TPTP Problem File: NUM925_1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925_1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 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_l192 [Bla11]
% Status : Theorem
% Rating : 0.00 v5.3.0
% Syntax : Number of formulae : 128 ( 66 unt; 21 typ; 0 def)
% Number of atoms : 163 ( 105 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 76 ( 20 ~; 4 |; 7 &)
% ( 36 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 18 ( 12 >; 6 *; 0 +; 0 <<)
% Number of predicates : 3 ( 2 usr; 0 prp; 2-2 aty)
% Number of functors : 17 ( 17 usr; 7 con; 0-2 aty)
% Number of variables : 118 ( 118 !; 0 ?; 118 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:29:12
%------------------------------------------------------------------------------
%----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 (19)
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_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_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_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: nat > int ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,
semiri984289939at_nat: nat > 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_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_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (106)
tff(fact_0_n1pos,axiom,
ord_less_int(zero_zero_int,plus_plus_int(one_one_int,semiri1621563631at_int(n))) ).
tff(fact_1_t1,axiom,
ord_less_int(one_one_int,t) ).
tff(fact_2_sum__power2__eq__zero__iff,axiom,
! [Xa: int,Ya: int] :
( ( plus_plus_int(power_power_int(Xa,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Ya,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int )
<=> ( ( Xa = zero_zero_int )
& ( Ya = zero_zero_int ) ) ) ).
tff(fact_3_one__power2,axiom,
power_power_int(one_one_int,number_number_of_nat(bit0(bit1(pls)))) = one_one_int ).
tff(fact_4_one__power2,axiom,
power_power_nat(one_one_nat,number_number_of_nat(bit0(bit1(pls)))) = one_one_nat ).
tff(fact_5_zero__power2,axiom,
power_power_int(zero_zero_int,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int ).
tff(fact_6_zero__power2,axiom,
power_power_nat(zero_zero_nat,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_nat ).
tff(fact_7_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_8_add__special_I2_J,axiom,
! [W_5: int] : plus_plus_int(one_one_int,number_number_of_int(W_5)) = number_number_of_int(plus_plus_int(bit1(pls),W_5)) ).
tff(fact_9_add__special_I3_J,axiom,
! [V_5: int] : plus_plus_int(number_number_of_int(V_5),one_one_int) = number_number_of_int(plus_plus_int(V_5,bit1(pls))) ).
tff(fact_10_one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_11_semiring__one__add__one__is__two,axiom,
plus_plus_int(one_one_int,one_one_int) = number_number_of_int(bit0(bit1(pls))) ).
tff(fact_12_semiring__one__add__one__is__two,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_13_quartic__square__square,axiom,
! [X: int] : power_power_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),number_number_of_nat(bit0(bit1(pls)))) = power_power_int(X,number_number_of_nat(bit0(bit0(bit1(pls))))) ).
tff(fact_14_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_4)) = one_one_int ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( power_power_int(zero_zero_int,number_number_of_nat(W_4)) = zero_zero_int ) ) ) ).
tff(fact_15_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( number_number_of_nat(W_4) = zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_4)) = one_one_nat ) )
& ( ( number_number_of_nat(W_4) != zero_zero_nat )
=> ( power_power_nat(zero_zero_nat,number_number_of_nat(W_4)) = zero_zero_nat ) ) ) ).
tff(fact_16_semiring__norm_I110_J,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_17_numeral__1__eq__1,axiom,
number_number_of_int(bit1(pls)) = one_one_int ).
tff(fact_18_n0,axiom,
ord_less_nat(zero_zero_nat,n) ).
tff(fact_19_zless__linear,axiom,
! [X: int,Y: int] :
( ord_less_int(X,Y)
| ( X = Y )
| ord_less_int(Y,X) ) ).
tff(fact_20_less__number__of__int__code,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(number_number_of_int(K_1),number_number_of_int(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_21_plus__numeral__code_I9_J,axiom,
! [V_3: int,W_3: int] : plus_plus_int(number_number_of_int(V_3),number_number_of_int(W_3)) = number_number_of_int(plus_plus_int(V_3,W_3)) ).
tff(fact_22_less__number__of,axiom,
! [Xa: int,Ya: int] :
( ord_less_int(number_number_of_int(Xa),number_number_of_int(Ya))
<=> ord_less_int(Xa,Ya) ) ).
tff(fact_23_zero__is__num__zero,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_24_zpower__int,axiom,
! [M: nat,N_1: nat] : power_power_int(semiri1621563631at_int(M),N_1) = semiri1621563631at_int(power_power_nat(M,N_1)) ).
tff(fact_25_int__power,axiom,
! [M: nat,N_1: nat] : semiri1621563631at_int(power_power_nat(M,N_1)) = power_power_int(semiri1621563631at_int(M),N_1) ).
tff(fact_26_zadd__int__left,axiom,
! [M: nat,N_1: nat,Z: int] : plus_plus_int(semiri1621563631at_int(M),plus_plus_int(semiri1621563631at_int(N_1),Z)) = plus_plus_int(semiri1621563631at_int(plus_plus_nat(M,N_1)),Z) ).
tff(fact_27_zadd__int,axiom,
! [M: nat,N_1: nat] : plus_plus_int(semiri1621563631at_int(M),semiri1621563631at_int(N_1)) = semiri1621563631at_int(plus_plus_nat(M,N_1)) ).
tff(fact_28_int__1,axiom,
semiri1621563631at_int(one_one_nat) = one_one_int ).
tff(fact_29_nat__number__of__Pls,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_30_semiring__norm_I113_J,axiom,
zero_zero_nat = number_number_of_nat(pls) ).
tff(fact_31_int__eq__0__conv,axiom,
! [Na: nat] :
( ( semiri1621563631at_int(Na) = zero_zero_int )
<=> ( Na = zero_zero_nat ) ) ).
tff(fact_32_int__0,axiom,
semiri1621563631at_int(zero_zero_nat) = zero_zero_int ).
tff(fact_33_nat__1__add__1,axiom,
plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) ).
tff(fact_34_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit1(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_35_rel__simps_I17_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit1(K_1),bit1(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_36_rel__simps_I2_J,axiom,
~ ord_less_int(pls,pls) ).
tff(fact_37_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit0(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_38_rel__simps_I14_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit0(K_1),bit0(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_39_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( ord_less_int(I,J)
=> ord_less_int(plus_plus_int(I,K),plus_plus_int(J,K)) ) ).
tff(fact_40_add__nat__number__of,axiom,
! [V_4: int,V_3: int] :
( ( ord_less_int(V_3,pls)
=> ( plus_plus_nat(number_number_of_nat(V_3),number_number_of_nat(V_4)) = number_number_of_nat(V_4) ) )
& ( ~ ord_less_int(V_3,pls)
=> ( ( ord_less_int(V_4,pls)
=> ( plus_plus_nat(number_number_of_nat(V_3),number_number_of_nat(V_4)) = number_number_of_nat(V_3) ) )
& ( ~ ord_less_int(V_4,pls)
=> ( plus_plus_nat(number_number_of_nat(V_3),number_number_of_nat(V_4)) = number_number_of_nat(plus_plus_int(V_3,V_4)) ) ) ) ) ) ).
tff(fact_41_one__is__num__one,axiom,
one_one_int = number_number_of_int(bit1(pls)) ).
tff(fact_42_nat__numeral__1__eq__1,axiom,
number_number_of_nat(bit1(pls)) = one_one_nat ).
tff(fact_43_Numeral1__eq1__nat,axiom,
one_one_nat = number_number_of_nat(bit1(pls)) ).
tff(fact_44_eq__number__of,axiom,
! [Xa: int,Ya: int] :
( ( number_number_of_int(Xa) = number_number_of_int(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_45_number__of__reorient,axiom,
! [Wa: int,Xa: nat] :
( ( number_number_of_nat(Wa) = Xa )
<=> ( Xa = number_number_of_nat(Wa) ) ) ).
tff(fact_46_number__of__reorient,axiom,
! [Wa: int,Xa: int] :
( ( number_number_of_int(Wa) = Xa )
<=> ( Xa = number_number_of_int(Wa) ) ) ).
tff(fact_47_rel__simps_I51_J,axiom,
! [K_1: int,L_1: int] :
( ( bit1(K_1) = bit1(L_1) )
<=> ( K_1 = L_1 ) ) ).
tff(fact_48_rel__simps_I48_J,axiom,
! [K_1: int,L_1: int] :
( ( bit0(K_1) = bit0(L_1) )
<=> ( K_1 = L_1 ) ) ).
tff(fact_49_even__less__0__iff,axiom,
! [A_1: int] :
( ord_less_int(plus_plus_int(A_1,A_1),zero_zero_int)
<=> ord_less_int(A_1,zero_zero_int) ) ).
tff(fact_50_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_51_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_52_zadd__commute,axiom,
! [Z: int,W_3: int] : plus_plus_int(Z,W_3) = plus_plus_int(W_3,Z) ).
tff(fact_53_int__int__eq,axiom,
! [Ma: nat,Na: nat] :
( ( semiri1621563631at_int(Ma) = semiri1621563631at_int(Na) )
<=> ( Ma = Na ) ) ).
tff(fact_54_less__special_I3_J,axiom,
! [Xa: int] :
( ord_less_int(number_number_of_int(Xa),zero_zero_int)
<=> ord_less_int(Xa,pls) ) ).
tff(fact_55_less__special_I1_J,axiom,
! [Ya: int] :
( ord_less_int(zero_zero_int,number_number_of_int(Ya))
<=> ord_less_int(pls,Ya) ) ).
tff(fact_56_rel__simps_I12_J,axiom,
! [K_1: int] :
( ord_less_int(bit1(K_1),pls)
<=> ord_less_int(K_1,pls) ) ).
tff(fact_57_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ord_less_int(bit1(K1),bit0(K2))
<=> ord_less_int(K1,K2) ) ).
tff(fact_58_rel__simps_I16_J,axiom,
! [K_1: int,L_1: int] :
( ord_less_int(bit1(K_1),bit0(L_1))
<=> ord_less_int(K_1,L_1) ) ).
tff(fact_59_rel__simps_I10_J,axiom,
! [K_1: int] :
( ord_less_int(bit0(K_1),pls)
<=> ord_less_int(K_1,pls) ) ).
tff(fact_60_rel__simps_I4_J,axiom,
! [K_1: int] :
( ord_less_int(pls,bit0(K_1))
<=> ord_less_int(pls,K_1) ) ).
tff(fact_61_bin__less__0__simps_I4_J,axiom,
! [Wa: int] :
( ord_less_int(bit1(Wa),zero_zero_int)
<=> ord_less_int(Wa,zero_zero_int) ) ).
tff(fact_62_bin__less__0__simps_I1_J,axiom,
~ ord_less_int(pls,zero_zero_int) ).
tff(fact_63_bin__less__0__simps_I3_J,axiom,
! [Wa: int] :
( ord_less_int(bit0(Wa),zero_zero_int)
<=> ord_less_int(Wa,zero_zero_int) ) ).
tff(fact_64_int__0__less__1,axiom,
ord_less_int(zero_zero_int,one_one_int) ).
tff(fact_65_zless__add1__eq,axiom,
! [Wa: int,Z_2: int] :
( ord_less_int(Wa,plus_plus_int(Z_2,one_one_int))
<=> ( ord_less_int(Wa,Z_2)
| ( Wa = Z_2 ) ) ) ).
tff(fact_66_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less_int(semiri1621563631at_int(K),zero_zero_int) ).
tff(fact_67_less__special_I4_J,axiom,
! [Xa: int] :
( ord_less_int(number_number_of_int(Xa),one_one_int)
<=> ord_less_int(Xa,bit1(pls)) ) ).
tff(fact_68_less__special_I2_J,axiom,
! [Ya: int] :
( ord_less_int(one_one_int,number_number_of_int(Ya))
<=> ord_less_int(bit1(pls),Ya) ) ).
tff(fact_69_odd__less__0,axiom,
! [Z_2: int] :
( ord_less_int(plus_plus_int(plus_plus_int(one_one_int,Z_2),Z_2),zero_zero_int)
<=> ord_less_int(Z_2,zero_zero_int) ) ).
tff(fact_70_double__eq__0__iff,axiom,
! [A_1: int] :
( ( plus_plus_int(A_1,A_1) = zero_zero_int )
<=> ( A_1 = zero_zero_int ) ) ).
tff(fact_71_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_72_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_73_rel__simps_I50_J,axiom,
! [K: int,L: int] : bit1(K) != bit0(L) ).
tff(fact_74_rel__simps_I49_J,axiom,
! [K: int,L: int] : bit0(K) != bit1(L) ).
tff(fact_75_rel__simps_I44_J,axiom,
! [K_1: int] :
( ( bit0(K_1) = pls )
<=> ( K_1 = pls ) ) ).
tff(fact_76_rel__simps_I38_J,axiom,
! [L_1: int] :
( ( pls = bit0(L_1) )
<=> ( pls = L_1 ) ) ).
tff(fact_77_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_78_Pls__def,axiom,
pls = zero_zero_int ).
tff(fact_79_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_80_add__Pls__right,axiom,
! [K: int] : plus_plus_int(K,pls) = K ).
tff(fact_81_add__Pls,axiom,
! [K: int] : plus_plus_int(pls,K) = K ).
tff(fact_82_add__Bit0__Bit0,axiom,
! [K: int,L: int] : plus_plus_int(bit0(K),bit0(L)) = bit0(plus_plus_int(K,L)) ).
tff(fact_83_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus_int(K,K) ).
tff(fact_84_zadd__0__right,axiom,
! [Z: int] : plus_plus_int(Z,zero_zero_int) = Z ).
tff(fact_85_zadd__0,axiom,
! [Z: int] : plus_plus_int(zero_zero_int,Z) = Z ).
tff(fact_86_semiring__numeral__0__eq__0,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_87_semiring__numeral__0__eq__0,axiom,
number_number_of_nat(pls) = zero_zero_nat ).
tff(fact_88_number__of__Pls,axiom,
number_number_of_int(pls) = zero_zero_int ).
tff(fact_89_semiring__norm_I112_J,axiom,
zero_zero_int = number_number_of_int(pls) ).
tff(fact_90_add__numeral__0,axiom,
! [A_3: int] : plus_plus_int(number_number_of_int(pls),A_3) = A_3 ).
tff(fact_91_add__numeral__0__right,axiom,
! [A_2: int] : plus_plus_int(A_2,number_number_of_int(pls)) = A_2 ).
tff(fact_92_power__eq__0__iff__number__of,axiom,
! [A_1: int,Wa: int] :
( ( power_power_int(A_1,number_number_of_nat(Wa)) = zero_zero_int )
<=> ( ( A_1 = zero_zero_int )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_93_power__eq__0__iff__number__of,axiom,
! [A_1: nat,Wa: int] :
( ( power_power_nat(A_1,number_number_of_nat(Wa)) = zero_zero_nat )
<=> ( ( A_1 = zero_zero_nat )
& ( number_number_of_nat(Wa) != zero_zero_nat ) ) ) ).
tff(fact_94_add__number__of__left,axiom,
! [V_2: int,W_2: int,Z_1: int] : plus_plus_int(number_number_of_int(V_2),plus_plus_int(number_number_of_int(W_2),Z_1)) = plus_plus_int(number_number_of_int(plus_plus_int(V_2,W_2)),Z_1) ).
tff(fact_95_add__number__of__eq,axiom,
! [V_1: int,W_1: int] : plus_plus_int(number_number_of_int(V_1),number_number_of_int(W_1)) = number_number_of_int(plus_plus_int(V_1,W_1)) ).
tff(fact_96_number__of__add,axiom,
! [V: int,W: int] : number_number_of_int(plus_plus_int(V,W)) = plus_plus_int(number_number_of_int(V),number_number_of_int(W)) ).
tff(fact_97_add__Bit1__Bit0,axiom,
! [K: int,L: int] : plus_plus_int(bit1(K),bit0(L)) = bit1(plus_plus_int(K,L)) ).
tff(fact_98_add__Bit0__Bit1,axiom,
! [K: int,L: int] : plus_plus_int(bit0(K),bit1(L)) = bit1(plus_plus_int(K,L)) ).
tff(fact_99_Bit1__def,axiom,
! [K: int] : bit1(K) = plus_plus_int(plus_plus_int(one_one_int,K),K) ).
tff(fact_100_odd__nonzero,axiom,
! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).
tff(fact_101_number__of__int,axiom,
! [N: nat] : number_number_of_nat(semiri1621563631at_int(N)) = semiri984289939at_nat(N) ).
tff(fact_102_number__of__int,axiom,
! [N: nat] : number_number_of_int(semiri1621563631at_int(N)) = semiri1621563631at_int(N) ).
tff(fact_103_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_104_power2__less__0,axiom,
! [A: int] : ~ ord_less_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),zero_zero_int) ).
tff(fact_105_sum__power2__gt__zero__iff,axiom,
! [Xa: int,Ya: int] :
( ord_less_int(zero_zero_int,plus_plus_int(power_power_int(Xa,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Ya,number_number_of_nat(bit0(bit1(pls))))))
<=> ( ( Xa != zero_zero_int )
| ( Ya != zero_zero_int ) ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
power_power_int(plus_plus_int(one_one_int,semiri1621563631at_int(n)),number_number_of_nat(bit0(bit1(pls)))) != zero_zero_int ).
%------------------------------------------------------------------------------