TPTP Problem File: NUM926_1.p
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%------------------------------------------------------------------------------
% File : NUM926_1 : 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_tff_l258 [Bla11]
% Status : Theorem
% Rating : 0.11 v8.2.0, 0.10 v8.1.0, 0.09 v7.5.0, 0.12 v7.4.0, 0.00 v6.0.0, 0.20 v5.5.0, 0.25 v5.4.0, 0.67 v5.3.0
% Syntax : Number of formulae : 148 ( 76 unt; 24 typ; 0 def)
% Number of atoms : 183 ( 87 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 67 ( 8 ~; 3 |; 3 &)
% ( 36 <=>; 17 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 26 ( 16 >; 10 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 16 ( 16 usr; 6 con; 0-2 aty)
% Number of variables : 244 ( 238 !; 6 ?; 244 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:32:47
%------------------------------------------------------------------------------
%----Should-be-implicit typings (2)
tff(ty_ty_tc__Int__Oint,type,
int: $tType ).
tff(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (22)
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: ( int * int ) > int ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: ( nat * nat ) > nat ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: ( int * int ) > int ).
tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: ( nat * nat ) > nat ).
tff(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
tff(sy_c_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: ( int * int ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: ( nat * nat ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: ( int * int ) > $o ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: ( nat * nat ) > $o ).
tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: ( int * nat ) > int ).
tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: ( nat * nat ) > nat ).
tff(sy_c_TwoSquares__Mirabelle__wxgjvpjuzc_Ois__sum2sq,type,
twoSqu1431725154sum2sq: int > $o ).
tff(sy_v_m,type,
m: int ).
tff(sy_v_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (123)
tff(fact_0_tpos,axiom,
ord_less_eq_int(one_one_int,t) ).
tff(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: int,Y: int] : 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) ) ).
tff(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: int,Y: int] : 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) ) ).
tff(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)) ).
tff(fact_4_p,axiom,
zprime(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(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) ).
tff(fact_6_qf1pt,axiom,
twoSqu1431725154sum2sq(times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),t)) ).
tff(fact_7_zadd__power2,axiom,
! [A_8: int,B_4: int] : power_power_int(plus_plus_int(A_8,B_4),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(power_power_int(A_8,number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A_8),B_4)),power_power_int(B_4,number_number_of_nat(bit0(bit1(pls))))) ).
tff(fact_8_zadd__power3,axiom,
! [A_8: int,B_4: int] : power_power_int(plus_plus_int(A_8,B_4),number_number_of_nat(bit1(bit1(pls)))) = plus_plus_int(plus_plus_int(plus_plus_int(power_power_int(A_8,number_number_of_nat(bit1(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),power_power_int(A_8,number_number_of_nat(bit0(bit1(pls))))),B_4)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A_8),power_power_int(B_4,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B_4,number_number_of_nat(bit1(bit1(pls))))) ).
tff(fact_9_power2__sum,axiom,
! [X_2: nat,Y_2: nat] : power_power_nat(plus_plus_nat(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(plus_plus_nat(power_power_nat(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),X_2),Y_2)) ).
tff(fact_10_power2__sum,axiom,
! [X_2: int,Y_2: int] : power_power_int(plus_plus_int(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),X_2),Y_2)) ).
tff(fact_11_power2__eq__square__number__of,axiom,
! [W_4: int] : power_power_int(number_number_of_int(W_4),number_number_of_nat(bit0(bit1(pls)))) = times_times_int(number_number_of_int(W_4),number_number_of_int(W_4)) ).
tff(fact_12_power2__eq__square__number__of,axiom,
! [W_4: int] : power_power_nat(number_number_of_nat(W_4),number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(number_number_of_nat(W_4),number_number_of_nat(W_4)) ).
tff(fact_13_cube__square,axiom,
! [A_8: int] : times_times_int(A_8,power_power_int(A_8,number_number_of_nat(bit0(bit1(pls))))) = power_power_int(A_8,number_number_of_nat(bit1(bit1(pls)))) ).
tff(fact_14_one__power2,axiom,
power_power_nat(one_one_nat,number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_15_one__power2,axiom,
power_power_int(one_one_int,number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_16_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_7: int] : times_times_int(X_7,X_7) = power_power_int(X_7,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_17_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_7: nat] : times_times_nat(X_7,X_7) = power_power_nat(X_7,number_number_of_nat(bit0(bit1(pls)))) ).
tff(fact_18_power2__eq__square,axiom,
! [A_7: int] : power_power_int(A_7,number_number_of_nat(bit0(bit1(pls)))) = times_times_int(A_7,A_7) ).
tff(fact_19_power2__eq__square,axiom,
! [A_7: nat] : power_power_nat(A_7,number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(A_7,A_7) ).
tff(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_6: int,N: nat] : power_power_int(X_6,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N)) = times_times_int(power_power_int(X_6,N),power_power_int(X_6,N)) ).
tff(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_6: nat,N: nat] : power_power_nat(X_6,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N)) = times_times_nat(power_power_nat(X_6,N),power_power_nat(X_6,N)) ).
tff(fact_22_add__special_I2_J,axiom,
! [W_3: int] : plus_plus_int(one_one_int,number_number_of_int(W_3)) = number_number_of_int(plus_plus_int(bit1(pls),W_3)) ).
tff(fact_23_add__special_I3_J,axiom,
! [V_3: int] : plus_plus_int(number_number_of_int(V_3),one_one_int) = number_number_of_int(plus_plus_int(V_3,bit1(pls))) ).
tff(fact_24_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_25__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: int] : 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) ).
tff(fact_26_zle__refl,axiom,
! [W: int] : ord_less_eq_int(W,W) ).
tff(fact_27_zle__linear,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
| ord_less_eq_int(W,Z) ) ).
tff(fact_28_zless__le,axiom,
! [Z_1: int,W_1: int] :
( ord_less_int(Z_1,W_1)
<=> ( ord_less_eq_int(Z_1,W_1)
& ( Z_1 != W_1 ) ) ) ).
tff(fact_29_zless__linear,axiom,
! [X_1: int,Y_1: int] :
( ord_less_int(X_1,Y_1)
| ( X_1 = Y_1 )
| ord_less_int(Y_1,X_1) ) ).
tff(fact_30_zle__trans,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ( ord_less_eq_int(J,K_1)
=> ord_less_eq_int(I,K_1) ) ) ).
tff(fact_31_zle__antisym,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
=> ( ord_less_eq_int(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_5: int,P_1: nat,Q_1: nat] : power_power_int(power_power_int(X_5,P_1),Q_1) = power_power_int(X_5,times_times_nat(P_1,Q_1)) ).
tff(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_5: nat,P_1: nat,Q_1: nat] : power_power_nat(power_power_nat(X_5,P_1),Q_1) = power_power_nat(X_5,times_times_nat(P_1,Q_1)) ).
tff(fact_34_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_4: int] : power_power_int(X_4,one_one_nat) = X_4 ).
tff(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_4: nat] : power_power_nat(X_4,one_one_nat) = X_4 ).
tff(fact_36_zpower__zpower,axiom,
! [X_1: int,Y_1: nat,Z: nat] : power_power_int(power_power_int(X_1,Y_1),Z) = power_power_int(X_1,times_times_nat(Y_1,Z)) ).
tff(fact_37_le__number__of__eq__not__less,axiom,
! [V_2: int,W_1: int] :
( ord_less_eq_nat(number_number_of_nat(V_2),number_number_of_nat(W_1))
<=> ~ ord_less_nat(number_number_of_nat(W_1),number_number_of_nat(V_2)) ) ).
tff(fact_38_le__number__of__eq__not__less,axiom,
! [V_2: int,W_1: int] :
( ord_less_eq_int(number_number_of_int(V_2),number_number_of_int(W_1))
<=> ~ ord_less_int(number_number_of_int(W_1),number_number_of_int(V_2)) ) ).
tff(fact_39_less__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_int(number_number_of_int(X_2),number_number_of_int(Y_2))
<=> ord_less_int(X_2,Y_2) ) ).
tff(fact_40_le__number__of,axiom,
! [X_2: int,Y_2: int] :
( ord_less_eq_int(number_number_of_int(X_2),number_number_of_int(Y_2))
<=> ord_less_eq_int(X_2,Y_2) ) ).
tff(fact_41_zadd__zless__mono,axiom,
! [Z_2: int,Z: int,W_2: int,W: int] :
( 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)) ) ) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_3: int,P: nat,Q: nat] : times_times_int(power_power_int(X_3,P),power_power_int(X_3,Q)) = power_power_int(X_3,plus_plus_nat(P,Q)) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_3: nat,P: nat,Q: nat] : times_times_nat(power_power_nat(X_3,P),power_power_nat(X_3,Q)) = power_power_nat(X_3,plus_plus_nat(P,Q)) ).
tff(fact_44_zpower__zadd__distrib,axiom,
! [X_1: int,Y_1: nat,Z: nat] : 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)) ).
tff(fact_45_nat__mult__2,axiom,
! [Z: nat] : times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z) = plus_plus_nat(Z,Z) ).
tff(fact_46_nat__mult__2__right,axiom,
! [Z: nat] : times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z) ).
tff(fact_47_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_48_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit1(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_49_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ord_less_int(bit1(K),bit1(L))
<=> ord_less_int(K,L) ) ).
tff(fact_50_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit1(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_51_rel__simps_I34_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit1(K),bit1(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_52_rel__simps_I2_J,axiom,
~ ord_less_int(pls,pls) ).
tff(fact_53_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_54_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ord_less_int(bit0(K),bit0(L))
<=> ord_less_int(K,L) ) ).
tff(fact_55_rel__simps_I19_J,axiom,
ord_less_eq_int(pls,pls) ).
tff(fact_56_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit0(K1),bit0(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_57_rel__simps_I31_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit0(K),bit0(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_58_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ord_less_int(number_number_of_int(K),number_number_of_int(L))
<=> ord_less_int(K,L) ) ).
tff(fact_59_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ord_less_eq_int(number_number_of_int(K),number_number_of_int(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_60_zadd__strict__right__mono,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_int(I,J)
=> ord_less_int(plus_plus_int(I,K_1),plus_plus_int(J,K_1)) ) ).
tff(fact_61_zadd__left__mono,axiom,
! [K_1: int,I: int,J: int] :
( ord_less_eq_int(I,J)
=> ord_less_eq_int(plus_plus_int(K_1,I),plus_plus_int(K_1,J)) ) ).
tff(fact_62_add__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( 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)) ) ) ) ) ) ).
tff(fact_63_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_64_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_65_rel__simps_I29_J,axiom,
! [K: int] :
( ord_less_eq_int(bit1(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_66_rel__simps_I5_J,axiom,
! [K: int] :
( ord_less_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_67_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_68_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit1(K),bit0(L))
<=> ord_less_int(K,L) ) ).
tff(fact_69_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_70_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ord_less_int(bit0(K),bit1(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_71_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ord_less_int(W,Z)
=> ord_less_eq_int(plus_plus_int(W,one_one_int),Z) ) ).
tff(fact_72_add1__zle__eq,axiom,
! [W_1: int,Z_1: int] :
( ord_less_eq_int(plus_plus_int(W_1,one_one_int),Z_1)
<=> ord_less_int(W_1,Z_1) ) ).
tff(fact_73_zle__add1__eq__le,axiom,
! [W_1: int,Z_1: int] :
( ord_less_int(W_1,plus_plus_int(Z_1,one_one_int))
<=> ord_less_eq_int(W_1,Z_1) ) ).
tff(fact_74_zprime__2,axiom,
zprime(number_number_of_int(bit0(bit1(pls)))) ).
tff(fact_75_is__mult__sum2sq,axiom,
! [Y_1: int,X_1: int] :
( twoSqu1431725154sum2sq(X_1)
=> ( twoSqu1431725154sum2sq(Y_1)
=> twoSqu1431725154sum2sq(times_times_int(X_1,Y_1)) ) ) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] : times_times_int(times_times_int(Lx_6,Ly_4),times_times_int(Rx_6,Ry_4)) = times_times_int(times_times_int(Lx_6,Rx_6),times_times_int(Ly_4,Ry_4)) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] : times_times_nat(times_times_nat(Lx_6,Ly_4),times_times_nat(Rx_6,Ry_4)) = times_times_nat(times_times_nat(Lx_6,Rx_6),times_times_nat(Ly_4,Ry_4)) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] : times_times_int(times_times_int(Lx_5,Ly_3),times_times_int(Rx_5,Ry_3)) = times_times_int(Rx_5,times_times_int(times_times_int(Lx_5,Ly_3),Ry_3)) ).
tff(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] : times_times_nat(times_times_nat(Lx_5,Ly_3),times_times_nat(Rx_5,Ry_3)) = times_times_nat(Rx_5,times_times_nat(times_times_nat(Lx_5,Ly_3),Ry_3)) ).
tff(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] : times_times_int(times_times_int(Lx_4,Ly_2),times_times_int(Rx_4,Ry_2)) = times_times_int(Lx_4,times_times_int(Ly_2,times_times_int(Rx_4,Ry_2))) ).
tff(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] : times_times_nat(times_times_nat(Lx_4,Ly_2),times_times_nat(Rx_4,Ry_2)) = times_times_nat(Lx_4,times_times_nat(Ly_2,times_times_nat(Rx_4,Ry_2))) ).
tff(fact_82_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] : times_times_int(times_times_int(Lx_3,Ly_1),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_1) ).
tff(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] : times_times_nat(times_times_nat(Lx_3,Ly_1),Rx_3) = times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_1) ).
tff(fact_84_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] : times_times_int(times_times_int(Lx_2,Ly),Rx_2) = times_times_int(Lx_2,times_times_int(Ly,Rx_2)) ).
tff(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] : times_times_nat(times_times_nat(Lx_2,Ly),Rx_2) = times_times_nat(Lx_2,times_times_nat(Ly,Rx_2)) ).
tff(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] : times_times_int(Lx_1,times_times_int(Rx_1,Ry_1)) = times_times_int(times_times_int(Lx_1,Rx_1),Ry_1) ).
tff(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] : times_times_nat(Lx_1,times_times_nat(Rx_1,Ry_1)) = times_times_nat(times_times_nat(Lx_1,Rx_1),Ry_1) ).
tff(fact_88_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] : times_times_int(Lx,times_times_int(Rx,Ry)) = times_times_int(Rx,times_times_int(Lx,Ry)) ).
tff(fact_89_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] : times_times_nat(Lx,times_times_nat(Rx,Ry)) = times_times_nat(Rx,times_times_nat(Lx,Ry)) ).
tff(fact_90_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_6: int,B_3: int] : times_times_int(A_6,B_3) = times_times_int(B_3,A_6) ).
tff(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_6: nat,B_3: nat] : times_times_nat(A_6,B_3) = times_times_nat(B_3,A_6) ).
tff(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_5: int,B_2: int,C_5: int,D_2: int] : plus_plus_int(plus_plus_int(A_5,B_2),plus_plus_int(C_5,D_2)) = plus_plus_int(plus_plus_int(A_5,C_5),plus_plus_int(B_2,D_2)) ).
tff(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_5: nat,B_2: nat,C_5: nat,D_2: nat] : plus_plus_nat(plus_plus_nat(A_5,B_2),plus_plus_nat(C_5,D_2)) = plus_plus_nat(plus_plus_nat(A_5,C_5),plus_plus_nat(B_2,D_2)) ).
tff(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_4: int,B_1: int,C_4: int] : plus_plus_int(plus_plus_int(A_4,B_1),C_4) = plus_plus_int(plus_plus_int(A_4,C_4),B_1) ).
tff(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_4: nat,B_1: nat,C_4: nat] : plus_plus_nat(plus_plus_nat(A_4,B_1),C_4) = plus_plus_nat(plus_plus_nat(A_4,C_4),B_1) ).
tff(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_3: int,B: int,C_3: int] : plus_plus_int(plus_plus_int(A_3,B),C_3) = plus_plus_int(A_3,plus_plus_int(B,C_3)) ).
tff(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_3: nat,B: nat,C_3: nat] : plus_plus_nat(plus_plus_nat(A_3,B),C_3) = plus_plus_nat(A_3,plus_plus_nat(B,C_3)) ).
tff(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_2: int,C_2: int,D_1: int] : plus_plus_int(A_2,plus_plus_int(C_2,D_1)) = plus_plus_int(plus_plus_int(A_2,C_2),D_1) ).
tff(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_2: nat,C_2: nat,D_1: nat] : plus_plus_nat(A_2,plus_plus_nat(C_2,D_1)) = plus_plus_nat(plus_plus_nat(A_2,C_2),D_1) ).
tff(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_1: int,C_1: int,D: int] : plus_plus_int(A_1,plus_plus_int(C_1,D)) = plus_plus_int(C_1,plus_plus_int(A_1,D)) ).
tff(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_1: nat,C_1: nat,D: nat] : plus_plus_nat(A_1,plus_plus_nat(C_1,D)) = plus_plus_nat(C_1,plus_plus_nat(A_1,D)) ).
tff(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: int,C: int] : plus_plus_int(A,C) = plus_plus_int(C,A) ).
tff(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: nat,C: nat] : plus_plus_nat(A,C) = plus_plus_nat(C,A) ).
tff(fact_104_eq__number__of,axiom,
! [X_2: int,Y_2: int] :
( ( number_number_of_int(X_2) = number_number_of_int(Y_2) )
<=> ( X_2 = Y_2 ) ) ).
tff(fact_105_number__of__reorient,axiom,
! [W_1: int,X_2: nat] :
( ( number_number_of_nat(W_1) = X_2 )
<=> ( X_2 = number_number_of_nat(W_1) ) ) ).
tff(fact_106_number__of__reorient,axiom,
! [W_1: int,X_2: int] :
( ( number_number_of_int(W_1) = X_2 )
<=> ( X_2 = number_number_of_int(W_1) ) ) ).
tff(fact_107_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_108_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_109_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
tff(fact_110_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_111_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_112_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
tff(fact_113_zadd__left__commute,axiom,
! [X_1: int,Y_1: int,Z: int] : plus_plus_int(X_1,plus_plus_int(Y_1,Z)) = plus_plus_int(Y_1,plus_plus_int(X_1,Z)) ).
tff(fact_114_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_115_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less_int(bit1(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_116_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_117_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ord_less_int(bit1(K),bit0(L))
<=> ord_less_int(K,L) ) ).
tff(fact_118_rel__simps_I10_J,axiom,
! [K: int] :
( ord_less_int(bit0(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_119_rel__simps_I4_J,axiom,
! [K: int] :
( ord_less_int(pls,bit0(K))
<=> ord_less_int(pls,K) ) ).
tff(fact_120_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_121_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ord_less_eq_int(bit0(K1),bit1(K2))
<=> ord_less_eq_int(K1,K2) ) ).
tff(fact_122_rel__simps_I32_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit0(K),bit1(L))
<=> ord_less_eq_int(K,L) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [X: int,Y: int] : 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) ).
%------------------------------------------------------------------------------