TPTP Problem File: NUM940_5.p
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%------------------------------------------------------------------------------
% File : NUM940_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 50
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_50 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 151 ( 60 unt; 34 typ; 0 def)
% Number of atoms : 198 ( 101 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 95 ( 14 ~; 10 |; 8 &)
% ( 20 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 20 ( 12 >; 8 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 5 con; 0-3 aty)
% Number of variables : 183 ( 163 !; 0 ?; 183 :)
% ( 20 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:07
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
%----Explicit typings (31)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[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_Omult__zero,type,
mult_zero:
!>[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_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : 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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Residues_OLegendre,type,
legendre: ( int * int ) > int ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > $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 ).
%----Relevant facts (97)
tff(fact_0__0964_A_K_Am_A_L_A1_Advd_A1_A_124_A4_A_K_Am_A_L_A1_Advd_A2_A_061_061_062_AFalse_096,axiom,
~ ( dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),one_one(int))
| dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,bit0(bit1(pls)))) ) ).
tff(fact_1__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_A_124_A4_A_K_Am_A_L_A1_Advd_A1_A_124_A4_A_K_Am_A_L_A1_Advd_A2_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) )
| dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),one_one(int))
| dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,bit0(bit1(pls)))) ) ).
tff(fact_2_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_3_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_4_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_5_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_6_mult__Bit1,axiom,
! [L1: int,K1: int] : ( times_times(int,bit1(K1),L1) = plus_plus(int,bit0(times_times(int,K1,L1)),L1) ) ).
tff(fact_7__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_A_124Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A0_A_124_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A_N1_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) )
| ( legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) = zero_zero(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))) = number_number_of(int,min) ) ) ).
tff(fact_8_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_9_add__Bit0__Bit1,axiom,
! [L1: int,K1: int] : ( plus_plus(int,bit0(K1),bit1(L1)) = bit1(plus_plus(int,K1,L1)) ) ).
tff(fact_10_add__Bit1__Bit0,axiom,
! [L1: int,K1: int] : ( plus_plus(int,bit1(K1),bit0(L1)) = bit1(plus_plus(int,K1,L1)) ) ).
tff(fact_11_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_12_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_13_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_14_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_15_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y: int,X: int] :
( ( number_number_of(A,X) = number_number_of(A,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_16_rel__simps_I51_J,axiom,
! [L: int,K: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_17_rel__simps_I48_J,axiom,
! [L: int,K: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_18_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( plus_plus(A,A2,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_19_rel__simps_I46_J,axiom,
! [K1: int] : ( bit1(K1) != pls ) ).
tff(fact_20_rel__simps_I39_J,axiom,
! [L1: int] : ( pls != bit1(L1) ) ).
tff(fact_21_rel__simps_I50_J,axiom,
! [L1: int,K1: int] : ( bit1(K1) != bit0(L1) ) ).
tff(fact_22_rel__simps_I49_J,axiom,
! [L1: int,K1: int] : ( bit0(K1) != bit1(L1) ) ).
tff(fact_23_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_24_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
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,
! [L1: int,K1: int] : ( times_times(int,bit0(K1),L1) = bit0(times_times(int,K1,L1)) ) ).
tff(fact_28_add__Bit0__Bit0,axiom,
! [L1: int,K1: int] : ( plus_plus(int,bit0(K1),bit0(L1)) = bit0(plus_plus(int,K1,L1)) ) ).
tff(fact_29_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_30_rel__simps_I43_J,axiom,
! [L: int] :
( ( min = bit1(L) )
<=> ( min = L ) ) ).
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,
! [L1: int] : ( min != bit0(L1) ) ).
tff(fact_36_left__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [V: int,B1: B,A1: B] : ( times_times(B,plus_plus(B,A1,B1),number_number_of(B,V)) = plus_plus(B,times_times(B,A1,number_number_of(B,V)),times_times(B,B1,number_number_of(B,V))) ) ) ).
tff(fact_37_right__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [C: B,B1: B,V: int] : ( times_times(B,number_number_of(B,V),plus_plus(B,B1,C)) = plus_plus(B,times_times(B,number_number_of(B,V),B1),times_times(B,number_number_of(B,V),C)) ) ) ).
tff(fact_38_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_39_zdvd__mult__cancel,axiom,
! [N1: int,M: int,K1: int] :
( dvd_dvd(int,times_times(int,K1,M),times_times(int,K1,N1))
=> ( ( K1 != zero_zero(int) )
=> dvd_dvd(int,M,N1) ) ) ).
tff(fact_40_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_41_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_42_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_43_odd__nonzero,axiom,
! [Z: int] : ( plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int) ) ).
tff(fact_44_number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : ( number_number_of(A,bit0(W)) = plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,W)),number_number_of(A,W)) ) ) ).
tff(fact_45_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X: A,W1: int] :
( ( number_number_of(A,W1) = X )
<=> ( X = number_number_of(A,W1) ) ) ) ).
tff(fact_46_number__of__is__id,axiom,
! [K1: int] : ( number_number_of(int,K1) = K1 ) ).
tff(fact_47_zdvd__period,axiom,
! [C1: int,T: int,X: int,D: int,A2: int] :
( dvd_dvd(int,A2,D)
=> ( dvd_dvd(int,A2,plus_plus(int,X,T))
<=> dvd_dvd(int,A2,plus_plus(int,plus_plus(int,X,times_times(int,C1,D)),T)) ) ) ).
tff(fact_48_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) ) ).
tff(fact_49_add__Pls__right,axiom,
! [K1: int] : ( plus_plus(int,K1,pls) = K1 ) ).
tff(fact_50_add__Pls,axiom,
! [K1: int] : ( plus_plus(int,pls,K1) = K1 ) ).
tff(fact_51_Bit0__def,axiom,
! [K1: int] : ( bit0(K1) = plus_plus(int,K1,K1) ) ).
tff(fact_52_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_53_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_54_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_55_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_56_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : ( plus_plus(A,number_number_of(A,pls),A1) = A1 ) ) ).
tff(fact_57_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : ( plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ) ).
tff(fact_58_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_59_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_60_Bit1__def,axiom,
! [K1: int] : ( bit1(K1) = plus_plus(int,plus_plus(int,one_one(int),K1),K1) ) ).
tff(fact_61_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_62_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : ( times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ) ).
tff(fact_63_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_64_semiring__numeral__1__eq__1,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_65_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74__0964_A_K_Am_A_L_A1_Advd_A1_A_N_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_096,axiom,
dvd_dvd(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),minus_minus(int,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))))) ).
tff(fact_75_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_76_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [B2: A,A2: A,C1: A] :
( dvd_dvd(A,times_times(A,C1,A2),times_times(A,C1,B2))
<=> ( ( C1 = zero_zero(A) )
| dvd_dvd(A,A2,B2) ) ) ) ).
tff(fact_77_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [B2: A,C1: A,A2: A] :
( dvd_dvd(A,times_times(A,A2,C1),times_times(A,B2,C1))
<=> ( ( C1 = zero_zero(A) )
| dvd_dvd(A,A2,B2) ) ) ) ).
tff(fact_78__096_0911_A_061_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong(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))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_79_p2,axiom,
ord_less(int,number_number_of(int,bit0(bit1(pls))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_80_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_81_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_82_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : ( times_times(A,zero_zero(A),A1) = zero_zero(A) ) ) ).
tff(fact_83_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : ( times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_84_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [B2: A,A2: A] :
( ( times_times(A,A2,B2) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
tff(fact_85_dvd__0__right,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : dvd_dvd(A,A1,zero_zero(A)) ) ).
tff(fact_86_rel__simps_I17_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit1(L))
<=> ord_less(int,K,L) ) ).
tff(fact_87_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_88_rel__simps_I14_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit0(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_89_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_90_diff__bin__simps_I7_J,axiom,
! [L1: int,K1: int] : ( minus_minus(int,bit0(K1),bit0(L1)) = bit0(minus_minus(int,K1,L1)) ) ).
tff(fact_91_rel__simps_I7_J,axiom,
~ ord_less(int,min,min) ).
tff(fact_92_right__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [C: B,B1: B,V: int] : ( times_times(B,number_number_of(B,V),minus_minus(B,B1,C)) = minus_minus(B,times_times(B,number_number_of(B,V),B1),times_times(B,number_number_of(B,V),C)) ) ) ).
tff(fact_93_left__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [V: int,B1: B,A1: B] : ( times_times(B,minus_minus(B,A1,B1),number_number_of(B,V)) = minus_minus(B,times_times(B,A1,number_number_of(B,V)),times_times(B,B1,number_number_of(B,V))) ) ) ).
tff(fact_94_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X: int] :
( ord_less(A,number_number_of(A,X),number_number_of(A,Y))
<=> ord_less(int,X,Y) ) ) ).
tff(fact_95_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_96_rel__simps_I16_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
%----Arities (17)
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
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___Rings_Omult__zero,axiom,
mult_zero(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___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Rings_Oidom,axiom,
idom(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___Rings_Omult__zero,axiom,
mult_zero(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,
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) ).
%------------------------------------------------------------------------------