TPTP Problem File: LCL772_5.p
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%------------------------------------------------------------------------------
% File : LCL772_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 77
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_77 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 153 ( 50 unt; 37 typ; 0 def)
% Number of atoms : 226 ( 88 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 139 ( 29 ~; 2 |; 4 &)
% ( 21 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 32 ( 15 >; 17 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 17 ( 17 usr; 4 con; 0-5 aty)
% Number of variables : 308 ( 286 !; 1 ?; 308 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:17
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
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_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (33)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[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_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__comm__monoid__add,type,
ordere216010020id_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_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[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_InductTermi_OIT,type,
it: dB > $o ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
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_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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
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_i,type,
i: nat ).
tff(sy_v_r,type,
r: dB ).
%----Relevant facts (99)
tff(fact_0_Var__IT,axiom,
! [N1: nat] : it(var(N1)) ).
tff(fact_1_Lambda,axiom,
! [R: dB] :
( it(R)
=> it(abs(R)) ) ).
tff(fact_2_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB: dB] :
( ( abs(DB) = abs(DB5) )
<=> ( DB = DB5 ) ) ).
tff(fact_3_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat1: nat] :
( ( var(Nat1) = var(Nat3) )
<=> ( Nat1 = Nat3 ) ) ).
tff(fact_4_dB_Osimps_I2_J,axiom,
! [DB24: dB,DB14: dB,DB2: dB,DB1: dB] :
( ( app(DB1,DB2) = app(DB14,DB24) )
<=> ( ( DB1 = DB14 )
& ( DB2 = DB24 ) ) ) ).
tff(fact_5_dB_Osimps_I7_J,axiom,
! [Nat: nat,DB4: dB] : abs(DB4) != var(Nat) ).
tff(fact_6_dB_Osimps_I6_J,axiom,
! [DB4: dB,Nat: nat] : var(Nat) != abs(DB4) ).
tff(fact_7_dB_Osimps_I9_J,axiom,
! [DB23: dB,DB13: dB,DB4: dB] : abs(DB4) != app(DB13,DB23) ).
tff(fact_8_dB_Osimps_I8_J,axiom,
! [DB4: dB,DB23: dB,DB13: dB] : app(DB13,DB23) != abs(DB4) ).
tff(fact_9_dB_Osimps_I4_J,axiom,
! [DB22: dB,DB12: dB,Nat: nat] : var(Nat) != app(DB12,DB22) ).
tff(fact_10_dB_Osimps_I5_J,axiom,
! [Nat: nat,DB22: dB,DB12: dB] : app(DB12,DB22) != var(Nat) ).
tff(fact_11_subst__Var__IT,axiom,
! [J: nat,I: nat,R: dB] :
( it(R)
=> it(subst(R,var(I),J)) ) ).
tff(fact_12_dB_Oexhaust,axiom,
! [Y: dB] :
( ! [Nat2: nat] : Y != var(Nat2)
=> ( ! [DB11: dB,DB21: dB] : Y != app(DB11,DB21)
=> ~ ! [DB3: dB] : Y != abs(DB3) ) ) ).
tff(fact_13_subst__App,axiom,
! [K1: nat,S: dB,U: dB,T: dB] : subst(app(T,U),S,K1) = app(subst(T,S,K1),subst(U,S,K1)) ).
tff(fact_14_subst__eq,axiom,
! [U: dB,K1: nat] : subst(var(K1),U,K1) = U ).
tff(fact_15_dB_Osimps_I12_J,axiom,
! [A: $tType,DB: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB)) = aa(dB,A,F3,DB) ).
tff(fact_16_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat1: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat1)) = aa(nat,A,F1,Nat1) ).
tff(fact_17_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_18_substn_Osimps_I2_J,axiom,
! [K1: nat,S: dB,U: dB,T: dB] : substn(app(T,U),S,K1) = app(substn(T,S,K1),substn(U,S,K1)) ).
tff(fact_19_liftn_Osimps_I2_J,axiom,
! [K1: nat,T: dB,S: dB,N1: nat] : liftn(N1,app(S,T),K1) = app(liftn(N1,S,K1),liftn(N1,T,K1)) ).
tff(fact_20_dB_Orecs_I3_J,axiom,
! [A: $tType,DB: 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,abs(DB)) = aa(A,A,aa(dB,fun(A,A),F3,DB),dB_rec(A,F1,F2,F3,DB)) ).
tff(fact_21_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat1: 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(Nat1)) = aa(nat,A,F1,Nat1) ).
tff(fact_22_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_23_subst__lt,axiom,
! [U: dB,I: nat,J: nat] :
( ord_less(nat,J,I)
=> ( subst(var(J),U,I) = var(J) ) ) ).
tff(fact_24_substn__subst__n,axiom,
! [N1: nat,S: dB,T: dB] : substn(T,S,N1) = subst(T,liftn(N1,S,zero_zero(nat)),N1) ).
tff(fact_25_liftn_Osimps_I1_J,axiom,
! [N1: nat,K1: nat,I: nat] :
( ( ord_less(nat,I,K1)
=> ( liftn(N1,var(I),K1) = var(I) ) )
& ( ~ ord_less(nat,I,K1)
=> ( liftn(N1,var(I),K1) = var(plus_plus(nat,I,N1)) ) ) ) ).
tff(fact_26_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),N: nat,M: nat] :
( ( ord_less(nat,M,N)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M)) )
=> ( ( ( M = N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M)) )
=> ( ( ord_less(nat,N,M)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M)) ) ) ) ).
tff(fact_27_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_28_less__not__refl2,axiom,
! [M1: nat,N1: nat] :
( ord_less(nat,N1,M1)
=> ( M1 != N1 ) ) ).
tff(fact_29_less__irrefl__nat,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_30_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_31_nat__add__left__cancel,axiom,
! [N: nat,M: nat,K2: nat] :
( ( plus_plus(nat,K2,M) = plus_plus(nat,K2,N) )
<=> ( M = N ) ) ).
tff(fact_32_nat__add__right__cancel,axiom,
! [N: nat,K2: nat,M: nat] :
( ( plus_plus(nat,M,K2) = plus_plus(nat,N,K2) )
<=> ( M = N ) ) ).
tff(fact_33_less__zeroE,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_34_less__nat__zero__code,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_35_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_36_add__is__0,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_37_nat__add__left__cancel__less,axiom,
! [N: nat,M: nat,K2: nat] :
( ord_less(nat,plus_plus(nat,K2,M),plus_plus(nat,K2,N))
<=> ord_less(nat,M,N) ) ).
tff(fact_38_add__gr__0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,zero_zero(nat),plus_plus(nat,M,N))
<=> ( ord_less(nat,zero_zero(nat),M)
| ord_less(nat,zero_zero(nat),N) ) ) ).
tff(fact_39_gr0I,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_40_add__lessD1,axiom,
! [K1: nat,J: nat,I: nat] :
( ord_less(nat,plus_plus(nat,I,J),K1)
=> ord_less(nat,I,K1) ) ).
tff(fact_41_less__add__eq__less,axiom,
! [N1: nat,M1: nat,L: nat,K1: nat] :
( ord_less(nat,K1,L)
=> ( ( plus_plus(nat,M1,L) = plus_plus(nat,K1,N1) )
=> ord_less(nat,M1,N1) ) ) ).
tff(fact_42_add__less__mono,axiom,
! [L: nat,K1: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ( ord_less(nat,K1,L)
=> ord_less(nat,plus_plus(nat,I,K1),plus_plus(nat,J,L)) ) ) ).
tff(fact_43_add__less__mono1,axiom,
! [K1: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,plus_plus(nat,I,K1),plus_plus(nat,J,K1)) ) ).
tff(fact_44_trans__less__add2,axiom,
! [M1: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,I,plus_plus(nat,M1,J)) ) ).
tff(fact_45_trans__less__add1,axiom,
! [M1: nat,J: nat,I: nat] :
( ord_less(nat,I,J)
=> ord_less(nat,I,plus_plus(nat,J,M1)) ) ).
tff(fact_46_gr__implies__not0,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,N1)
=> ( N1 != zero_zero(nat) ) ) ).
tff(fact_47_not__add__less2,axiom,
! [I: nat,J: nat] : ~ ord_less(nat,plus_plus(nat,J,I),I) ).
tff(fact_48_not__add__less1,axiom,
! [J: nat,I: nat] : ~ ord_less(nat,plus_plus(nat,I,J),I) ).
tff(fact_49_not__less0,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_50_plus__nat_Oadd__0,axiom,
! [N1: nat] : plus_plus(nat,zero_zero(nat),N1) = N1 ).
tff(fact_51_Nat_Oadd__0__right,axiom,
! [M1: nat] : plus_plus(nat,M1,zero_zero(nat)) = M1 ).
tff(fact_52_nat__add__commute,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,N1) = plus_plus(nat,N1,M1) ).
tff(fact_53_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_54_nat__add__assoc,axiom,
! [K1: nat,N1: nat,M1: nat] : plus_plus(nat,plus_plus(nat,M1,N1),K1) = plus_plus(nat,M1,plus_plus(nat,N1,K1)) ).
tff(fact_55_add__eq__self__zero,axiom,
! [N1: nat,M1: nat] :
( ( plus_plus(nat,M1,N1) = M1 )
=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_56_liftn__0,axiom,
! [K1: nat,T: dB] : liftn(zero_zero(nat),T,K1) = T ).
tff(fact_57_substn__subst__0,axiom,
! [S: dB,T: dB] : substn(T,S,zero_zero(nat)) = subst(T,S,zero_zero(nat)) ).
tff(fact_58_less__not__refl,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_59_nat__neq__iff,axiom,
! [N: nat,M: nat] :
( ( M != N )
<=> ( ord_less(nat,M,N)
| ord_less(nat,N,M) ) ) ).
tff(fact_60_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_61_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_62_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_63_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_64_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = plus_plus(A,A2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_65_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_66_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_67_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_68_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_69_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_70_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_71_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_72_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_73_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_74_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_75_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_76_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_77_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_78_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_79_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_80_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_81_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_82_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_83_add__pos__pos,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,zero_zero(A),B)
=> ord_less(A,zero_zero(A),plus_plus(A,A1,B)) ) ) ) ).
tff(fact_84_add__neg__neg,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,plus_plus(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_85_less__imp__add__positive,axiom,
! [J: nat,I: nat] :
( ord_less(nat,I,J)
=> ? [K: nat] :
( ord_less(nat,zero_zero(nat),K)
& ( plus_plus(nat,I,K) = J ) ) ) ).
tff(fact_86_pos__add__strict,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B,C)
=> ord_less(A,B,plus_plus(A,A1,C)) ) ) ) ).
tff(fact_87_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(var(Nat)) = zero_zero(nat) ).
tff(fact_88_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_89_termination__basic__simps_I2_J,axiom,
! [Y: nat,Z: nat,X: nat] :
( ord_less(nat,X,Z)
=> ord_less(nat,X,plus_plus(nat,Y,Z)) ) ).
tff(fact_90_termination__basic__simps_I1_J,axiom,
! [Z: nat,Y: nat,X: nat] :
( ord_less(nat,X,Y)
=> ord_less(nat,X,plus_plus(nat,Y,Z)) ) ).
tff(fact_91_add__0__iff,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [A2: A,B1: A] :
( ( B1 = plus_plus(A,B1,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_93_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_94_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D)) = plus_plus(A,C,plus_plus(A,A1,D)) ) ).
tff(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A1,C),D) ) ).
tff(fact_96_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_97_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_98_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ).
%----Arities (12)
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___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
it(r) ).
tff(conj_1,hypothesis,
it(app(r,var(i))) ).
tff(conj_2,conjecture,
it(app(abs(r),var(i))) ).
%------------------------------------------------------------------------------