TPTP Problem File: NUM978_5.p
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%------------------------------------------------------------------------------
% File : NUM978_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 123
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_123 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 186 ( 67 unt; 46 typ; 0 def)
% Number of atoms : 262 ( 76 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 147 ( 25 ~; 3 |; 5 &)
% ( 30 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 16 ( 11 >; 5 *; 0 +; 0 <<)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-3 aty)
% Number of functors : 17 ( 17 usr; 7 con; 0-4 aty)
% Number of variables : 247 ( 214 !; 1 ?; 247 :)
% ( 32 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:47
%------------------------------------------------------------------------------
%----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_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (42)
tff(sy_cl_Groups_Oone,type,
one:
!>[A1: $tType] : $o ).
tff(sy_cl_Int_Onumber,type,
number:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A1: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A1: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A1: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A1: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A1: $tType] : $o ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A1: $tType] : A1 ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A1: $tType] : ( ( A1 * A1 ) > A1 ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A1: $tType] : ( ( A1 * A1 ) > A1 ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A1: $tType] : A1 ).
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,type,
number_number_of:
!>[A1: $tType] : ( int > A1 ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A1: $tType] : ( nat > A1 ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A1: $tType] : ( ( A1 * A1 ) > $o ) ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Ois__sum2sq,type,
twoSqu1567020053sum2sq: int > $o ).
tff(sy_c_aa,type,
aa:
!>[A1: $tType,B: $tType] : ( ( fun(A1,B) * A1 ) > B ) ).
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_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
tff(sy_v_tn____,type,
tn: nat ).
%----Relevant facts (96)
tff(fact_0_tn0,axiom,
ord_less(nat,zero_zero(nat),tn) ).
tff(fact_1_smaller_I1_J,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_2_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_3_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_4_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_5_of__nat__0__less__iff,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [Na: nat] :
( ord_less(A1,zero_zero(A1),semiring_1_of_nat(A1,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_6_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_7_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_8_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_9_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A1: $tType] :
( linord219039673up_add(A1)
=> ! [A2: A1] :
( ord_less(A1,zero_zero(A1),plus_plus(A1,A2,A2))
<=> ord_less(A1,zero_zero(A1),A2) ) ) ).
tff(fact_10_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A1: $tType] :
( linord219039673up_add(A1)
=> ! [A2: A1] :
( ord_less(A1,plus_plus(A1,A2,A2),zero_zero(A1))
<=> ord_less(A1,A2,zero_zero(A1)) ) ) ).
tff(fact_11_of__nat__less__iff,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [Na: nat,Ma: nat] :
( ord_less(A1,semiring_1_of_nat(A1,Ma),semiring_1_of_nat(A1,Na))
<=> ord_less(nat,Ma,Na) ) ) ).
tff(fact_12_of__nat__0,axiom,
! [A1: $tType] :
( semiring_1(A1)
=> ( semiring_1_of_nat(A1,zero_zero(nat)) = zero_zero(A1) ) ) ).
tff(fact_13_t1,axiom,
ord_less(int,one_one(int),t) ).
tff(fact_14_add__left__cancel,axiom,
! [A1: $tType] :
( cancel_semigroup_add(A1)
=> ! [C1: A1,B2: A1,A2: A1] :
( ( plus_plus(A1,A2,B2) = plus_plus(A1,A2,C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_15_add__right__cancel,axiom,
! [A1: $tType] :
( cancel_semigroup_add(A1)
=> ! [C1: A1,A2: A1,B2: A1] :
( ( plus_plus(A1,B2,A2) = plus_plus(A1,C1,A2) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_16_add__is__0,axiom,
! [Na: nat,Ma: nat] :
( ( plus_plus(nat,Ma,Na) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
& ( Na = zero_zero(nat) ) ) ) ).
tff(fact_17_nat__add__left__cancel__less,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ord_less(nat,plus_plus(nat,K1,Ma),plus_plus(nat,K1,Na))
<=> ord_less(nat,Ma,Na) ) ).
tff(fact_18_of__nat__eq__iff,axiom,
! [A1: $tType] :
( semiring_char_0(A1)
=> ! [Na: nat,Ma: nat] :
( ( semiring_1_of_nat(A1,Ma) = semiring_1_of_nat(A1,Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_19_double__zero__sym,axiom,
! [A1: $tType] :
( linord219039673up_add(A1)
=> ! [A2: A1] :
( ( zero_zero(A1) = plus_plus(A1,A2,A2) )
<=> ( A2 = zero_zero(A1) ) ) ) ).
tff(fact_20_add__less__cancel__left,axiom,
! [A1: $tType] :
( ordere236663937imp_le(A1)
=> ! [B2: A1,A2: A1,C1: A1] :
( ord_less(A1,plus_plus(A1,C1,A2),plus_plus(A1,C1,B2))
<=> ord_less(A1,A2,B2) ) ) ).
tff(fact_21_add__less__cancel__right,axiom,
! [A1: $tType] :
( ordere236663937imp_le(A1)
=> ! [B2: A1,C1: A1,A2: A1] :
( ord_less(A1,plus_plus(A1,A2,C1),plus_plus(A1,B2,C1))
<=> ord_less(A1,A2,B2) ) ) ).
tff(fact_22_add__gr__0,axiom,
! [Na: nat,Ma: nat] :
( ord_less(nat,zero_zero(nat),plus_plus(nat,Ma,Na))
<=> ( ord_less(nat,zero_zero(nat),Ma)
| ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_23_of__nat__add,axiom,
! [A1: $tType] :
( semiring_1(A1)
=> ! [N: nat,M: nat] : semiring_1_of_nat(A1,plus_plus(nat,M,N)) = plus_plus(A1,semiring_1_of_nat(A1,M),semiring_1_of_nat(A1,N)) ) ).
tff(fact_24_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A1: $tType] :
( ab_semigroup_add(A1)
=> ! [C: A1,B1: A1,A: A1] : plus_plus(A1,plus_plus(A1,A,B1),C) = plus_plus(A1,A,plus_plus(A1,B1,C)) ) ).
tff(fact_25_add__left__imp__eq,axiom,
! [A1: $tType] :
( cancel_semigroup_add(A1)
=> ! [C: A1,B1: A1,A: A1] :
( ( plus_plus(A1,A,B1) = plus_plus(A1,A,C) )
=> ( B1 = C ) ) ) ).
tff(fact_26_add__imp__eq,axiom,
! [A1: $tType] :
( cancel146912293up_add(A1)
=> ! [C: A1,B1: A1,A: A1] :
( ( plus_plus(A1,A,B1) = plus_plus(A1,A,C) )
=> ( B1 = C ) ) ) ).
tff(fact_27_add__right__imp__eq,axiom,
! [A1: $tType] :
( cancel_semigroup_add(A1)
=> ! [C: A1,A: A1,B1: A1] :
( ( plus_plus(A1,B1,A) = plus_plus(A1,C,A) )
=> ( B1 = C ) ) ) ).
tff(fact_28_add__eq__self__zero,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = M )
=> ( N = zero_zero(nat) ) ) ).
tff(fact_29_Nat_Oadd__0__right,axiom,
! [M: nat] : plus_plus(nat,M,zero_zero(nat)) = M ).
tff(fact_30_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus(nat,zero_zero(nat),N) = N ).
tff(fact_31_add__lessD1,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less(nat,plus_plus(nat,I,J),K)
=> ord_less(nat,I,K) ) ).
tff(fact_32_less__add__eq__less,axiom,
! [N: nat,M: nat,L: nat,K: nat] :
( ord_less(nat,K,L)
=> ( ( plus_plus(nat,M,L) = plus_plus(nat,K,N) )
=> ord_less(nat,M,N) ) ) ).
tff(fact_33_add__less__mono,axiom,
! [L: nat,K: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ( ord_less(nat,K,L)
=> ord_less(nat,plus_plus(nat,I,K),plus_plus(nat,J,L)) ) ) ).
tff(fact_34_add__less__mono1,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,plus_plus(nat,I,K),plus_plus(nat,J,K)) ) ).
tff(fact_35_trans__less__add2,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,I,plus_plus(nat,M,J)) ) ).
tff(fact_36_trans__less__add1,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,I,plus_plus(nat,J,M)) ) ).
tff(fact_37_not__add__less2,axiom,
! [I: nat,J: nat] : ~ ord_less(nat,plus_plus(nat,J,I),I) ).
tff(fact_38_not__add__less1,axiom,
! [J: nat,I: nat] : ~ ord_less(nat,plus_plus(nat,I,J),I) ).
tff(fact_39_add_Ocomm__neutral,axiom,
! [A1: $tType] :
( comm_monoid_add(A1)
=> ! [A: A1] : plus_plus(A1,A,zero_zero(A1)) = A ) ).
tff(fact_40_add__0__right,axiom,
! [A1: $tType] :
( monoid_add(A1)
=> ! [A: A1] : plus_plus(A1,A,zero_zero(A1)) = A ) ).
tff(fact_41_add__0,axiom,
! [A1: $tType] :
( comm_monoid_add(A1)
=> ! [A: A1] : plus_plus(A1,zero_zero(A1),A) = A ) ).
tff(fact_42_add__0__left,axiom,
! [A1: $tType] :
( monoid_add(A1)
=> ! [A: A1] : plus_plus(A1,zero_zero(A1),A) = A ) ).
tff(fact_43_add__less__imp__less__left,axiom,
! [A1: $tType] :
( ordere236663937imp_le(A1)
=> ! [B1: A1,A: A1,C: A1] :
( ord_less(A1,plus_plus(A1,C,A),plus_plus(A1,C,B1))
=> ord_less(A1,A,B1) ) ) ).
tff(fact_44_add__less__imp__less__right,axiom,
! [A1: $tType] :
( ordere236663937imp_le(A1)
=> ! [B1: A1,C: A1,A: A1] :
( ord_less(A1,plus_plus(A1,A,C),plus_plus(A1,B1,C))
=> ord_less(A1,A,B1) ) ) ).
tff(fact_45_add__strict__mono,axiom,
! [A1: $tType] :
( ordere223160158up_add(A1)
=> ! [D: A1,C: A1,B1: A1,A: A1] :
( ord_less(A1,A,B1)
=> ( ord_less(A1,C,D)
=> ord_less(A1,plus_plus(A1,A,C),plus_plus(A1,B1,D)) ) ) ) ).
tff(fact_46_add__strict__left__mono,axiom,
! [A1: $tType] :
( ordere223160158up_add(A1)
=> ! [C: A1,B1: A1,A: A1] :
( ord_less(A1,A,B1)
=> ord_less(A1,plus_plus(A1,C,A),plus_plus(A1,C,B1)) ) ) ).
tff(fact_47_add__strict__right__mono,axiom,
! [A1: $tType] :
( ordere223160158up_add(A1)
=> ! [C: A1,B1: A1,A: A1] :
( ord_less(A1,A,B1)
=> ord_less(A1,plus_plus(A1,A,C),plus_plus(A1,B1,C)) ) ) ).
tff(fact_48_zero__reorient,axiom,
! [A1: $tType] :
( zero(A1)
=> ! [X: A1] :
( ( zero_zero(A1) = X )
<=> ( X = zero_zero(A1) ) ) ) ).
tff(fact_49_of__nat__less__0__iff,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [M: nat] : ~ ord_less(A1,semiring_1_of_nat(A1,M),zero_zero(A1)) ) ).
tff(fact_50_of__nat__less__imp__less,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [N: nat,M: nat] :
( ord_less(A1,semiring_1_of_nat(A1,M),semiring_1_of_nat(A1,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_51_less__imp__of__nat__less,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(A1,semiring_1_of_nat(A1,M),semiring_1_of_nat(A1,N)) ) ) ).
tff(fact_52_add__neg__neg,axiom,
! [A1: $tType] :
( ordere216010020id_add(A1)
=> ! [B1: A1,A: A1] :
( ord_less(A1,A,zero_zero(A1))
=> ( ord_less(A1,B1,zero_zero(A1))
=> ord_less(A1,plus_plus(A1,A,B1),zero_zero(A1)) ) ) ) ).
tff(fact_53_add__pos__pos,axiom,
! [A1: $tType] :
( ordere216010020id_add(A1)
=> ! [B1: A1,A: A1] :
( ord_less(A1,zero_zero(A1),A)
=> ( ord_less(A1,zero_zero(A1),B1)
=> ord_less(A1,zero_zero(A1),plus_plus(A1,A,B1)) ) ) ) ).
tff(fact_54_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),Na: nat,Ma: nat] :
( ( ord_less(nat,Ma,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ( Ma = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ord_less(nat,Na,Ma)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) ) ) ) ).
tff(fact_55_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_56_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_57_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_58_linorder__neqE__nat,axiom,
! [Y1: nat,X1: nat] :
( ( X1 != Y1 )
=> ( ~ ord_less(nat,X1,Y1)
=> ord_less(nat,Y1,X1) ) ) ).
tff(fact_59_nat__neq__iff,axiom,
! [Na: nat,Ma: nat] :
( ( Ma != Na )
<=> ( ord_less(nat,Ma,Na)
| ord_less(nat,Na,Ma) ) ) ).
tff(fact_60_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_61_double__eq__0__iff,axiom,
! [A1: $tType] :
( linord219039673up_add(A1)
=> ! [A2: A1] :
( ( plus_plus(A1,A2,A2) = zero_zero(A1) )
<=> ( A2 = zero_zero(A1) ) ) ) ).
tff(fact_62_smaller_I2_J,axiom,
~ ( ord_less(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
=> ~ twoSqu1567020053sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))) ) ).
tff(fact_63_pos__add__strict,axiom,
! [A1: $tType] :
( linordered_semidom(A1)
=> ! [C: A1,B1: A1,A: A1] :
( ord_less(A1,zero_zero(A1),A)
=> ( ord_less(A1,B1,C)
=> ord_less(A1,B1,plus_plus(A1,A,C)) ) ) ) ).
tff(fact_64_even__less__0__iff,axiom,
! [A1: $tType] :
( linordered_idom(A1)
=> ! [A2: A1] :
( ord_less(A1,plus_plus(A1,A2,A2),zero_zero(A1))
<=> ord_less(A1,A2,zero_zero(A1)) ) ) ).
tff(fact_65_less__imp__add__positive,axiom,
! [J: nat,I: nat] :
( ord_less(nat,I,J)
=> ? [K2: nat] :
( ord_less(nat,zero_zero(nat),K2)
& ( plus_plus(nat,I,K2) = J ) ) ) ).
tff(fact_66_zero__less__int__conv,axiom,
! [Na: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_67_add__0__iff,axiom,
! [A1: $tType] :
( semiri456707255roduct(A1)
=> ! [A2: A1,B2: A1] :
( ( B2 = plus_plus(A1,B2,A2) )
<=> ( A2 = zero_zero(A1) ) ) ) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A1: $tType] :
( comm_semiring_1(A1)
=> ! [A: A1] : plus_plus(A1,A,zero_zero(A1)) = A ) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A1: $tType] :
( comm_semiring_1(A1)
=> ! [A: A1] : plus_plus(A1,zero_zero(A1),A) = A ) ).
tff(fact_70_eq__number__of,axiom,
! [A1: $tType] :
( ( number_ring(A1)
& ring_char_0(A1) )
=> ! [Y: int,X: int] :
( ( number_number_of(A1,X) = number_number_of(A1,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_71_rel__simps_I51_J,axiom,
! [L1: int,K1: int] :
( ( bit1(K1) = bit1(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_72_rel__simps_I48_J,axiom,
! [L1: int,K1: int] :
( ( bit0(K1) = bit0(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_73_nat__add__left__cancel,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ( plus_plus(nat,K1,Ma) = plus_plus(nat,K1,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_74_nat__add__right__cancel,axiom,
! [Na: nat,K1: nat,Ma: nat] :
( ( plus_plus(nat,Ma,K1) = plus_plus(nat,Na,K1) )
<=> ( Ma = Na ) ) ).
tff(fact_75_nQ1,axiom,
~ twoSqu1567020053sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,zero_zero(nat))))) ).
tff(fact_76_mult__zero__left,axiom,
! [A1: $tType] :
( mult_zero(A1)
=> ! [A: A1] : times_times(A1,zero_zero(A1),A) = zero_zero(A1) ) ).
tff(fact_77_mult__zero__right,axiom,
! [A1: $tType] :
( mult_zero(A1)
=> ! [A: A1] : times_times(A1,A,zero_zero(A1)) = zero_zero(A1) ) ).
tff(fact_78_mult__eq__0__iff,axiom,
! [A1: $tType] :
( ring_n68954251visors(A1)
=> ! [B2: A1,A2: A1] :
( ( times_times(A1,A2,B2) = zero_zero(A1) )
<=> ( ( A2 = zero_zero(A1) )
| ( B2 = zero_zero(A1) ) ) ) ) ).
tff(fact_79_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_80_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_81_rel__simps_I50_J,axiom,
! [L: int,K: int] : bit1(K) != bit0(L) ).
tff(fact_82_rel__simps_I49_J,axiom,
! [L: int,K: int] : bit0(K) != bit1(L) ).
tff(fact_83_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_84_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_85_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_86_rel__simps_I17_J,axiom,
! [L1: int,K1: int] :
( ord_less(int,bit1(K1),bit1(L1))
<=> ord_less(int,K1,L1) ) ).
tff(fact_87_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_88_rel__simps_I14_J,axiom,
! [L1: int,K1: int] :
( ord_less(int,bit0(K1),bit0(L1))
<=> ord_less(int,K1,L1) ) ).
tff(fact_89_of__nat__1,axiom,
! [A1: $tType] :
( semiring_1(A1)
=> ( semiring_1_of_nat(A1,one_one(nat)) = one_one(A1) ) ) ).
tff(fact_90_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_91_mult__Bit0,axiom,
! [L: int,K: int] : times_times(int,bit0(K),L) = bit0(times_times(int,K,L)) ).
tff(fact_92_add__Bit0__Bit0,axiom,
! [L: int,K: int] : plus_plus(int,bit0(K),bit0(L)) = bit0(plus_plus(int,K,L)) ).
tff(fact_93_p0,axiom,
ord_less(int,zero_zero(int),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_94_left__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [V: int,B1: B,A: B] : times_times(B,plus_plus(B,A,B1),number_number_of(B,V)) = plus_plus(B,times_times(B,A,number_number_of(B,V)),times_times(B,B1,number_number_of(B,V))) ) ).
tff(fact_95_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)) ) ).
%----Arities (41)
tff(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(int) ).
tff(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(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___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(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___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(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___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(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,
ord_less(nat,zero_zero(nat),n) ).
%------------------------------------------------------------------------------