TPTP Problem File: NUM924_1.p
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%------------------------------------------------------------------------------
% File : NUM924_1 : 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_tff_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.00 v8.2.0, 0.10 v8.1.0, 0.18 v7.5.0, 0.25 v7.4.0, 0.00 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 133 ( 50 unt; 26 typ; 0 def)
% Number of atoms : 184 ( 73 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 90 ( 13 ~; 4 |; 7 &)
% ( 43 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 8 ( 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 : 18 ( 18 usr; 8 con; 0-2 aty)
% Number of variables : 160 ( 160 !; 0 ?; 160 :)
% SPC : TF0_CAX_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:24:03
%------------------------------------------------------------------------------
%----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 (24)
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_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_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 (106)
tff(fact_0__096t_A_060_A0_096,axiom,
ord_less_int(t,zero_zero_int) ).
tff(fact_1_calculation_I1_J,axiom,
ord_less_int(t,one_one_int) ).
tff(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)) ).
tff(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) ).
tff(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 ) ) ).
tff(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq_int(one_one_int,t) ).
tff(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)) ).
tff(fact_7_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] : ~ ord_less_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))))),zero_zero_int) ).
tff(fact_8_sum__power2__gt__zero__iff,axiom,
! [X_1: int,Y_1: int] :
( ord_less_int(zero_zero_int,plus_plus_int(power_power_int(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_1,number_number_of_nat(bit0(bit1(pls))))))
<=> ( ( X_1 != zero_zero_int )
| ( Y_1 != zero_zero_int ) ) ) ).
tff(fact_9_sum__power2__eq__zero__iff,axiom,
! [X_1: int,Y_1: int] :
( ( plus_plus_int(power_power_int(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_1,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int )
<=> ( ( X_1 = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_10_power2__less__0,axiom,
! [A_2: int] : ~ ord_less_int(power_power_int(A_2,number_number_of_nat(bit0(bit1(pls)))),zero_zero_int) ).
tff(fact_11_zero__less__power2,axiom,
! [A_1: int] :
( ord_less_int(zero_zero_int,power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))))
<=> ( A_1 != zero_zero_int ) ) ).
tff(fact_12_one__power2,axiom,
power_power_nat(one_one_nat,number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_13_one__power2,axiom,
power_power_int(one_one_int,number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_14_zero__power2,axiom,
power_power_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_15_zero__power2,axiom,
power_power_int(zero_zero_int,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_16_zero__eq__power2,axiom,
! [A_1: int] :
( ( power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_17_add__special_I2_J,axiom,
! [W_6: int] : plus_plus_int(one_one_int,number_number_of_int(W_6)) = number_number_of_int(plus_plus_int(bit1(pls),W_6)) ).
tff(fact_18_add__special_I3_J,axiom,
! [V_13: int] : plus_plus_int(number_number_of_int(V_13),one_one_int) = number_number_of_int(plus_plus_int(V_13,bit1(pls))) ).
tff(fact_19_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_20__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_21_p,axiom,
zprime(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) ).
tff(fact_22_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_23_zle__refl,axiom,
! [W: int] : ord_less_eq_int(W,W) ).
tff(fact_24_number__of__is__id,axiom,
! [K_1: int] : number_number_of_int(K_1) = K_1 ).
tff(fact_25_zmult__commute,axiom,
! [Z: int,W: int] : times_times_int(Z,W) = times_times_int(W,Z) ).
tff(fact_26_zle__linear,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
| ord_less_eq_int(W,Z) ) ).
tff(fact_27_times__numeral__code_I5_J,axiom,
! [V_2: int,W: int] : times_times_int(number_number_of_int(V_2),number_number_of_int(W)) = number_number_of_int(times_times_int(V_2,W)) ).
tff(fact_28_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_29_le__number__of,axiom,
! [X_1: int,Y_1: int] :
( ord_less_eq_int(number_number_of_int(X_1),number_number_of_int(Y_1))
<=> ord_less_eq_int(X_1,Y_1) ) ).
tff(fact_30_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_31_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_32_zle__antisym,axiom,
! [Z: int,W: int] :
( ord_less_eq_int(Z,W)
=> ( ord_less_eq_int(W,Z)
=> ( Z = W ) ) ) ).
tff(fact_33_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z: nat] : power_power_int(X,plus_plus_nat(Y,Z)) = times_times_int(power_power_int(X,Y),power_power_int(X,Z)) ).
tff(fact_34_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_35_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_36_rel__simps_I19_J,axiom,
ord_less_eq_int(pls,pls) ).
tff(fact_37_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_38_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_39_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_40_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_41_eq__number__of__0,axiom,
! [V_7: int] :
( ( number_number_of_nat(V_7) = zero_zero_nat )
<=> ord_less_eq_int(V_7,pls) ) ).
tff(fact_42_eq__0__number__of,axiom,
! [V_7: int] :
( ( zero_zero_nat = number_number_of_nat(V_7) )
<=> ord_less_eq_int(V_7,pls) ) ).
tff(fact_43_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ord_less_eq_int(pls,V_11)
=> ( ord_less_eq_int(pls,V_12)
=> ( times_times_nat(number_number_of_nat(V_11),number_number_of_nat(V_12)) = number_number_of_nat(times_times_int(V_11,V_12)) ) ) ) ).
tff(fact_44_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ord_less_eq_int(pls,V_11)
=> ( ord_less_eq_int(pls,V_12)
=> ( times_times_int(number_number_of_int(V_11),number_number_of_int(V_12)) = number_number_of_int(times_times_int(V_11,V_12)) ) ) ) ).
tff(fact_45_mult__number__of__left,axiom,
! [V_10: int,W_5: int,Z_3: int] : times_times_int(number_number_of_int(V_10),times_times_int(number_number_of_int(W_5),Z_3)) = times_times_int(number_number_of_int(times_times_int(V_10,W_5)),Z_3) ).
tff(fact_46_arith__simps_I32_J,axiom,
! [V_9: int,W_4: int] : times_times_int(number_number_of_int(V_9),number_number_of_int(W_4)) = number_number_of_int(times_times_int(V_9,W_4)) ).
tff(fact_47_number__of__mult,axiom,
! [V_8: int,W_3: int] : number_number_of_int(times_times_int(V_8,W_3)) = times_times_int(number_number_of_int(V_8),number_number_of_int(W_3)) ).
tff(fact_48_sum__squares__le__zero__iff,axiom,
! [X_1: int,Y_1: int] :
( ord_less_eq_int(plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_1,Y_1)),zero_zero_int)
<=> ( ( X_1 = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_49_sum__squares__ge__zero,axiom,
! [X_2: int,Y_2: int] : ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2))) ).
tff(fact_50_le__special_I3_J,axiom,
! [X_1: int] :
( ord_less_eq_int(number_number_of_int(X_1),zero_zero_int)
<=> ord_less_eq_int(X_1,pls) ) ).
tff(fact_51_le__special_I1_J,axiom,
! [Y_1: int] :
( ord_less_eq_int(zero_zero_int,number_number_of_int(Y_1))
<=> ord_less_eq_int(pls,Y_1) ) ).
tff(fact_52_less__0__number__of,axiom,
! [V_7: int] :
( ord_less_nat(zero_zero_nat,number_number_of_nat(V_7))
<=> ord_less_int(pls,V_7) ) ).
tff(fact_53_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( ord_less_eq_int(number_number_of_int(V_7),number_number_of_int(W_1))
<=> ~ ord_less_int(number_number_of_int(W_1),number_number_of_int(V_7)) ) ).
tff(fact_54_le__number__of__eq__not__less,axiom,
! [V_7: int,W_1: int] :
( ord_less_eq_nat(number_number_of_nat(V_7),number_number_of_nat(W_1))
<=> ~ ord_less_nat(number_number_of_nat(W_1),number_number_of_nat(V_7)) ) ).
tff(fact_55_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_56_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_57_rel__simps_I32_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit0(K),bit1(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_58_rel__simps_I27_J,axiom,
! [K: int] :
( ord_less_eq_int(bit0(K),pls)
<=> ord_less_eq_int(K,pls) ) ).
tff(fact_59_rel__simps_I21_J,axiom,
! [K: int] :
( ord_less_eq_int(pls,bit0(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_60_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_61_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_62_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_63_le__special_I4_J,axiom,
! [X_1: int] :
( ord_less_eq_int(number_number_of_int(X_1),one_one_int)
<=> ord_less_eq_int(X_1,bit1(pls)) ) ).
tff(fact_64_le__special_I2_J,axiom,
! [Y_1: int] :
( ord_less_eq_int(one_one_int,number_number_of_int(Y_1))
<=> ord_less_eq_int(bit1(pls),Y_1) ) ).
tff(fact_65_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_66_mult__Pls,axiom,
! [W: int] : times_times_int(pls,W) = pls ).
tff(fact_67_mult__Bit0,axiom,
! [K_1: int,L_1: int] : times_times_int(bit0(K_1),L_1) = bit0(times_times_int(K_1,L_1)) ).
tff(fact_68_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_69_zmult__1__right,axiom,
! [Z: int] : times_times_int(Z,one_one_int) = Z ).
tff(fact_70_zmult__1,axiom,
! [Z: int] : times_times_int(one_one_int,Z) = Z ).
tff(fact_71_plus__numeral__code_I9_J,axiom,
! [V_2: int,W: int] : plus_plus_int(number_number_of_int(V_2),number_number_of_int(W)) = number_number_of_int(plus_plus_int(V_2,W)) ).
tff(fact_72_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
tff(fact_73_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
tff(fact_74_rel__simps_I29_J,axiom,
! [K: int] :
( ord_less_eq_int(bit1(K),pls)
<=> ord_less_int(K,pls) ) ).
tff(fact_75_rel__simps_I5_J,axiom,
! [K: int] :
( ord_less_int(pls,bit1(K))
<=> ord_less_eq_int(pls,K) ) ).
tff(fact_76_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_77_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ord_less_eq_int(bit1(K),bit0(L))
<=> ord_less_int(K,L) ) ).
tff(fact_78_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_79_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ord_less_int(bit0(K),bit1(L))
<=> ord_less_eq_int(K,L) ) ).
tff(fact_80_less__nat__number__of,axiom,
! [V_7: int,V_6: int] :
( ord_less_nat(number_number_of_nat(V_7),number_number_of_nat(V_6))
<=> ( ( ord_less_int(V_7,V_6)
=> ord_less_int(pls,V_6) )
& ord_less_int(V_7,V_6) ) ) ).
tff(fact_81_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( ord_less_eq_int(one_one_int,Z_1)
<=> ord_less_int(zero_zero_int,Z_1) ) ).
tff(fact_82_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_83_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_84_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_85_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_86_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_87_semiring__add__number__of,axiom,
! [V_5: int,V_4: int] :
( ord_less_eq_int(pls,V_4)
=> ( ord_less_eq_int(pls,V_5)
=> ( plus_plus_int(number_number_of_int(V_4),number_number_of_int(V_5)) = number_number_of_int(plus_plus_int(V_4,V_5)) ) ) ) ).
tff(fact_88_semiring__add__number__of,axiom,
! [V_5: int,V_4: int] :
( ord_less_eq_int(pls,V_4)
=> ( ord_less_eq_int(pls,V_5)
=> ( plus_plus_nat(number_number_of_nat(V_4),number_number_of_nat(V_5)) = number_number_of_nat(plus_plus_int(V_4,V_5)) ) ) ) ).
tff(fact_89_add__nat__number__of,axiom,
! [V_3: int,V_2: int] :
( ( ord_less_int(V_2,pls)
=> ( plus_plus_nat(number_number_of_nat(V_2),number_number_of_nat(V_3)) = number_number_of_nat(V_3) ) )
& ( ~ ord_less_int(V_2,pls)
=> ( ( ord_less_int(V_3,pls)
=> ( plus_plus_nat(number_number_of_nat(V_2),number_number_of_nat(V_3)) = number_number_of_nat(V_2) ) )
& ( ~ ord_less_int(V_3,pls)
=> ( plus_plus_nat(number_number_of_nat(V_2),number_number_of_nat(V_3)) = number_number_of_nat(plus_plus_int(V_2,V_3)) ) ) ) ) ) ).
tff(fact_90_le__imp__0__less,axiom,
! [Z: int] :
( ord_less_eq_int(zero_zero_int,Z)
=> ord_less_int(zero_zero_int,plus_plus_int(one_one_int,Z)) ) ).
tff(fact_91_eq__number__of,axiom,
! [X_1: int,Y_1: int] :
( ( number_number_of_int(X_1) = number_number_of_int(Y_1) )
<=> ( X_1 = Y_1 ) ) ).
tff(fact_92_number__of__reorient,axiom,
! [W_1: int,X_1: nat] :
( ( number_number_of_nat(W_1) = X_1 )
<=> ( X_1 = number_number_of_nat(W_1) ) ) ).
tff(fact_93_number__of__reorient,axiom,
! [W_1: int,X_1: int] :
( ( number_number_of_int(W_1) = X_1 )
<=> ( X_1 = number_number_of_int(W_1) ) ) ).
tff(fact_94_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_95_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_96_zless__linear,axiom,
! [X: int,Y: int] :
( ord_less_int(X,Y)
| ( X = Y )
| ord_less_int(Y,X) ) ).
tff(fact_97_sum__squares__eq__zero__iff,axiom,
! [X_1: int,Y_1: int] :
( ( plus_plus_int(times_times_int(X_1,X_1),times_times_int(Y_1,Y_1)) = zero_zero_int )
<=> ( ( X_1 = zero_zero_int )
& ( Y_1 = zero_zero_int ) ) ) ).
tff(fact_98_left__distrib__number__of,axiom,
! [A: int,B_1: int,V_1: int] : times_times_int(plus_plus_int(A,B_1),number_number_of_int(V_1)) = plus_plus_int(times_times_int(A,number_number_of_int(V_1)),times_times_int(B_1,number_number_of_int(V_1))) ).
tff(fact_99_left__distrib__number__of,axiom,
! [A: nat,B_1: nat,V_1: int] : times_times_nat(plus_plus_nat(A,B_1),number_number_of_nat(V_1)) = plus_plus_nat(times_times_nat(A,number_number_of_nat(V_1)),times_times_nat(B_1,number_number_of_nat(V_1))) ).
tff(fact_100_right__distrib__number__of,axiom,
! [V: int,B: int,C: int] : times_times_int(number_number_of_int(V),plus_plus_int(B,C)) = plus_plus_int(times_times_int(number_number_of_int(V),B),times_times_int(number_number_of_int(V),C)) ).
tff(fact_101_right__distrib__number__of,axiom,
! [V: int,B: nat,C: nat] : times_times_nat(number_number_of_nat(V),plus_plus_nat(B,C)) = plus_plus_nat(times_times_nat(number_number_of_nat(V),B),times_times_nat(number_number_of_nat(V),C)) ).
tff(fact_102_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_103_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] : plus_plus_int(X,plus_plus_int(Y,Z)) = plus_plus_int(Y,plus_plus_int(X,Z)) ).
tff(fact_104_zadd__commute,axiom,
! [Z: int,W: int] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
tff(fact_105_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
%----Conjectures (1)
tff(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) ).
%------------------------------------------------------------------------------