TPTP Problem File: NUM924^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_thf_l102 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.3.0
% Syntax : Number of formulae : 133 ( 50 unt; 26 typ; 0 def)
% Number of atoms : 184 ( 73 equ; 0 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 814 ( 13 ~; 4 |; 7 &; 724 @)
% ( 43 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% Number of symbols : 25 ( 24 usr; 8 con; 0-2 aty)
% Number of variables : 160 ( 0 ^; 160 !; 0 ?; 160 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:06:31
%------------------------------------------------------------------------------
%----Should-be-implicit typings (2)
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (24)
thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
thf(sy_c_Int_OBit0,type,
bit0: int > int ).
thf(sy_c_Int_OBit1,type,
bit1: int > int ).
thf(sy_c_Int_OPls,type,
pls: int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_TwoSquares__Mirabelle__fqdbopfjxb_Ois__sum2sq,type,
twoSqu1013291560sum2sq: int > $o ).
thf(sy_v_m,type,
m: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_t____,type,
t: int ).
%----Relevant facts (106)
thf(fact_0__096t_A_060_A0_096,axiom,
ord_less_int @ t @ zero_zero_int ).
thf(fact_1_calculation_I1_J,axiom,
ord_less_int @ t @ one_one_int ).
thf(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 ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ( ord_less_eq_int @ one_one_int @ t ) ).
thf(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 ) ).
thf(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 ) ).
thf(fact_8_sum__power2__gt__zero__iff,axiom,
! [X_10: int,Y_9: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_10 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_9 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_10 != zero_zero_int )
| ( Y_9 != zero_zero_int ) ) ) ).
thf(fact_9_sum__power2__eq__zero__iff,axiom,
! [X_9: int,Y_8: int] :
( ( ( plus_plus_int @ ( power_power_int @ X_9 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_int )
<=> ( ( X_9 = zero_zero_int )
& ( Y_8 = zero_zero_int ) ) ) ).
thf(fact_10_power2__less__0,axiom,
! [A_3: int] :
~ ( ord_less_int @ ( power_power_int @ A_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_11_zero__less__power2,axiom,
! [A_2: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_2 != zero_zero_int ) ) ).
thf(fact_12_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_13_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_14_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_15_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(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 ) ) ).
thf(fact_17_add__special_I2_J,axiom,
! [W_7: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_7 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_7 ) ) ) ).
thf(fact_18_add__special_I3_J,axiom,
! [V_12: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_12 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_12 @ ( bit1 @ pls ) ) ) ) ).
thf(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 ) ).
thf(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 ) ) ).
thf(fact_21_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_22_qf1pt,axiom,
twoSqu1013291560sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ).
thf(fact_23_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_24_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_25_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_26_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ).
thf(fact_29_le__number__of,axiom,
! [X_8: int,Y_7: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_8 ) @ ( number_number_of_int @ Y_7 ) )
<=> ( ord_less_eq_int @ X_8 @ Y_7 ) ) ).
thf(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 ) ) ) ).
thf(fact_31_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_32_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(fact_36_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(fact_39_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_40_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_41_eq__number__of__0,axiom,
! [V_2: int] :
( ( ( number_number_of_nat @ V_2 )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V_2 @ pls ) ) ).
thf(fact_42_eq__0__number__of,axiom,
! [V_2: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V_2 ) )
<=> ( ord_less_eq_int @ V_2 @ pls ) ) ).
thf(fact_43_semiring__mult__number__of,axiom,
! [V_11: int,V_10: int] :
( ( ord_less_eq_int @ pls @ V_10 )
=> ( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_10 ) @ ( number_number_of_nat @ V_11 ) )
= ( number_number_of_nat @ ( times_times_int @ V_10 @ V_11 ) ) ) ) ) ).
thf(fact_44_semiring__mult__number__of,axiom,
! [V_11: int,V_10: int] :
( ( ord_less_eq_int @ pls @ V_10 )
=> ( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( times_times_int @ ( number_number_of_int @ V_10 ) @ ( number_number_of_int @ V_11 ) )
= ( number_number_of_int @ ( times_times_int @ V_10 @ V_11 ) ) ) ) ) ).
thf(fact_45_mult__number__of__left,axiom,
! [V_9: int,W_6: int,Z_2: int] :
( ( times_times_int @ ( number_number_of_int @ V_9 ) @ ( times_times_int @ ( number_number_of_int @ W_6 ) @ Z_2 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_9 @ W_6 ) ) @ Z_2 ) ) ).
thf(fact_46_arith__simps_I32_J,axiom,
! [V_8: int,W_5: int] :
( ( times_times_int @ ( number_number_of_int @ V_8 ) @ ( number_number_of_int @ W_5 ) )
= ( number_number_of_int @ ( times_times_int @ V_8 @ W_5 ) ) ) ).
thf(fact_47_number__of__mult,axiom,
! [V_7: int,W_4: int] :
( ( number_number_of_int @ ( times_times_int @ V_7 @ W_4 ) )
= ( times_times_int @ ( number_number_of_int @ V_7 ) @ ( number_number_of_int @ W_4 ) ) ) ).
thf(fact_48_sum__squares__le__zero__iff,axiom,
! [X_7: int,Y_6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_7 @ X_7 ) @ ( times_times_int @ Y_6 @ Y_6 ) ) @ zero_zero_int )
<=> ( ( X_7 = zero_zero_int )
& ( Y_6 = zero_zero_int ) ) ) ).
thf(fact_49_sum__squares__ge__zero,axiom,
! [X_6: int,Y_5: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_6 @ X_6 ) @ ( times_times_int @ Y_5 @ Y_5 ) ) ) ).
thf(fact_50_le__special_I3_J,axiom,
! [X_5: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_5 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_5 @ pls ) ) ).
thf(fact_51_le__special_I1_J,axiom,
! [Y_4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_4 ) )
<=> ( ord_less_eq_int @ pls @ Y_4 ) ) ).
thf(fact_52_less__0__number__of,axiom,
! [V_2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V_2 ) )
<=> ( ord_less_int @ pls @ V_2 ) ) ).
thf(fact_53_le__number__of__eq__not__less,axiom,
! [V_6: int,W_3: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_6 ) @ ( number_number_of_int @ W_3 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_3 ) @ ( number_number_of_int @ V_6 ) ) ) ).
thf(fact_54_le__number__of__eq__not__less,axiom,
! [V_6: int,W_3: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_6 ) @ ( number_number_of_nat @ W_3 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_3 ) @ ( number_number_of_nat @ V_6 ) ) ) ).
thf(fact_55_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(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 ) ) ).
thf(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 ) ) ).
thf(fact_58_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_59_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_60_zadd__zless__mono,axiom,
! [Z_1: int,Z: int,W_2: int,W: int] :
( ( ord_less_int @ W_2 @ W )
=> ( ( ord_less_eq_int @ Z_1 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_2 @ Z_1 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_61_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_62_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_63_le__special_I4_J,axiom,
! [X_4: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_4 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_4 @ ( bit1 @ pls ) ) ) ).
thf(fact_64_le__special_I2_J,axiom,
! [Y_3: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_3 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_3 ) ) ).
thf(fact_65_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_66_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_67_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(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 ) ) ).
thf(fact_69_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_70_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_74_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_75_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(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 ) ) ).
thf(fact_77_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(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 ) ) ).
thf(fact_79_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_80_less__nat__number__of,axiom,
! [V_2: int,V_3: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V_2 ) @ ( number_number_of_nat @ V_3 ) )
<=> ( ( ( ord_less_int @ V_2 @ V_3 )
=> ( ord_less_int @ pls @ V_3 ) )
& ( ord_less_int @ V_2 @ V_3 ) ) ) ).
thf(fact_81_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_82_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_83_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(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 ) ) ).
thf(fact_85_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_86_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ).
thf(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 ) ) ) ) ) ) ) ).
thf(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 ) ) ) ).
thf(fact_91_eq__number__of,axiom,
! [X_3: int,Y_2: int] :
( ( ( number_number_of_int @ X_3 )
= ( number_number_of_int @ Y_2 ) )
<=> ( X_3 = Y_2 ) ) ).
thf(fact_92_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 ) ) ) ).
thf(fact_93_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 ) ) ) ).
thf(fact_94_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_95_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_96_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(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 ) ) ) ).
thf(fact_104_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_105_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
%----Conjectures (1)
thf(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 ).
%------------------------------------------------------------------------------