TPTP Problem File: LCL774_5.p
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%------------------------------------------------------------------------------
% File : LCL774_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 79
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_79 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 147 ( 60 unt; 37 typ; 0 def)
% Number of atoms : 193 ( 124 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 107 ( 24 ~; 4 |; 15 &)
% ( 21 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 36 ( 17 >; 19 *; 0 +; 0 <<)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 6 con; 0-5 aty)
% Number of variables : 288 ( 263 !; 7 ?; 288 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:21
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (32)
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_Groups_Osemigroup__add,type,
semigroup_add:
!>[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_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_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_InductTermi_OIT,type,
it: fun(dB,bool) ).
tff(sy_c_Lambda_Obeta,type,
beta: ( dB * dB ) > $o ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
tff(sy_c_Lambda_OdB_OApp,type,
app: fun(dB,fun(dB,dB)) ).
tff(sy_c_Lambda_OdB_OVar,type,
var: nat > dB ).
tff(sy_c_Lambda_OdB_OdB__case,type,
dB_case:
!>[T2: $tType] : ( ( fun(nat,T2) * fun(dB,fun(dB,T2)) * fun(dB,T2) * dB ) > T2 ) ).
tff(sy_c_Lambda_OdB_OdB__rec,type,
dB_rec:
!>[T2: $tType] : ( ( fun(nat,T2) * fun(dB,fun(dB,fun(T2,fun(T2,T2)))) * fun(dB,fun(T2,T2)) * dB ) > T2 ) ).
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_List_Ofoldl,type,
foldl:
!>[B1: $tType,A: $tType] : ( ( fun(B1,fun(A,B1)) * B1 * list(A) ) > B1 ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
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 (98)
tff(fact_0_Var__IT,axiom,
! [N1: nat] : pp(aa(dB,bool,it,var(N1))) ).
tff(fact_1_dB_Osimps_I1_J,axiom,
! [Nat5: nat,Nat4: nat] :
( ( var(Nat4) = var(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_2_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB21: dB,DB11: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) = aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) )
<=> ( ( DB11 = DB13 )
& ( DB21 = DB23 ) ) ) ).
tff(fact_3_dB_Osimps_I4_J,axiom,
! [DB22: dB,DB12: dB,Nat3: nat] : var(Nat3) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) ).
tff(fact_4_dB_Osimps_I5_J,axiom,
! [Nat3: nat,DB22: dB,DB12: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) != var(Nat3) ).
tff(fact_5_subst__Var__IT,axiom,
! [J: nat,I: nat,R1: dB] :
( pp(aa(dB,bool,it,R1))
=> pp(aa(dB,bool,it,subst(R1,var(I),J))) ) ).
tff(fact_6_lift__IT,axiom,
! [I: nat,T1: dB] :
( pp(aa(dB,bool,it,T1))
=> pp(aa(dB,bool,it,lift(T1,I))) ) ).
tff(fact_7_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat4: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat4)) = aa(nat,A,F1,Nat4) ).
tff(fact_8_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),N: nat,Rs1: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs1) = foldl(dB,dB,app,var(N),Ss) )
<=> ( ( M = N )
& ( Rs1 = Ss ) ) ) ).
tff(fact_9_dB_Osimps_I11_J,axiom,
! [A: $tType,DB21: dB,DB11: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21)) = aa(dB,A,aa(dB,fun(dB,A),F2,DB11),DB21) ).
tff(fact_10_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U: dB,T1: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T1,S1,K)),substn(U,S1,K)) ).
tff(fact_11_IT_OVar,axiom,
! [N: nat,Rs1: list(dB)] :
( listsp(dB,it,Rs1)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(N),Rs1))) ) ).
tff(fact_12_liftn_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB,N1: nat] : liftn(N1,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N1,S1,K)),liftn(N1,T1,K)) ).
tff(fact_13_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),R: dB] :
( ( foldl(dB,dB,app,R,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( R = S ) ) ).
tff(fact_14_subst__App,axiom,
! [K: nat,S1: dB,U: dB,T1: dB] : subst(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,subst(T1,S1,K)),subst(U,S1,K)) ).
tff(fact_15_lift_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB] : lift(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,lift(S1,K)),lift(T1,K)) ).
tff(fact_16_subst__lift,axiom,
! [S1: dB,K: nat,T1: dB] : subst(lift(T1,K),S1,K) = T1 ).
tff(fact_17_subst__eq,axiom,
! [U: dB,K: nat] : subst(var(K),U,K) = U ).
tff(fact_18_substn__subst__n,axiom,
! [N1: nat,S1: dB,T1: dB] : substn(T1,S1,N1) = subst(T1,liftn(N1,S1,zero_zero(nat)),N1) ).
tff(fact_19_Var__eq__apps__conv,axiom,
! [Ss: list(dB),S: dB,M: nat] :
( ( var(M) = foldl(dB,dB,app,S,Ss) )
<=> ( ( var(M) = S )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_20_Var__apps__neq__Abs__apps,axiom,
! [Ss: list(dB),R: dB,Ts: list(dB),N: nat] : foldl(dB,dB,app,var(N),Ts) != foldl(dB,dB,app,abs(R),Ss) ).
tff(fact_21_Abs__App__neq__Var__apps,axiom,
! [Ss: list(dB),N: nat,Ta: dB,S: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S)),Ta) != foldl(dB,dB,app,var(N),Ss) ).
tff(fact_22_liftn__lift,axiom,
! [K: nat,T1: dB,N1: nat] : liftn(suc(N1),T1,K) = lift(liftn(N1,T1,K),K) ).
tff(fact_23_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs1: list(dB),R: dB] :
( ( foldl(dB,dB,app,abs(R),Rs1) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R = S )
& ( Rs1 = Ss ) ) ) ).
tff(fact_24_substn__subst__0,axiom,
! [S1: dB,T1: dB] : substn(T1,S1,zero_zero(nat)) = subst(T1,S1,zero_zero(nat)) ).
tff(fact_25_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat4: 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(Nat4)) = aa(nat,A,F1,Nat4) ).
tff(fact_26_dB_Orecs_I2_J,axiom,
! [A: $tType,DB21: dB,DB11: 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,aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21)) = 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,DB11),DB21),dB_rec(A,F1,F2,F3,DB11)),dB_rec(A,F1,F2,F3,DB21)) ).
tff(fact_27_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB3: dB] :
( ( abs(DB3) = abs(DB5) )
<=> ( DB3 = DB5 ) ) ).
tff(fact_28_Lambda,axiom,
! [R1: dB] :
( pp(aa(dB,bool,it,R1))
=> pp(aa(dB,bool,it,abs(R1))) ) ).
tff(fact_29_apps__eq__Abs__conv,axiom,
! [R: dB,Ss: list(dB),S: dB] :
( ( foldl(dB,dB,app,S,Ss) = abs(R) )
<=> ( ( S = abs(R) )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_30_Abs__eq__apps__conv,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( ( abs(R) = foldl(dB,dB,app,S,Ss) )
<=> ( ( abs(R) = S )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_31_Beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R,S,zero_zero(nat)),Ss)))
=> ( pp(aa(dB,bool,it,S))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R)),S),Ss))) ) ) ).
tff(fact_32_dB_Orecs_I3_J,axiom,
! [A: $tType,DB3: 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(DB3)) = aa(A,A,aa(dB,fun(A,A),F3,DB3),dB_rec(A,F1,F2,F3,DB3)) ).
tff(fact_33_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB4: dB] : abs(DB4) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_34_dB_Osimps_I8_J,axiom,
! [DB4: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB4) ).
tff(fact_35_dB_Osimps_I7_J,axiom,
! [Nat3: nat,DB4: dB] : abs(DB4) != var(Nat3) ).
tff(fact_36_dB_Osimps_I6_J,axiom,
! [DB4: dB,Nat3: nat] : var(Nat3) != abs(DB4) ).
tff(fact_37_liftn__0,axiom,
! [K: nat,T1: dB] : liftn(zero_zero(nat),T1,K) = T1 ).
tff(fact_38_dB_Osimps_I12_J,axiom,
! [A: $tType,DB3: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB3)) = aa(dB,A,F3,DB3) ).
tff(fact_39_listsp_ONil,axiom,
! [A: $tType,A3: fun(A,bool)] : listsp(A,A3,nil(A)) ).
tff(fact_40_IT_Osimps,axiom,
! [A1: dB] :
( pp(aa(dB,bool,it,A1))
<=> ( ? [Rs: list(dB),N3: nat] :
( ( A1 = foldl(dB,dB,app,var(N3),Rs) )
& listsp(dB,it,Rs) )
| ? [R2: dB] :
( ( A1 = abs(R2) )
& pp(aa(dB,bool,it,R2)) )
| ? [R2: dB,S2: dB,Ss1: list(dB)] :
( ( A1 = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R2)),S2),Ss1) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R2,S2,zero_zero(nat)),Ss1)))
& pp(aa(dB,bool,it,S2)) ) ) ) ).
tff(fact_41_nat_Oinject,axiom,
! [Nat5: nat,Nat4: nat] :
( ( suc(Nat4) = suc(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_42_dB_Osize_I1_J,axiom,
! [Nat3: nat] : dB_size(var(Nat3)) = zero_zero(nat) ).
tff(fact_43_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_44_n__not__Suc__n,axiom,
! [N1: nat] : N1 != suc(N1) ).
tff(fact_45_Suc__n__not__n,axiom,
! [N1: nat] : suc(N1) != N1 ).
tff(fact_46_Suc__inject,axiom,
! [Y: nat,X1: nat] :
( ( suc(X1) = suc(Y) )
=> ( X1 = Y ) ) ).
tff(fact_47_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_48_nat_Osimps_I2_J,axiom,
! [Nat2: nat] : zero_zero(nat) != suc(Nat2) ).
tff(fact_49_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_50_nat_Osimps_I3_J,axiom,
! [Nat1: nat] : suc(Nat1) != zero_zero(nat) ).
tff(fact_51_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_52_Suc__neq__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_53_foldl__Nil,axiom,
! [B1: $tType,A: $tType,A1: A,F: fun(A,fun(B1,A))] : foldl(A,B1,F,A1,nil(B1)) = A1 ).
tff(fact_54_not0__implies__Suc,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> ? [M2: nat] : N1 = suc(M2) ) ).
tff(fact_55_nat__induct,axiom,
! [N: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N2: nat] :
( pp(aa(nat,bool,P,N2))
=> pp(aa(nat,bool,P,suc(N2))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
tff(fact_56_zero__induct,axiom,
! [K1: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,K1))
=> ( ! [N2: nat] :
( pp(aa(nat,bool,P,suc(N2)))
=> pp(aa(nat,bool,P,N2)) )
=> pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).
tff(fact_57_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat: nat] : Y != suc(Nat) ) ).
tff(fact_58_dB_Osize_I3_J,axiom,
! [DB: dB] : dB_size(abs(DB)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),dB_size(DB)),suc(zero_zero(nat))) ).
tff(fact_59_dB_Osize_I2_J,axiom,
! [DB2: dB,DB1: dB] : dB_size(aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),dB_size(DB1)),dB_size(DB2))),suc(zero_zero(nat))) ).
tff(fact_60_beta,axiom,
! [T1: dB,S1: dB] : beta(aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T1),subst(S1,T1,zero_zero(nat))) ).
tff(fact_61_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A1: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_62_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_63_nat__add__right__cancel,axiom,
! [N: nat,K1: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K1) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K1) )
<=> ( M = N ) ) ).
tff(fact_64_nat__add__left__cancel,axiom,
! [N: nat,M: nat,K1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K1),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K1),N) )
<=> ( M = N ) ) ).
tff(fact_65_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_66_add__is__0,axiom,
! [N: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_67_add__Suc__right,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),suc(N1)) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)) ).
tff(fact_68_add__Suc,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),suc(M1)),N1) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)) ).
tff(fact_69_appL,axiom,
! [U: dB,T1: dB,S1: dB] :
( beta(S1,T1)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U),aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U)) ) ).
tff(fact_70_appR,axiom,
! [U: dB,T1: dB,S1: dB] :
( beta(S1,T1)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,U),S1),aa(dB,dB,aa(dB,fun(dB,dB),app,U),T1)) ) ).
tff(fact_71_beta__cases_I1_J,axiom,
! [T1: dB,I: nat] : ~ beta(var(I),T1) ).
tff(fact_72_abs,axiom,
! [T1: dB,S1: dB] :
( beta(S1,T1)
=> beta(abs(S1),abs(T1)) ) ).
tff(fact_73_ext,axiom,
! [B1: $tType,A: $tType,G: fun(A,B1),F: fun(A,B1)] :
( ! [X2: A] : aa(A,B1,F,X2) = aa(A,B1,G,X2)
=> ( F = G ) ) ).
tff(fact_74_foldl__assoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [Zs: list(A),Y1: A,X: A] : foldl(A,A,plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y1),Zs) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),foldl(A,A,plus_plus(A),Y1,Zs)) ) ).
tff(fact_75_subst__preserves__beta,axiom,
! [I: nat,T1: dB,S1: dB,R1: dB] :
( beta(R1,S1)
=> beta(subst(R1,T1,I),subst(S1,T1,I)) ) ).
tff(fact_76_lift__preserves__beta,axiom,
! [I: nat,S1: dB,R1: dB] :
( beta(R1,S1)
=> beta(lift(R1,I),lift(S1,I)) ) ).
tff(fact_77_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A2: A,B: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C),A2) )
=> ( B = C ) ) ) ).
tff(fact_78_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C) )
=> ( B = C ) ) ) ).
tff(fact_79_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C) )
=> ( B = C ) ) ) ).
tff(fact_80_nat__add__assoc,axiom,
! [K: nat,N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),K)) ).
tff(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B),C)) ) ).
tff(fact_82_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),Z)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X1),Z)) ).
tff(fact_83_nat__add__commute,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),M1) ).
tff(fact_84_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).
tff(fact_85_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).
tff(fact_86_plus__nat_Oadd__0,axiom,
! [N1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N1) = N1 ).
tff(fact_87_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).
tff(fact_88_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).
tff(fact_89_Nat_Oadd__0__right,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),zero_zero(nat)) = M1 ).
tff(fact_90_add__Suc__shift,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),suc(M1)),N1) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),suc(N1)) ).
tff(fact_91_add__eq__self__zero,axiom,
! [N1: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1) = M1 )
=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_92_foldl__absorb0,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Zs: list(A),X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),foldl(A,A,plus_plus(A),zero_zero(A),Zs)) = foldl(A,A,plus_plus(A),X,Zs) ) ).
tff(fact_93_one__is__add,axiom,
! [N: nat,M: nat] :
( ( suc(zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_94_add__is__1,axiom,
! [N: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = suc(zero_zero(nat)) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_95_beta__cases_I2_J,axiom,
! [S1: dB,R1: dB] :
( beta(abs(R1),S1)
=> ~ ! [T: dB] :
( ( S1 = abs(T) )
=> ~ beta(R1,T) ) ) ).
tff(fact_96_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_97_apps__preserves__beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( beta(R,S)
=> beta(foldl(dB,dB,app,R,Ss),foldl(dB,dB,app,S,Ss)) ) ).
%----Arities (7)
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__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add(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,
pp(aa(dB,bool,it,r)) ).
tff(conj_1,hypothesis,
pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun(dB,dB),app,r),var(i)))) ).
tff(conj_2,conjecture,
pp(aa(dB,bool,it,var(i))) ).
%------------------------------------------------------------------------------