TPTP Problem File: NUM942_5.p
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%------------------------------------------------------------------------------
% File : NUM942_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 59
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_59 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 136 ( 49 unt; 24 typ; 0 def)
% Number of atoms : 190 ( 119 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 91 ( 13 ~; 3 |; 8 &)
% ( 14 <=>; 53 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 11 ( 8 >; 3 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-2 aty)
% Number of functors : 12 ( 12 usr; 5 con; 0-3 aty)
% Number of variables : 222 ( 211 !; 0 ?; 222 :)
% ( 11 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:12
%------------------------------------------------------------------------------
%----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 (21)
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_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_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[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_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_Residues_OLegendre,type,
legendre: ( 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 ).
%----Relevant facts (97)
tff(fact_0_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_1_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : plus_plus(A,one_one(A),number_number_of(A,W1)) = number_number_of(A,plus_plus(int,bit1(pls),W1)) ) ).
tff(fact_2_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_3_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_4_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_5_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_6_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_7_add__Bit0__Bit1,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit1(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_8_add__Bit1__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit1(K1),bit0(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_9_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,plus_plus(int,V,W1)) ) ).
tff(fact_10_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z1: A,W1: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W1),Z1)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W1)),Z1) ) ).
tff(fact_11_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,times_times(int,V,W1)) ) ).
tff(fact_12_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z1: A,W1: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W1),Z1)) = times_times(A,number_number_of(A,times_times(int,V,W1)),Z1) ) ).
tff(fact_13_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_14_rel__simps_I51_J,axiom,
! [L1: int,K: int] :
( ( bit1(K) = bit1(L1) )
<=> ( K = L1 ) ) ).
tff(fact_15_rel__simps_I48_J,axiom,
! [L1: int,K: int] :
( ( bit0(K) = bit0(L1) )
<=> ( K = L1 ) ) ).
tff(fact_16_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_17_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_18_rel__simps_I50_J,axiom,
! [L: int,K1: int] : bit1(K1) != bit0(L) ).
tff(fact_19_rel__simps_I49_J,axiom,
! [L: int,K1: int] : bit0(K1) != bit1(L) ).
tff(fact_20_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_21_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_22_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_23_mult__Pls,axiom,
! [W1: int] : times_times(int,pls,W1) = pls ).
tff(fact_24_mult__Bit0,axiom,
! [L: int,K1: int] : times_times(int,bit0(K1),L) = bit0(times_times(int,K1,L)) ).
tff(fact_25_add__Bit0__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit0(L)) = bit0(plus_plus(int,K1,L)) ).
tff(fact_26_rel__simps_I47_J,axiom,
! [K: int] :
( ( bit1(K) = min )
<=> ( K = min ) ) ).
tff(fact_27_rel__simps_I43_J,axiom,
! [L1: int] :
( ( min = bit1(L1) )
<=> ( min = L1 ) ) ).
tff(fact_28_Bit1__Min,axiom,
bit1(min) = min ).
tff(fact_29_rel__simps_I37_J,axiom,
pls != min ).
tff(fact_30_rel__simps_I40_J,axiom,
min != pls ).
tff(fact_31_rel__simps_I45_J,axiom,
! [K1: int] : bit0(K1) != min ).
tff(fact_32_rel__simps_I42_J,axiom,
! [L: int] : min != bit0(L) ).
tff(fact_33_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_34_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_35_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X: A,W: int] :
( ( number_number_of(A,W) = X )
<=> ( X = number_number_of(A,W) ) ) ) ).
tff(fact_36_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_37_add__Pls__right,axiom,
! [K1: int] : plus_plus(int,K1,pls) = K1 ).
tff(fact_38_add__Pls,axiom,
! [K1: int] : plus_plus(int,pls,K1) = K1 ).
tff(fact_39_Bit0__def,axiom,
! [K1: int] : bit0(K1) = plus_plus(int,K1,K1) ).
tff(fact_40_times__numeral__code_I5_J,axiom,
! [W1: int,V: int] : times_times(int,number_number_of(int,V),number_number_of(int,W1)) = number_number_of(int,times_times(int,V,W1)) ).
tff(fact_41_int__distrib_I1_J,axiom,
! [W1: int,Z2: int,Z11: int] : times_times(int,plus_plus(int,Z11,Z2),W1) = plus_plus(int,times_times(int,Z11,W1),times_times(int,Z2,W1)) ).
tff(fact_42_int__distrib_I2_J,axiom,
! [Z2: int,Z11: int,W1: int] : times_times(int,W1,plus_plus(int,Z11,Z2)) = plus_plus(int,times_times(int,W1,Z11),times_times(int,W1,Z2)) ).
tff(fact_43_plus__numeral__code_I9_J,axiom,
! [W1: int,V: int] : plus_plus(int,number_number_of(int,V),number_number_of(int,W1)) = number_number_of(int,plus_plus(int,V,W1)) ).
tff(fact_44_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,number_number_of(A,pls),A1) = A1 ) ).
tff(fact_45_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ).
tff(fact_46_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,times_times(int,V,W1)) = times_times(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_47_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,plus_plus(int,V,W1)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_48_Bit1__def,axiom,
! [K1: int] : bit1(K1) = plus_plus(int,plus_plus(int,one_one(int),K1),K1) ).
tff(fact_49_number__of__Bit1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : number_number_of(A,bit1(W1)) = plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,W1)),number_number_of(A,W1)) ) ).
tff(fact_50_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_51_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_52_semiring__numeral__1__eq__1,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_53_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_54_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_55_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_56_double__number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : times_times(A,plus_plus(A,one_one(A),one_one(A)),number_number_of(A,W1)) = number_number_of(A,bit0(W1)) ) ).
tff(fact_57_semiring__mult__2,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z1: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z1) = plus_plus(A,Z1,Z1) ) ).
tff(fact_58_mult__2,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z1: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z1) = plus_plus(A,Z1,Z1) ) ).
tff(fact_59_semiring__mult__2__right,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z1: A] : times_times(A,Z1,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z1,Z1) ) ).
tff(fact_60_mult__2__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z1: A] : times_times(A,Z1,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z1,Z1) ) ).
tff(fact_61_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_62_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_63_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_64_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_65_semiring__norm_I110_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( one_one(A) = number_number_of(A,bit1(pls)) ) ) ).
tff(fact_66_eq__number__of__Pls__Min,axiom,
number_number_of(int,pls) != number_number_of(int,min) ).
tff(fact_67_nat__mult__2,axiom,
! [Z1: nat] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z1) = plus_plus(nat,Z1,Z1) ).
tff(fact_68_nat__mult__2__right,axiom,
! [Z1: nat] : times_times(nat,Z1,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z1,Z1) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A,A1: A] : plus_plus(A,times_times(A,A1,M),M) = times_times(A,plus_plus(A,A1,one_one(A)),M) ) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A,M: A] : plus_plus(A,M,times_times(A,A1,M)) = times_times(A,plus_plus(A,A1,one_one(A)),M) ) ).
tff(fact_71_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_72_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_73_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_74_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)) ) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I16_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,times_times(A,Lx,Rx),Ly) ) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Lx,times_times(A,Ly,times_times(A,Rx,Ry))) ) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,times_times(A,Lx,Ly),Ry)) ) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),times_times(A,Ly,Ry)) ) ).
tff(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,A1: A] : plus_plus(A,A1,C) = plus_plus(A,C,A1) ) ).
tff(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D1)) = plus_plus(A,C,plus_plus(A,A1,D1)) ) ).
tff(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D1)) = plus_plus(A,plus_plus(A,A1,C),D1) ) ).
tff(fact_82_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,plus_plus(A,A1,C),B) ) ).
tff(fact_84_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),plus_plus(A,C,D1)) = plus_plus(A,plus_plus(A,A1,C),plus_plus(A,B,D1)) ) ).
tff(fact_85_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z1: A,Y1: A,X1: A] : times_times(A,X1,plus_plus(A,Y1,Z1)) = plus_plus(A,times_times(A,X1,Y1),times_times(A,X1,Z1)) ) ).
tff(fact_86_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [D: A,C1: A,B1: A,A2: A] :
( ( ( A2 != B1 )
& ( C1 != D ) )
<=> ( plus_plus(A,times_times(A,A2,C1),times_times(A,B1,D)) != plus_plus(A,times_times(A,A2,D),times_times(A,B1,C1)) ) ) ) ).
tff(fact_87_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : times_times(A,plus_plus(A,A1,B),C) = plus_plus(A,times_times(A,A1,C),times_times(A,B,C)) ) ).
tff(fact_88_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,M: A,A1: A] : plus_plus(A,times_times(A,A1,M),times_times(A,B,M)) = times_times(A,plus_plus(A,A1,B),M) ) ).
tff(fact_89_crossproduct__eq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [Z: A,X: A,Y: A,W: A] :
( ( plus_plus(A,times_times(A,W,Y),times_times(A,X,Z)) = plus_plus(A,times_times(A,W,Z),times_times(A,X,Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ) ).
tff(fact_90_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_92_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A] : plus_plus(A,M,M) = times_times(A,plus_plus(A,one_one(A),one_one(A)),M) ) ).
tff(fact_94_nat__1__eq__mult__iff,axiom,
! [N: nat,Ma: nat] :
( ( one_one(nat) = times_times(nat,Ma,N) )
<=> ( ( Ma = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
tff(fact_95_nat__mult__eq__1__iff,axiom,
! [N: nat,Ma: nat] :
( ( times_times(nat,Ma,N) = one_one(nat) )
<=> ( ( Ma = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
tff(fact_96_nat__add__right__cancel,axiom,
! [N: nat,K: nat,Ma: nat] :
( ( plus_plus(nat,Ma,K) = plus_plus(nat,N,K) )
<=> ( Ma = N ) ) ).
%----Arities (12)
tff(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(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___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___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(nat) ).
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_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) ).
%------------------------------------------------------------------------------