TPTP Problem File: LCL793_5.p
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%------------------------------------------------------------------------------
% File : LCL793_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 137
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_137 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 160 ( 61 unt; 43 typ; 0 def)
% Number of atoms : 211 ( 86 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 116 ( 22 ~; 4 |; 4 &)
% ( 20 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 38 ( 20 >; 18 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 5 con; 0-5 aty)
% Number of variables : 259 ( 237 !; 1 ?; 259 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:59
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_String_Ochar,type,
char: $tType ).
tff(ty_tc_String_Oliteral,type,
literal: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (37)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[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_Olinordered__ab__group__add,type,
linord219039673up_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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: bool > nat ).
tff(sy_c_InductTermi_OIT,type,
it: dB > $o ).
tff(sy_c_Lambda_OdB_OApp,type,
app: ( dB * dB ) > dB ).
tff(sy_c_Lambda_OdB_OVar,type,
var: nat > dB ).
tff(sy_c_Lambda_OdB_OdB__case,type,
dB_case:
!>[T1: $tType] : ( ( fun(nat,T1) * fun(dB,fun(dB,T1)) * fun(dB,T1) * dB ) > T1 ) ).
tff(sy_c_Lambda_OdB_OdB__rec,type,
dB_rec:
!>[T1: $tType] : ( ( fun(nat,T1) * fun(dB,fun(dB,fun(T1,fun(T1,T1)))) * fun(dB,fun(T1,T1)) * dB ) > T1 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * nat ) > dB ).
tff(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_String_Ochar_Ochar__size,type,
char_size: char > nat ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B2: $tType] : ( ( fun(A,B2) * A ) > B2 ) ).
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_t____,type,
t: dB ).
tff(sy_v_u____,type,
u: dB ).
tff(sy_v_ua______,type,
ua: dB ).
%----Relevant facts (99)
tff(fact_0__096IT_At_096,axiom,
it(t) ).
tff(fact_1_Var_I3_J,axiom,
it(ua) ).
tff(fact_2_uIT,axiom,
it(u) ).
tff(fact_3__096IT_A_Ilift_Au_A0_J_096,axiom,
it(lift(u,zero_zero(nat))) ).
tff(fact_4_lift__IT,axiom,
! [I: nat,T: dB] :
( it(T)
=> it(lift(T,I)) ) ).
tff(fact_5_Var__IT,axiom,
! [N: nat] : it(var(N)) ).
tff(fact_6_app__Var__IT,axiom,
! [I: nat,T: dB] :
( it(T)
=> it(app(T,var(I))) ) ).
tff(fact_7_lift_Osimps_I2_J,axiom,
! [K: nat,T: dB,S: dB] : lift(app(S,T),K) = app(lift(S,K),lift(T,K)) ).
tff(fact_8_dB_Osimps_I1_J,axiom,
! [Nat2: nat,Nat: nat] :
( ( var(Nat) = var(Nat2) )
<=> ( Nat = Nat2 ) ) ).
tff(fact_9_dB_Osimps_I2_J,axiom,
! [DB22: dB,DB12: dB,DB2: dB,DB1: dB] :
( ( app(DB1,DB2) = app(DB12,DB22) )
<=> ( ( DB1 = DB12 )
& ( DB2 = DB22 ) ) ) ).
tff(fact_10_dB_Osimps_I4_J,axiom,
! [DB21: dB,DB11: dB,Nat1: nat] : var(Nat1) != app(DB11,DB21) ).
tff(fact_11_dB_Osimps_I5_J,axiom,
! [Nat1: nat,DB21: dB,DB11: dB] : app(DB11,DB21) != var(Nat1) ).
tff(fact_12_dB_Osize_I1_J,axiom,
! [Nat1: nat] : dB_size(var(Nat1)) = zero_zero(nat) ).
tff(fact_13_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_14_dB_Osize_I4_J,axiom,
! [Nat1: nat] : size_size(dB,var(Nat1)) = zero_zero(nat) ).
tff(fact_15_subst__Var__IT,axiom,
! [J: nat,I: nat,R: dB] :
( it(R)
=> it(subst(R,var(I),J)) ) ).
tff(fact_16_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_17_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_18_subst__App,axiom,
! [K: nat,S: dB,U: dB,T: dB] : subst(app(T,U),S,K) = app(subst(T,S,K),subst(U,S,K)) ).
tff(fact_19_subst__eq,axiom,
! [U: dB,K: nat] : subst(var(K),U,K) = U ).
tff(fact_20_subst__lift,axiom,
! [S: dB,K: nat,T: dB] : subst(lift(T,K),S,K) = T ).
tff(fact_21_nat_Osize_I1_J,axiom,
nat_size(zero_zero(nat)) = zero_zero(nat) ).
tff(fact_22_bool_Osize_I4_J,axiom,
size_size(bool,fFalse) = zero_zero(nat) ).
tff(fact_23_bool_Osize_I3_J,axiom,
size_size(bool,fTrue) = zero_zero(nat) ).
tff(fact_24_nat_Osize_I3_J,axiom,
size_size(nat,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_25_substn__subst__0,axiom,
! [S: dB,T: dB] : substn(T,S,zero_zero(nat)) = subst(T,S,zero_zero(nat)) ).
tff(fact_26_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat)) = aa(nat,A,F1,Nat) ).
tff(fact_27_dB_Osimps_I11_J,axiom,
! [A: $tType,DB2: dB,DB1: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,app(DB1,DB2)) = aa(dB,A,aa(dB,fun(dB,A),F2,DB1),DB2) ).
tff(fact_28_substn_Osimps_I2_J,axiom,
! [K: nat,S: dB,U: dB,T: dB] : substn(app(T,U),S,K) = app(substn(T,S,K),substn(U,S,K)) ).
tff(fact_29_nat__size,axiom,
! [N: nat] : size_size(nat,N) = N ).
tff(fact_30_size__bool,axiom,
! [B1: bool] : size_size(bool,B1) = zero_zero(nat) ).
tff(fact_31_substn__subst__n,axiom,
! [N: nat,S: dB,T: dB] : substn(T,S,N) = subst(T,liftn(N,S,zero_zero(nat)),N) ).
tff(fact_32_liftn_Osimps_I2_J,axiom,
! [K: nat,T: dB,S: dB,N: nat] : liftn(N,app(S,T),K) = app(liftn(N,S,K),liftn(N,T,K)) ).
tff(fact_33_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat: nat,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,var(Nat)) = aa(nat,A,F1,Nat) ).
tff(fact_34_dB_Orecs_I2_J,axiom,
! [A: $tType,DB2: dB,DB1: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,app(DB1,DB2)) = aa(A,A,aa(A,fun(A,A),aa(dB,fun(A,fun(A,A)),aa(dB,fun(dB,fun(A,fun(A,A))),F2,DB1),DB2),dB_rec(A,F1,F2,F3,DB1)),dB_rec(A,F1,F2,F3,DB2)) ).
tff(fact_35_subst__lt,axiom,
! [U: dB,I: nat,J: nat] :
( ord_less(nat,J,I)
=> ( subst(var(J),U,I) = var(J) ) ) ).
tff(fact_36_char__size,axiom,
! [C: char] : char_size(C) = zero_zero(nat) ).
tff(fact_37_liftn__0,axiom,
! [K: nat,T: dB] : liftn(zero_zero(nat),T,K) = T ).
tff(fact_38_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_39_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_40_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_41_size__char,axiom,
! [C: char] : size_size(char,C) = zero_zero(nat) ).
tff(fact_42_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_43_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( ord_less(nat,M1,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ord_less(nat,Na,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) ) ) ) ).
tff(fact_44_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_45_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_46_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_47_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_48_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( ord_less(nat,M1,Na)
| ord_less(nat,Na,M1) ) ) ).
tff(fact_49_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_50_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_51_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_52_size__literal__def,axiom,
! [S: literal] : size_size(literal,S) = zero_zero(nat) ).
tff(fact_53_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_54_liftn_Osimps_I1_J,axiom,
! [N: nat,K: nat,I: nat] :
( ( ord_less(nat,I,K)
=> ( liftn(N,var(I),K) = var(I) ) )
& ( ~ ord_less(nat,I,K)
=> ( liftn(N,var(I),K) = var(plus_plus(nat,I,N)) ) ) ) ).
tff(fact_55_infinite__descent0,axiom,
! [Na: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N1: nat] :
( ord_less(nat,zero_zero(nat),N1)
=> ( ~ pp(aa(nat,bool,P1,N1))
=> ? [M2: nat] :
( ord_less(nat,M2,N1)
& ~ pp(aa(nat,bool,P1,M2)) ) ) )
=> pp(aa(nat,bool,P1,Na)) ) ) ).
tff(fact_56_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Aa: A,B1: A] :
( ( plus_plus(A,B1,Aa) = plus_plus(A,C1,Aa) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_57_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,Aa: A] :
( ( plus_plus(A,Aa,B1) = plus_plus(A,Aa,C1) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_58_nat__add__right__cancel,axiom,
! [Na: nat,K1: nat,M1: nat] :
( ( plus_plus(nat,M1,K1) = plus_plus(nat,Na,K1) )
<=> ( M1 = Na ) ) ).
tff(fact_59_nat__add__left__cancel,axiom,
! [Na: nat,M1: nat,K1: nat] :
( ( plus_plus(nat,K1,M1) = plus_plus(nat,K1,Na) )
<=> ( M1 = Na ) ) ).
tff(fact_60_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Na: nat,M1: nat] :
( ( semiring_1_of_nat(A,M1) = semiring_1_of_nat(A,Na) )
<=> ( M1 = Na ) ) ) ).
tff(fact_61_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ( zero_zero(A) = plus_plus(A,Aa,Aa) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_62_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,Aa: A,C1: A] :
( ord_less(A,plus_plus(A,C1,Aa),plus_plus(A,C1,B1))
<=> ord_less(A,Aa,B1) ) ) ).
tff(fact_63_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C1: A,Aa: A] :
( ord_less(A,plus_plus(A,Aa,C1),plus_plus(A,B1,C1))
<=> ord_less(A,Aa,B1) ) ) ).
tff(fact_64_add__is__0,axiom,
! [Na: nat,M1: nat] :
( ( plus_plus(nat,M1,Na) = zero_zero(nat) )
<=> ( ( M1 = zero_zero(nat) )
& ( Na = zero_zero(nat) ) ) ) ).
tff(fact_65_nat__add__left__cancel__less,axiom,
! [Na: nat,M1: nat,K1: nat] :
( ord_less(nat,plus_plus(nat,K1,M1),plus_plus(nat,K1,Na))
<=> ord_less(nat,M1,Na) ) ).
tff(fact_66_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ord_less(A,plus_plus(A,Aa,Aa),zero_zero(A))
<=> ord_less(A,Aa,zero_zero(A)) ) ) ).
tff(fact_67_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ord_less(A,zero_zero(A),plus_plus(A,Aa,Aa))
<=> ord_less(A,zero_zero(A),Aa) ) ) ).
tff(fact_68_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_69_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,M1: nat] :
( ord_less(A,semiring_1_of_nat(A,M1),semiring_1_of_nat(A,Na))
<=> ord_less(nat,M1,Na) ) ) ).
tff(fact_70_add__gr__0,axiom,
! [Na: nat,M1: nat] :
( ord_less(nat,zero_zero(nat),plus_plus(nat,M1,Na))
<=> ( ord_less(nat,zero_zero(nat),M1)
| ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_71_of__nat__add,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,M: nat] : semiring_1_of_nat(A,plus_plus(nat,M,N)) = plus_plus(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ).
tff(fact_72_not__add__less1,axiom,
! [J: nat,I: nat] : ~ ord_less(nat,plus_plus(nat,I,J),I) ).
tff(fact_73_not__add__less2,axiom,
! [I: nat,J: nat] : ~ ord_less(nat,plus_plus(nat,J,I),I) ).
tff(fact_74_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_75_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_76_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_77_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_78_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_79_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_80_add__eq__self__zero,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = M )
=> ( N = zero_zero(nat) ) ) ).
tff(fact_81_Nat_Oadd__0__right,axiom,
! [M: nat] : plus_plus(nat,M,zero_zero(nat)) = M ).
tff(fact_82_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus(nat,zero_zero(nat),N) = N ).
tff(fact_83_nat__add__assoc,axiom,
! [K: nat,N: nat,M: nat] : plus_plus(nat,plus_plus(nat,M,N),K) = plus_plus(nat,M,plus_plus(nat,N,K)) ).
tff(fact_84_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
tff(fact_85_nat__add__commute,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
tff(fact_86_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_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__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_89_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_90_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_91_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_92_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_93_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_94_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_95_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_96_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_97_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_98_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) ) ) ).
%----Arities (11)
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___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___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___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (6)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
it(app(lift(u,zero_zero(nat)),var(zero_zero(nat)))) ).
%------------------------------------------------------------------------------