ord_less(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2))
<=> ord_less(int,X_2,Y_2) ) ) ).
fof(fact_31_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_2,Y_2] :
( ord_less_eq(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2))
<=> ord_less_eq(int,X_2,Y_2) ) ) ).
fof(fact_32_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_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1,P,Q] : times_times(X_a,power_power(X_a,X_1,P),power_power(X_a,X_1,Q)) = power_power(X_a,X_1,plus_plus(nat,P,Q)) ) ).
fof(fact_34_zpower__zadd__distrib,axiom,
! [X_1,Y_1,Z] : power_power(int,X_1,plus_plus(nat,Y_1,Z)) = times_times(int,power_power(int,X_1,Y_1),power_power(int,X_1,Z)) ).
fof(fact_35_nat__mult__2,axiom,
! [Z] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
fof(fact_36_nat__mult__2__right,axiom,
! [Z] : times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z) ).
fof(fact_37_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
fof(fact_38_less__int__code_I16_J,axiom,
! [K1,K2] :
( ord_less(int,bit1(K1),bit1(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_39_rel__simps_I17_J,axiom,
! [K,L] :
( ord_less(int,bit1(K),bit1(L))
<=> ord_less(int,K,L) ) ).
fof(fact_40_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_41_rel__simps_I34_J,axiom,
! [K,L] :
( ord_less_eq(int,bit1(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_42_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
fof(fact_43_less__int__code_I13_J,axiom,
! [K1,K2] :
( ord_less(int,bit0(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_44_rel__simps_I14_J,axiom,
! [K,L] :
( ord_less(int,bit0(K),bit0(L))
<=> ord_less(int,K,L) ) ).
fof(fact_45_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
fof(fact_46_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_47_rel__simps_I31_J,axiom,
! [K,L] :
( ord_less_eq(int,bit0(K),bit0(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_48_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_49_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_50_zadd__strict__right__mono,axiom,
! [K_1,I,J] :
( ord_less(int,I,J)
=> ord_less(int,plus_plus(int,I,K_1),plus_plus(int,J,K_1)) ) ).
fof(fact_51_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_52_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_53_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
fof(fact_54_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
fof(fact_55_rel__simps_I29_J,axiom,
! [K] :
( ord_less_eq(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
fof(fact_56_rel__simps_I5_J,axiom,
! [K] :
( ord_less(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
fof(fact_57_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( ord_less_eq(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_58_rel__simps_I33_J,axiom,
! [K,L] :
( ord_less_eq(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
fof(fact_59_less__int__code_I14_J,axiom,
! [K1,K2] :
( ord_less(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
fof(fact_60_rel__simps_I15_J,axiom,
! [K,L] :
( ord_less(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
fof(fact_61_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_62_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_63_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_64_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
fof(fact_65_is__mult__sum2sq,axiom,
! [Y_1,X_1] :
( twoSqu33214720sum2sq(X_1)
=> ( twoSqu33214720sum2sq(Y_1)
=> twoSqu33214720sum2sq(times_times(int,X_1,Y_1)) ) ) ).
fof(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),times_times(X_a,Ly,Ry)) ) ).
fof(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,times_times(X_a,Lx,Ly),Ry)) ) ).
fof(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,Lx,times_times(X_a,Ly,times_times(X_a,Rx,Ry))) ) ).
fof(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,times_times(X_a,Lx,Rx),Ly) ) ).
fof(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,Lx,times_times(X_a,Ly,Rx)) ) ).
fof(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),Ry) ) ).
fof(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,Lx,Ry)) ) ).
fof(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : times_times(X_a,A_1,B) = times_times(X_a,B,A_1) ) ).
fof(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D] : plus_plus(X_a,plus_plus(X_a,A_1,B),plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D)) ) ).
fof(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B) ) ).
fof(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,A_1,plus_plus(X_a,B,C)) ) ).
fof(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : plus_plus(X_a,A_1,plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),D) ) ).
fof(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : plus_plus(X_a,A_1,plus_plus(X_a,C,D)) = plus_plus(X_a,C,plus_plus(X_a,A_1,D)) ) ).
fof(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C] : plus_plus(X_a,A_1,C) = plus_plus(X_a,C,A_1) ) ).
fof(fact_80_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [X_2,Y_2] :
( number_number_of(X_a,X_2) = number_number_of(X_a,Y_2)
<=> X_2 = Y_2 ) ) ).
fof(fact_81_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [W_1,X_2] :
( number_number_of(X_a,W_1) = ti(X_a,X_2)
<=> ti(X_a,X_2) = number_number_of(X_a,W_1) ) ) ).
fof(fact_82_rel__simps_I51_J,axiom,
! [K,L] :
( bit1(K) = bit1(L)
<=> K = L ) ).
fof(fact_83_rel__simps_I48_J,axiom,
! [K,L] :
( bit0(K) = bit0(L)
<=> K = L ) ).
fof(fact_84_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_85_zmult__commute,axiom,
! [Z,W] : times_times(int,Z,W) = times_times(int,W,Z) ).
fof(fact_86_number__of__is__id,axiom,
! [K_1] : number_number_of(int,K_1) = K_1 ).
fof(fact_87_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_88_zadd__left__commute,axiom,
! [X_1,Y_1,Z] : plus_plus(int,X_1,plus_plus(int,Y_1,Z)) = plus_plus(int,Y_1,plus_plus(int,X_1,Z)) ).
fof(fact_89_zadd__commute,axiom,
! [Z,W] : plus_plus(int,Z,W) = plus_plus(int,W,Z) ).
fof(fact_90_rel__simps_I12_J,axiom,
! [K] :
( ord_less(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
fof(fact_91_less__int__code_I15_J,axiom,
! [K1,K2] :
( ord_less(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
fof(fact_92_rel__simps_I16_J,axiom,
! [K,L] :
( ord_less(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
fof(fact_93_rel__simps_I10_J,axiom,
! [K] :
( ord_less(int,bit0(K),pls)
<=> ord_less(int,K,pls) ) ).
fof(fact_94_rel__simps_I4_J,axiom,
! [K] :
( ord_less(int,pls,bit0(K))
<=> ord_less(int,pls,K) ) ).
fof(fact_95_rel__simps_I22_J,axiom,
! [K] :
( ord_less_eq(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
fof(fact_96_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_97_rel__simps_I32_J,axiom,
! [K,L] :
( ord_less_eq(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
%----Arities (15)
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___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
fof(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
fof(arity_Int_Oint___Rings_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___Int_Onumber,axiom,
number(int) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___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,
? [X,Y] : plus_plus(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),power_power(int,Y,number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) ).
%------------------------------------------------------------------------------