TPTP Problem File: NUM939_5.p
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%------------------------------------------------------------------------------
% File : NUM939_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 49
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_49 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 160 ( 59 unt; 33 typ; 0 def)
% Number of atoms : 230 ( 82 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 111 ( 8 ~; 2 |; 9 &)
% ( 28 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 14 ( 10 >; 4 *; 0 +; 0 <<)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 4 con; 0-3 aty)
% Number of variables : 227 ( 206 !; 0 ?; 227 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:05
%------------------------------------------------------------------------------
%----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 (30)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
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_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_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[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_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_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Ois__sum2sq,type,
twoSqu1567020053sum2sq: 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 ).
%----Relevant facts (97)
tff(fact_0_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_1__0964_A_K_Am_A_L_A1_Advd_A1_A_124_A4_A_K_Am_A_L_A1_Advd_A2_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_2_is__mult__sum2sq,axiom,
! [Y1: int,X1: int] :
( twoSqu1567020053sum2sq(X1)
=> ( twoSqu1567020053sum2sq(Y1)
=> twoSqu1567020053sum2sq(times_times(int,X1,Y1)) ) ) ).
tff(fact_3_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_4_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_5_add__special_I3_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V2: int] : plus_plus(A,number_number_of(A,V2),one_one(A)) = number_number_of(A,plus_plus(int,V2,bit1(pls))) ) ).
tff(fact_6_less__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less(A,one_one(A),number_number_of(A,Y))
<=> ord_less(int,bit1(pls),Y) ) ) ).
tff(fact_7_less__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X: int] :
( ord_less(A,number_number_of(A,X),one_one(A))
<=> ord_less(int,X,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,
! [L1: int,K1: int] : times_times(int,bit1(K1),L1) = plus_plus(int,bit0(times_times(int,K1,L1)),L1) ).
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_add__Bit0__Bit1,axiom,
! [L1: int,K1: int] : plus_plus(int,bit0(K1),bit1(L1)) = bit1(plus_plus(int,K1,L1)) ).
tff(fact_12_add__Bit1__Bit0,axiom,
! [L1: int,K1: int] : plus_plus(int,bit1(K1),bit0(L1)) = bit1(plus_plus(int,K1,L1)) ).
tff(fact_13_rel__simps_I4_J,axiom,
! [K: int] :
( ord_less(int,pls,bit0(K))
<=> ord_less(int,pls,K) ) ).
tff(fact_14_rel__simps_I10_J,axiom,
! [K: int] :
( ord_less(int,bit0(K),pls)
<=> ord_less(int,K,pls) ) ).
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_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_19_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_20_rel__simps_I50_J,axiom,
! [L1: int,K1: int] : bit1(K1) != bit0(L1) ).
tff(fact_21_rel__simps_I49_J,axiom,
! [L1: int,K1: int] : bit0(K1) != bit1(L1) ).
tff(fact_22_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_23_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_24_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_25_rel__simps_I17_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit1(L))
<=> ord_less(int,K,L) ) ).
tff(fact_26_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_27_rel__simps_I14_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit0(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_28_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_29_mult__Bit0,axiom,
! [L1: int,K1: int] : times_times(int,bit0(K1),L1) = bit0(times_times(int,K1,L1)) ).
tff(fact_30_add__Bit0__Bit0,axiom,
! [L1: int,K1: int] : plus_plus(int,bit0(K1),bit0(L1)) = bit0(plus_plus(int,K1,L1)) ).
tff(fact_31_left__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [V2: int,B: B2,A1: B2] : times_times(B2,plus_plus(B2,A1,B),number_number_of(B2,V2)) = plus_plus(B2,times_times(B2,A1,number_number_of(B2,V2)),times_times(B2,B,number_number_of(B2,V2))) ) ).
tff(fact_32_right__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [C: B2,B: B2,V2: int] : times_times(B2,number_number_of(B2,V2),plus_plus(B2,B,C)) = plus_plus(B2,times_times(B2,number_number_of(B2,V2),B),times_times(B2,number_number_of(B2,V2),C)) ) ).
tff(fact_33_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_34_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V2: int] : times_times(A,number_number_of(A,V2),times_times(A,number_number_of(A,W),Z)) = times_times(A,number_number_of(A,times_times(int,V2,W)),Z) ) ).
tff(fact_35_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V2: int] : times_times(A,number_number_of(A,V2),number_number_of(A,W)) = number_number_of(A,times_times(int,V2,W)) ) ).
tff(fact_36_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V2: int] : plus_plus(A,number_number_of(A,V2),plus_plus(A,number_number_of(A,W),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V2,W)),Z) ) ).
tff(fact_37_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V2: int] : plus_plus(A,number_number_of(A,V2),number_number_of(A,W)) = number_number_of(A,plus_plus(int,V2,W)) ) ).
tff(fact_38_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_39_rel__simps_I16_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_40_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_41_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_42_less__int__code_I16_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit1(K11),bit1(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_43_less__int__code_I13_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit0(K11),bit0(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_44_add__Pls__right,axiom,
! [K1: int] : plus_plus(int,K1,pls) = K1 ).
tff(fact_45_add__Pls,axiom,
! [K1: int] : plus_plus(int,pls,K1) = K1 ).
tff(fact_46_Bit0__def,axiom,
! [K1: int] : bit0(K1) = plus_plus(int,K1,K1) ).
tff(fact_47_less__number__of__int__code,axiom,
! [L: int,K: int] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less(int,K,L) ) ).
tff(fact_48_times__numeral__code_I5_J,axiom,
! [W: int,V2: int] : times_times(int,number_number_of(int,V2),number_number_of(int,W)) = number_number_of(int,times_times(int,V2,W)) ).
tff(fact_49_int__distrib_I1_J,axiom,
! [W: int,Z2: int,Z11: int] : times_times(int,plus_plus(int,Z11,Z2),W) = plus_plus(int,times_times(int,Z11,W),times_times(int,Z2,W)) ).
tff(fact_50_int__distrib_I2_J,axiom,
! [Z2: int,Z11: int,W: int] : times_times(int,W,plus_plus(int,Z11,Z2)) = plus_plus(int,times_times(int,W,Z11),times_times(int,W,Z2)) ).
tff(fact_51_plus__numeral__code_I9_J,axiom,
! [W: int,V2: int] : plus_plus(int,number_number_of(int,V2),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V2,W)) ).
tff(fact_52_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,number_number_of(A,pls),A1) = A1 ) ).
tff(fact_53_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ).
tff(fact_54_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V2: int] : number_number_of(A,times_times(int,V2,W)) = times_times(A,number_number_of(A,V2),number_number_of(A,W)) ) ).
tff(fact_55_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V2: int] : number_number_of(A,plus_plus(int,V2,W)) = plus_plus(A,number_number_of(A,V2),number_number_of(A,W)) ) ).
tff(fact_56_less__int__code_I15_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit1(K11),bit0(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_57_Bit1__def,axiom,
! [K1: int] : bit1(K1) = plus_plus(int,plus_plus(int,one_one(int),K1),K1) ).
tff(fact_58_zless__add1__eq,axiom,
! [Z1: int,W1: int] :
( ord_less(int,W1,plus_plus(int,Z1,one_one(int)))
<=> ( ord_less(int,W1,Z1)
| ( W1 = Z1 ) ) ) ).
tff(fact_59_zdvd__period,axiom,
! [C1: int,T: int,X: int,D1: int,A2: int] :
( dvd_dvd(int,A2,D1)
=> ( dvd_dvd(int,A2,plus_plus(int,X,T))
<=> dvd_dvd(int,A2,plus_plus(int,plus_plus(int,X,times_times(int,C1,D1)),T)) ) ) ).
tff(fact_60_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_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_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_67_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_68_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_69_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_70_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_71_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_72_add__nat__number__of,axiom,
! [V3: int,V2: int] :
( ( ord_less(int,V2,pls)
=> ( plus_plus(nat,number_number_of(nat,V2),number_number_of(nat,V3)) = number_number_of(nat,V3) ) )
& ( ~ ord_less(int,V2,pls)
=> ( ( ord_less(int,V3,pls)
=> ( plus_plus(nat,number_number_of(nat,V2),number_number_of(nat,V3)) = number_number_of(nat,V2) ) )
& ( ~ ord_less(int,V3,pls)
=> ( plus_plus(nat,number_number_of(nat,V2),number_number_of(nat,V3)) = number_number_of(nat,plus_plus(int,V2,V3)) ) ) ) ) ) ).
tff(fact_73_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_74_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_75_less__nat__number__of,axiom,
! [V1: int,V: int] :
( ord_less(nat,number_number_of(nat,V),number_number_of(nat,V1))
<=> ( ( ord_less(int,V,V1)
=> ord_less(int,pls,V1) )
& ord_less(int,V,V1) ) ) ).
tff(fact_76_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_77_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,A2: A,C1: A] :
( ord_less(A,plus_plus(A,C1,A2),plus_plus(A,C1,B1))
<=> ord_less(A,A2,B1) ) ) ).
tff(fact_78_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C1: A,A2: A] :
( ord_less(A,plus_plus(A,A2,C1),plus_plus(A,B1,C1))
<=> ord_less(A,A2,B1) ) ) ).
tff(fact_79_semiring__norm_I110_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( one_one(A) = number_number_of(A,bit1(pls)) ) ) ).
tff(fact_80_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,A2: A] :
( ( plus_plus(A,A2,B1) = plus_plus(A,A2,C1) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_81_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B1: A] :
( ( plus_plus(A,B1,A2) = plus_plus(A,C1,A2) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_82_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_83_nat__mult__2,axiom,
! [Z: nat] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
tff(fact_84_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B: A,A1: A] : times_times(A,times_times(A,A1,B),C) = times_times(A,A1,times_times(A,B,C)) ) ).
tff(fact_85_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(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_86_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_87_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_88_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
=> ( B = C ) ) ) ).
tff(fact_89_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
tff(fact_90_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C)) ) ) ).
tff(fact_91_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B)) ) ) ).
tff(fact_92_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ) ) ).
tff(fact_93_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C))
=> ord_less(A,A1,B) ) ) ).
tff(fact_94_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B))
=> ord_less(A,A1,B) ) ) ).
tff(fact_95_mult__1__left,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_96_mult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
%----Arities (27)
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_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(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___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_Int_Oint___Groups_Oone,axiom,
one(int) ).
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_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___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(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,axiom,
semiring(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,
$false ).
%------------------------------------------------------------------------------