TPTP Problem File: NUM945_5.p
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%------------------------------------------------------------------------------
% File : NUM945_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 64
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_64 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 153 ( 67 unt; 30 typ; 0 def)
% Number of atoms : 189 ( 107 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 78 ( 12 ~; 3 |; 6 &)
% ( 11 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 18 ( 11 >; 7 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 1 prp; 0-3 aty)
% Number of functors : 13 ( 13 usr; 5 con; 0-3 aty)
% Number of variables : 185 ( 170 !; 2 ?; 185 :)
% ( 13 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:18
%------------------------------------------------------------------------------
%----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_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (26)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
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:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_Residues_OLegendre,type,
legendre: ( int * int ) > int ).
tff(sy_c_Residues_OQuadRes,type,
quadRes: ( int * int ) > $o ).
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_thesis____,type,
thesis: $o ).
%----Relevant facts (97)
tff(fact_0__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_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__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_3__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_4_power2__eq__square__number__of,axiom,
! [B2: $tType] :
( ( monoid_mult(B2)
& number(B2) )
=> ! [W: int] : power_power(B2,number_number_of(B2,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B2,number_number_of(B2,W),number_number_of(B2,W)) ) ).
tff(fact_5_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) ) ) ).
tff(fact_6_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : plus_plus(A,one_one(A),number_number_of(A,W)) = number_number_of(A,plus_plus(int,bit1(pls),W)) ) ).
tff(fact_7_add__special_I3_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : plus_plus(A,number_number_of(A,V),one_one(A)) = number_number_of(A,plus_plus(int,V,bit1(pls))) ) ).
tff(fact_8_one__add__one__is__two,axiom,
! [A: $tType] :
( number_ring(A)
=> ( plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_9_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_10_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_11_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_12_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_13_power2__sum,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Y: A,X: A] : power_power(A,plus_plus(A,X,Y),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(A,plus_plus(A,power_power(A,X,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Y,number_number_of(nat,bit0(bit1(pls))))),times_times(A,times_times(A,number_number_of(A,bit0(bit1(pls))),X),Y)) ) ).
tff(fact_14_add__Bit0__Bit1,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit1(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_15_add__Bit1__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit1(K1),bit0(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_16_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y2: int,X1: int] :
( ( number_number_of(A,X1) = number_number_of(A,Y2) )
<=> ( X1 = Y2 ) ) ) ).
tff(fact_17_rel__simps_I51_J,axiom,
! [L1: int,K2: int] :
( ( bit1(K2) = bit1(L1) )
<=> ( K2 = L1 ) ) ).
tff(fact_18_rel__simps_I48_J,axiom,
! [L1: int,K2: int] :
( ( bit0(K2) = bit0(L1) )
<=> ( K2 = L1 ) ) ).
tff(fact_19_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_20_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_21_rel__simps_I50_J,axiom,
! [L: int,K1: int] : bit1(K1) != bit0(L) ).
tff(fact_22_rel__simps_I49_J,axiom,
! [L: int,K1: int] : bit0(K1) != bit1(L) ).
tff(fact_23_rel__simps_I44_J,axiom,
! [K2: int] :
( ( bit0(K2) = pls )
<=> ( K2 = pls ) ) ).
tff(fact_24_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_25_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_26_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_27_mult__Bit0,axiom,
! [L: int,K1: int] : times_times(int,bit0(K1),L) = bit0(times_times(int,K1,L)) ).
tff(fact_28_add__Bit0__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit0(L)) = bit0(plus_plus(int,K1,L)) ).
tff(fact_29_rel__simps_I47_J,axiom,
! [K2: int] :
( ( bit1(K2) = min )
<=> ( K2 = min ) ) ).
tff(fact_30_rel__simps_I43_J,axiom,
! [L1: int] :
( ( min = bit1(L1) )
<=> ( min = L1 ) ) ).
tff(fact_31_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_32_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_33_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_34_rel__simps_I45_J,axiom,
! [K1: int] : bit0(K1) != min ).
tff(fact_35_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_36_left__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [V: int,B: B2,A1: B2] : times_times(B2,plus_plus(B2,A1,B),number_number_of(B2,V)) = plus_plus(B2,times_times(B2,A1,number_number_of(B2,V)),times_times(B2,B,number_number_of(B2,V))) ) ).
tff(fact_37_right__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [C: B2,B: B2,V: int] : times_times(B2,number_number_of(B2,V),plus_plus(B2,B,C)) = plus_plus(B2,times_times(B2,number_number_of(B2,V),B),times_times(B2,number_number_of(B2,V),C)) ) ).
tff(fact_38_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W),Z)) = times_times(A,number_number_of(A,times_times(int,V,W)),Z) ) ).
tff(fact_39_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,times_times(int,V,W)) ) ).
tff(fact_40_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W)),Z) ) ).
tff(fact_41_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,plus_plus(int,V,W)) ) ).
tff(fact_42_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_43_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_44_zpower__zpower,axiom,
! [Z: nat,Y: nat,X: int] : power_power(int,power_power(int,X,Y),Z) = power_power(int,X,times_times(nat,Y,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_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_48_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_49_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X1: A,W1: int] :
( ( number_number_of(A,W1) = X1 )
<=> ( X1 = number_number_of(A,W1) ) ) ) ).
tff(fact_50_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_51_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A1: A] : power_power(A,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = power_power(A,power_power(A,A1,N),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_52_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_53_power__m1__even,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [N: nat] : power_power(A,number_number_of(A,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = one_one(A) ) ).
tff(fact_54_add__Pls__right,axiom,
! [K1: int] : plus_plus(int,K1,pls) = K1 ).
tff(fact_55_add__Pls,axiom,
! [K1: int] : plus_plus(int,pls,K1) = K1 ).
tff(fact_56_Bit0__def,axiom,
! [K1: int] : bit0(K1) = plus_plus(int,K1,K1) ).
tff(fact_57_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_58_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_59_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_60_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_61_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,number_number_of(A,pls),A1) = A1 ) ).
tff(fact_62_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ).
tff(fact_63_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,times_times(int,V,W)) = times_times(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_64_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,plus_plus(int,V,W)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_65_Bit1__def,axiom,
! [K1: int] : bit1(K1) = plus_plus(int,plus_plus(int,one_one(int),K1),K1) ).
tff(fact_66_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
tff(fact_67_number__of__Bit1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : number_number_of(A,bit1(W)) = plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,W)),number_number_of(A,W)) ) ).
tff(fact_68_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_69_mult__numeral__1__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,A1,number_number_of(A,bit1(pls))) = A1 ) ).
tff(fact_70_semiring__numeral__1__eq__1,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_71_semiring__norm_I110_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( one_one(A) = number_number_of(A,bit1(pls)) ) ) ).
tff(fact_72_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_73_pos__zmult__eq__1__iff__lemma,axiom,
! [N: int,M: int] :
( ( times_times(int,M,N) = one_one(int) )
=> ( ( M = one_one(int) )
| ( M = number_number_of(int,min) ) ) ) ).
tff(fact_74_zmult__eq__1__iff,axiom,
! [N1: int,Ma: int] :
( ( times_times(int,Ma,N1) = one_one(int) )
<=> ( ( ( Ma = one_one(int) )
& ( N1 = one_one(int) ) )
| ( ( Ma = number_number_of(int,min) )
& ( N1 = number_number_of(int,min) ) ) ) ) ).
tff(fact_75_double__number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : times_times(A,plus_plus(A,one_one(A),one_one(A)),number_number_of(A,W)) = number_number_of(A,bit0(W)) ) ).
tff(fact_76_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit1(bit1(pls)))) = times_times(A,times_times(A,A1,A1),A1) ) ).
tff(fact_77_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_78_semiring__mult__2,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z) = plus_plus(A,Z,Z) ) ).
tff(fact_79_mult__2,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z) = plus_plus(A,Z,Z) ) ).
tff(fact_80_semiring__mult__2__right,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z: A] : times_times(A,Z,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z,Z) ) ).
tff(fact_81_mult__2__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A] : times_times(A,Z,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z,Z) ) ).
tff(fact_82_semiring__one__add__one__is__two,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_83_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) = times_times(A,A1,A1) ) ).
tff(fact_84_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_85_QuadRes__def,axiom,
! [X1: int,Ma: int] :
( quadRes(Ma,X1)
<=> ? [Y1: int] : zcong(power_power(int,Y1,number_number_of(nat,bit0(bit1(pls)))),X1,Ma) ) ).
tff(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] : power_power(A,X,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(A,power_power(A,X,N),power_power(A,X,N)) ) ).
tff(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : times_times(A,X,X) = power_power(A,X,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_88_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : power_power(A,one_one(A),N) = one_one(A) ) ).
tff(fact_89_neg__one__power,axiom,
! [N: nat] :
( ( power_power(int,number_number_of(int,min),N) = one_one(int) )
| ( power_power(int,number_number_of(int,min),N) = number_number_of(int,min) ) ) ).
tff(fact_90_four__x__squared,axiom,
! [X: real] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) = power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_91_zcong__zpower__zmult,axiom,
! [Z: nat,P: int,Y: nat,X: int] :
( zcong(power_power(int,X,Y),one_one(int),P)
=> zcong(power_power(int,X,times_times(nat,Y,Z)),one_one(int),P) ) ).
tff(fact_92_zcong__iff__lin,axiom,
! [Ma: int,B1: int,A2: int] :
( zcong(A2,B1,Ma)
<=> ? [K: int] : B1 = plus_plus(int,A2,times_times(int,Ma,K)) ) ).
tff(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,A1: A] : times_times(A,A1,B) = times_times(A,B,A1) ) ).
tff(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,Lx,Ry)) ) ).
tff(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),Ry) ) ).
tff(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,Lx,times_times(A,Ly,Rx)) ) ).
%----Arities (22)
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
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___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
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) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
tff(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
monoid_mult(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring,axiom,
semiring(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
! [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)))
=> thesis ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------