TPTP Problem File: NUM954_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM954_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 77
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_77 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v6.4.0
% Syntax : Number of formulae : 147 ( 64 unt; 35 typ; 0 def)
% Number of atoms : 172 ( 100 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 76 ( 16 ~; 2 |; 6 &)
% ( 12 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 26 ( 16 >; 10 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 19 ( 19 usr; 8 con; 0-4 aty)
% Number of variables : 166 ( 149 !; 2 ?; 166 :)
% ( 15 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:45
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (31)
tff(sy_cl_Int_Onumber,type,
number:
!>[A2: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A2: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A2: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A2: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A2: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A2: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A2: $tType] : $o ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A2: $tType] : A2 ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A2: $tType] : ( ( A2 * A2 ) > A2 ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A2: $tType] : ( ( A2 * A2 ) > A2 ) ).
tff(sy_c_IntPrimes_Ozcong,type,
zcong: ( int * int * int ) > $o ).
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_OMin,type,
min: int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A2: $tType] : ( int > A2 ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A2: $tType] : ( ( A2 * nat ) > A2 ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A2: $tType,B1: $tType] : ( ( A2 * B1 ) > product_prod(A2,B1) ) ).
tff(sy_c_Residues_OLegendre,type,
legendre: ( int * int ) > int ).
tff(sy_c_Residues_OQuadRes,type,
quadRes: ( int * int ) > $o ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A2: $tType] : ( ( A2 * A2 ) > $o ) ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Ois__sum2sq,type,
twoSqu1567020053sum2sq: int > $o ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Osum2sq,type,
twoSqu196287499sum2sq: product_prod(int,int) > int ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_m,type,
m: int ).
tff(sy_v_s1____,type,
s1: int ).
tff(sy_v_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (97)
tff(fact_0__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
twoSqu196287499sum2sq(product_Pair(int,int,s,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_1_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_2_is__mult__sum2sq,axiom,
! [Y: int,X1: int] :
( twoSqu1567020053sum2sq(X1)
=> ( twoSqu1567020053sum2sq(Y)
=> twoSqu1567020053sum2sq(times_times(int,X1,Y)) ) ) ).
tff(fact_3_add__special_I2_J,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int] : plus_plus(A2,one_one(A2),number_number_of(A2,W)) = number_number_of(A2,plus_plus(int,bit1(pls),W)) ) ).
tff(fact_4_add__special_I3_J,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [V: int] : plus_plus(A2,number_number_of(A2,V),one_one(A2)) = number_number_of(A2,plus_plus(int,V,bit1(pls))) ) ).
tff(fact_5_one__add__one__is__two,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ( plus_plus(A2,one_one(A2),one_one(A2)) = number_number_of(A2,bit0(bit1(pls))) ) ) ).
tff(fact_6_mult__Bit1,axiom,
! [L: int,K1: int] : times_times(int,bit1(K1),L) = plus_plus(int,bit0(times_times(int,K1,L)),L) ).
tff(fact_7_numeral__1__eq__1,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ( number_number_of(A2,bit1(pls)) = one_one(A2) ) ) ).
tff(fact_8_add__Bit0__Bit1,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit1(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_9_add__Bit1__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit1(K1),bit0(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_10_add__number__of__eq,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int,V: int] : plus_plus(A2,number_number_of(A2,V),number_number_of(A2,W)) = number_number_of(A2,plus_plus(int,V,W)) ) ).
tff(fact_11_add__number__of__left,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [Z: A2,W: int,V: int] : plus_plus(A2,number_number_of(A2,V),plus_plus(A2,number_number_of(A2,W),Z)) = plus_plus(A2,number_number_of(A2,plus_plus(int,V,W)),Z) ) ).
tff(fact_12_arith__simps_I32_J,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int,V: int] : times_times(A2,number_number_of(A2,V),number_number_of(A2,W)) = number_number_of(A2,times_times(int,V,W)) ) ).
tff(fact_13_mult__number__of__left,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [Z: A2,W: int,V: int] : times_times(A2,number_number_of(A2,V),times_times(A2,number_number_of(A2,W),Z)) = times_times(A2,number_number_of(A2,times_times(int,V,W)),Z) ) ).
tff(fact_14_right__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [C1: B1,B: B1,V: int] : times_times(B1,number_number_of(B1,V),plus_plus(B1,B,C1)) = plus_plus(B1,times_times(B1,number_number_of(B1,V),B),times_times(B1,number_number_of(B1,V),C1)) ) ).
tff(fact_15_eq__number__of,axiom,
! [A2: $tType] :
( ( number_ring(A2)
& ring_char_0(A2) )
=> ! [Y1: int,X: int] :
( ( number_number_of(A2,X) = number_number_of(A2,Y1) )
<=> ( X = Y1 ) ) ) ).
tff(fact_16_rel__simps_I51_J,axiom,
! [L1: int,K: int] :
( ( bit1(K) = bit1(L1) )
<=> ( K = L1 ) ) ).
tff(fact_17_rel__simps_I48_J,axiom,
! [L1: int,K: int] :
( ( bit0(K) = bit0(L1) )
<=> ( K = L1 ) ) ).
tff(fact_18_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_19_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_20_rel__simps_I50_J,axiom,
! [L: int,K1: int] : bit1(K1) != bit0(L) ).
tff(fact_21_rel__simps_I49_J,axiom,
! [L: int,K1: int] : bit0(K1) != bit1(L) ).
tff(fact_22_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_23_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_24_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_25_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_26_mult__Bit0,axiom,
! [L: int,K1: int] : times_times(int,bit0(K1),L) = bit0(times_times(int,K1,L)) ).
tff(fact_27_add__Bit0__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit0(L)) = bit0(plus_plus(int,K1,L)) ).
tff(fact_28_left__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [V: int,B: B1,A1: B1] : times_times(B1,plus_plus(B1,A1,B),number_number_of(B1,V)) = plus_plus(B1,times_times(B1,A1,number_number_of(B1,V)),times_times(B1,B,number_number_of(B1,V))) ) ).
tff(fact_29_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_30_number__of__reorient,axiom,
! [A2: $tType] :
( number(A2)
=> ! [X: A2,W1: int] :
( ( number_number_of(A2,W1) = X )
<=> ( X = number_number_of(A2,W1) ) ) ) ).
tff(fact_31_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_32_add__Pls__right,axiom,
! [K1: int] : plus_plus(int,K1,pls) = K1 ).
tff(fact_33_add__Pls,axiom,
! [K1: int] : plus_plus(int,pls,K1) = K1 ).
tff(fact_34_Bit0__def,axiom,
! [K1: int] : bit0(K1) = plus_plus(int,K1,K1) ).
tff(fact_35_times__numeral__code_I5_J,axiom,
! [W: int,V: int] : times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W)) ).
tff(fact_36_int__distrib_I1_J,axiom,
! [W: int,Z2: int,Z1: 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_37_int__distrib_I2_J,axiom,
! [Z2: int,Z1: int,W: 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_38_plus__numeral__code_I9_J,axiom,
! [W: int,V: int] : plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W)) ).
tff(fact_39_add__numeral__0,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [A1: A2] : plus_plus(A2,number_number_of(A2,pls),A1) = A1 ) ).
tff(fact_40_add__numeral__0__right,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [A1: A2] : plus_plus(A2,A1,number_number_of(A2,pls)) = A1 ) ).
tff(fact_41_number__of__mult,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int,V: int] : number_number_of(A2,times_times(int,V,W)) = times_times(A2,number_number_of(A2,V),number_number_of(A2,W)) ) ).
tff(fact_42_number__of__add,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int,V: int] : number_number_of(A2,plus_plus(int,V,W)) = plus_plus(A2,number_number_of(A2,V),number_number_of(A2,W)) ) ).
tff(fact_43_Bit1__def,axiom,
! [K1: int] : bit1(K1) = plus_plus(int,plus_plus(int,one_one(int),K1),K1) ).
tff(fact_44_number__of__Bit1,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int] : number_number_of(A2,bit1(W)) = plus_plus(A2,plus_plus(A2,one_one(A2),number_number_of(A2,W)),number_number_of(A2,W)) ) ).
tff(fact_45_mult__numeral__1,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [A1: A2] : times_times(A2,number_number_of(A2,bit1(pls)),A1) = A1 ) ).
tff(fact_46_mult__numeral__1__right,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [A1: A2] : times_times(A2,A1,number_number_of(A2,bit1(pls))) = A1 ) ).
tff(fact_47_semiring__numeral__1__eq__1,axiom,
! [A2: $tType] :
( number_semiring(A2)
=> ( number_number_of(A2,bit1(pls)) = one_one(A2) ) ) ).
tff(fact_48_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_49_double__number__of__Bit0,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [W: int] : times_times(A2,plus_plus(A2,one_one(A2),one_one(A2)),number_number_of(A2,W)) = number_number_of(A2,bit0(W)) ) ).
tff(fact_50_semiring__mult__2,axiom,
! [A2: $tType] :
( number_semiring(A2)
=> ! [Z: A2] : times_times(A2,number_number_of(A2,bit0(bit1(pls))),Z) = plus_plus(A2,Z,Z) ) ).
tff(fact_51_mult__2,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [Z: A2] : times_times(A2,number_number_of(A2,bit0(bit1(pls))),Z) = plus_plus(A2,Z,Z) ) ).
tff(fact_52_semiring__mult__2__right,axiom,
! [A2: $tType] :
( number_semiring(A2)
=> ! [Z: A2] : times_times(A2,Z,number_number_of(A2,bit0(bit1(pls)))) = plus_plus(A2,Z,Z) ) ).
tff(fact_53_mult__2__right,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [Z: A2] : times_times(A2,Z,number_number_of(A2,bit0(bit1(pls)))) = plus_plus(A2,Z,Z) ) ).
tff(fact_54_semiring__one__add__one__is__two,axiom,
! [A2: $tType] :
( number_semiring(A2)
=> ( plus_plus(A2,one_one(A2),one_one(A2)) = number_number_of(A2,bit0(bit1(pls))) ) ) ).
tff(fact_55_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_56_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_57_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_58_is__sum2sq__def,axiom,
! [X: int] :
( twoSqu1567020053sum2sq(X)
<=> ? [A3: int,B2: int] : twoSqu196287499sum2sq(product_Pair(int,int,A3,B2)) = X ) ).
tff(fact_59__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_J_A_K_At_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,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_60_semiring__norm_I110_J,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ( one_one(A2) = number_number_of(A2,bit1(pls)) ) ) ).
tff(fact_61__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) = one_one(int) ).
tff(fact_62_nat__mult__2,axiom,
! [Z: nat] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
tff(fact_63_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_64_rel__simps_I43_J,axiom,
! [L1: int] :
( ( min = bit1(L1) )
<=> ( min = L1 ) ) ).
tff(fact_65_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_66_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_67_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_68_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_69_rel__simps_I45_J,axiom,
! [K1: int] : bit0(K1) != min ).
tff(fact_70_one__power2,axiom,
! [A2: $tType] :
( semiring_1(A2)
=> ( power_power(A2,one_one(A2),number_number_of(nat,bit0(bit1(pls)))) = one_one(A2) ) ) ).
tff(fact_71_power2__eq__square__number__of,axiom,
! [B1: $tType] :
( ( monoid_mult(B1)
& number(B1) )
=> ! [W: int] : power_power(B1,number_number_of(B1,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B1,number_number_of(B1,W),number_number_of(B1,W)) ) ).
tff(fact_72_zpower__zpower,axiom,
! [Z: nat,Y: nat,X1: int] : power_power(int,power_power(int,X1,Y),Z) = power_power(int,X1,times_times(nat,Y,Z)) ).
tff(fact_73_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
tff(fact_74_power__m1__even,axiom,
! [A2: $tType] :
( number_ring(A2)
=> ! [N1: nat] : power_power(A2,number_number_of(A2,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N1)) = one_one(A2) ) ).
tff(fact_75_power3__eq__cube,axiom,
! [A2: $tType] :
( monoid_mult(A2)
=> ! [A1: A2] : power_power(A2,A1,number_number_of(nat,bit1(bit1(pls)))) = times_times(A2,times_times(A2,A1,A1),A1) ) ).
tff(fact_76_power__even__eq,axiom,
! [A2: $tType] :
( monoid_mult(A2)
=> ! [N1: nat,A1: A2] : power_power(A2,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N1)) = power_power(A2,power_power(A2,A1,N1),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_77_quartic__square__square,axiom,
! [X1: int] : power_power(int,power_power(int,X1,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X1,number_number_of(nat,bit0(bit0(bit1(pls))))) ).
tff(fact_78_power2__eq__square,axiom,
! [A2: $tType] :
( monoid_mult(A2)
=> ! [A1: A2] : power_power(A2,A1,number_number_of(nat,bit0(bit1(pls)))) = times_times(A2,A1,A1) ) ).
tff(fact_79_cube__square,axiom,
! [A1: int] : times_times(int,A1,power_power(int,A1,number_number_of(nat,bit0(bit1(pls))))) = power_power(int,A1,number_number_of(nat,bit1(bit1(pls)))) ).
tff(fact_80_zmult__eq__1__iff,axiom,
! [N: int,Ma: int] :
( ( times_times(int,Ma,N) = one_one(int) )
<=> ( ( ( Ma = one_one(int) )
& ( N = one_one(int) ) )
| ( ( Ma = number_number_of(int,min) )
& ( N = number_number_of(int,min) ) ) ) ) ).
tff(fact_81_pos__zmult__eq__1__iff__lemma,axiom,
! [N1: int,M: int] :
( ( times_times(int,M,N1) = one_one(int) )
=> ( ( M = one_one(int) )
| ( M = number_number_of(int,min) ) ) ) ).
tff(fact_82_power2__sum,axiom,
! [A2: $tType] :
( number_semiring(A2)
=> ! [Y: A2,X1: A2] : power_power(A2,plus_plus(A2,X1,Y),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(A2,plus_plus(A2,power_power(A2,X1,number_number_of(nat,bit0(bit1(pls)))),power_power(A2,Y,number_number_of(nat,bit0(bit1(pls))))),times_times(A2,times_times(A2,number_number_of(A2,bit0(bit1(pls))),X1),Y)) ) ).
tff(fact_83_zadd__power2,axiom,
! [B: int,A1: int] : power_power(int,plus_plus(int,A1,B),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,plus_plus(int,power_power(int,A1,number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A1),B)),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_84_zadd__power3,axiom,
! [B: int,A1: int] : power_power(int,plus_plus(int,A1,B),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(int,power_power(int,A1,number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),power_power(int,A1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) ).
tff(fact_85_Legendre__1mod4,axiom,
! [M: int] :
( zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int)))
=> ( legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))) = one_one(int) ) ) ).
tff(fact_86_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_87_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_88__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_126_061_A1_096,axiom,
( ~ quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,min))
=> ( legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) != one_one(int) ) ) ).
tff(fact_89__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,min)) ).
tff(fact_90_s,axiom,
zcong(power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_91_s1,axiom,
zcong(power_power(int,s1,number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_92__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_A_K_Am_A_L_A1_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [S1: int] : ~ zcong(power_power(int,S1,number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_93__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong(power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),power_power(int,s1,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_94__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int))) ).
tff(fact_95_zdvd__period,axiom,
! [C: int,Ta: int,X: int,D: int,A: int] :
( dvd_dvd(int,A,D)
=> ( dvd_dvd(int,A,plus_plus(int,X,Ta))
<=> dvd_dvd(int,A,plus_plus(int,plus_plus(int,X,times_times(int,C,D)),Ta)) ) ) ).
tff(fact_96_zdvd__reduce,axiom,
! [Ma: int,N: int,K: int] :
( dvd_dvd(int,K,plus_plus(int,N,times_times(int,K,Ma)))
<=> dvd_dvd(int,K,N) ) ).
%----Arities (12)
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
twoSqu1567020053sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t)) ).
%------------------------------------------------------------------------------