TPTP Problem File: LCL760_5.p
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%------------------------------------------------------------------------------
% File : LCL760_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 59
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_59 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 167 ( 66 unt; 47 typ; 0 def)
% Number of atoms : 211 ( 126 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 123 ( 32 ~; 8 |; 13 &)
% ( 19 <=>; 51 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 33 ( 18 >; 15 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 32 ( 32 usr; 12 con; 0-5 aty)
% Number of variables : 321 ( 283 !; 7 ?; 321 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:33
%------------------------------------------------------------------------------
%----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 (42)
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_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_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
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] : 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__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: fun(dB,fun(nat,dB)) ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: fun(dB,fun(dB,fun(nat,dB))) ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * nat ) > dB ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > list(B) ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_ia,type,
ia: nat ).
tff(sy_v_ja,type,
ja: nat ).
tff(sy_v_ra,type,
ra: dB ).
tff(sy_v_s,type,
s: dB ).
tff(sy_v_ss,type,
ss: list(dB) ).
%----Relevant facts (98)
tff(fact_0_subst__map,axiom,
! [Ia: nat,U1: dB,Ts: list(dB),T2: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,T2,Ts)),U1),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T2),U1),Ia),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,U1),Ia),Ts)) ).
tff(fact_1_Var__apps__eq__Var__apps__conv,axiom,
! [Ss1: list(dB),N2: nat,Rs1: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs1) = foldl(dB,dB,app,var(N2),Ss1) )
<=> ( ( M = N2 )
& ( Rs1 = Ss1 ) ) ) ).
tff(fact_2_subst__App,axiom,
! [K: nat,S: dB,U: dB,T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(dB,dB,aa(dB,fun(dB,dB),app,T),U)),S),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),S),K)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U),S),K)) ).
tff(fact_3_apps__eq__tail__conv,axiom,
! [S3: dB,Ts: list(dB),Ra: dB] :
( ( foldl(dB,dB,app,Ra,Ts) = foldl(dB,dB,app,S3,Ts) )
<=> ( Ra = S3 ) ) ).
tff(fact_4_map__ident,axiom,
! [A: $tType,X3: list(A)] : map(A,A,combi(A),X3) = X3 ).
tff(fact_5_dB_Osimps_I1_J,axiom,
! [Nat5: nat,Nat4: nat] :
( ( var(Nat4) = var(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_6_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB22: dB,DB12: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) = aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) )
<=> ( ( DB12 = DB13 )
& ( DB22 = DB23 ) ) ) ).
tff(fact_7_nat_Oinject,axiom,
! [Nat5: nat,Nat4: nat] :
( ( suc(Nat4) = suc(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_8_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun(C,B),A1: A,G: fun(A,fun(B,A))] : foldl(A,B,G,A1,map(C,B,F,Xs)) = foldl(A,C,combc(A,fun(C,B),fun(C,A),aa(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A))),aa(fun(fun(B,A),fun(fun(C,B),fun(C,A))),fun(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A)))),combb(fun(B,A),fun(fun(C,B),fun(C,A)),A),combb(B,A,C)),G),F),A1,Xs) ).
tff(fact_9_subst__eq,axiom,
! [U: dB,K: nat] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,var(K)),U),K) = U ).
tff(fact_10_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_11_Suc__inject,axiom,
! [Y1: nat,X2: nat] :
( ( suc(X2) = suc(Y1) )
=> ( X2 = Y1 ) ) ).
tff(fact_12_Suc__n__not__n,axiom,
! [N: nat] : suc(N) != N ).
tff(fact_13_n__not__Suc__n,axiom,
! [N: nat] : N != suc(N) ).
tff(fact_14_Suc__neq__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_15_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_16_nat_Osimps_I3_J,axiom,
! [Nat3: nat] : suc(Nat3) != zero_zero(nat) ).
tff(fact_17_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_18_nat_Osimps_I2_J,axiom,
! [Nat2: nat] : zero_zero(nat) != suc(Nat2) ).
tff(fact_19_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_20_dB_Osimps_I5_J,axiom,
! [Nat: nat,DB21: dB,DB11: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) != var(Nat) ).
tff(fact_21_dB_Osimps_I4_J,axiom,
! [DB21: dB,DB11: dB,Nat: nat] : var(Nat) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) ).
tff(fact_22_nat_Oexhaust,axiom,
! [Y1: nat] :
( ( Y1 != zero_zero(nat) )
=> ~ ! [Nat1: nat] : Y1 != suc(Nat1) ) ).
tff(fact_23_zero__induct,axiom,
! [K1: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,K1))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P1,suc(N3)))
=> pp(aa(nat,bool,P1,N3)) )
=> pp(aa(nat,bool,P1,zero_zero(nat))) ) ) ).
tff(fact_24_nat__induct,axiom,
! [N2: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P1,N3))
=> pp(aa(nat,bool,P1,suc(N3))) )
=> pp(aa(nat,bool,P1,N2)) ) ) ).
tff(fact_25_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M2: nat] : N = suc(M2) ) ).
tff(fact_26_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(var(Nat)) = zero_zero(nat) ).
tff(fact_27_lift__map,axiom,
! [Ia: nat,Ts: list(dB),T2: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,foldl(dB,dB,app,T2,Ts)),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),lift,T2),Ia),map(dB,dB,combc(dB,nat,dB,lift,Ia),Ts)) ).
tff(fact_28_dB_Osize_I4_J,axiom,
! [Nat: nat] : size_size(dB,var(Nat)) = zero_zero(nat) ).
tff(fact_29_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Ia: A,N2: nat,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,suc(N2),Ia) = semiri532925092at_aux(A,Inc,N2,aa(A,A,Inc,Ia)) ) ).
tff(fact_30_Beta,axiom,
! [Ss1: list(dB),S3: dB,Ra: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ra),S3),zero_zero(nat)),Ss1)))
=> ( pp(aa(dB,bool,it,S3))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(Ra)),S3),Ss1))) ) ) ).
tff(fact_31_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB4: dB] :
( ( abs(DB4) = abs(DB5) )
<=> ( DB4 = DB5 ) ) ).
tff(fact_32_lift_Osimps_I2_J,axiom,
! [K: nat,T: dB,S: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(dB,dB,aa(dB,fun(dB,dB),app,S),T)),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),lift,S),K)),aa(nat,dB,aa(dB,fun(nat,dB),lift,T),K)) ).
tff(fact_33_Lambda,axiom,
! [R2: dB] :
( pp(aa(dB,bool,it,R2))
=> pp(aa(dB,bool,it,abs(R2))) ) ).
tff(fact_34_lift__IT,axiom,
! [I1: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),I1))) ) ).
tff(fact_35_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss1: list(dB),S3: dB,Rs1: list(dB),Ra: dB] :
( ( foldl(dB,dB,app,abs(Ra),Rs1) = foldl(dB,dB,app,abs(S3),Ss1) )
<=> ( ( Ra = S3 )
& ( Rs1 = Ss1 ) ) ) ).
tff(fact_36_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB3: dB] : abs(DB3) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_37_dB_Osimps_I8_J,axiom,
! [DB3: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB3) ).
tff(fact_38_dB_Osimps_I7_J,axiom,
! [Nat: nat,DB3: dB] : abs(DB3) != var(Nat) ).
tff(fact_39_dB_Osimps_I6_J,axiom,
! [DB3: dB,Nat: nat] : var(Nat) != abs(DB3) ).
tff(fact_40_subst__lift,axiom,
! [S: dB,K: nat,T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),K)),S),K) = T ).
tff(fact_41_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Ia: A,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,zero_zero(nat),Ia) = Ia ) ).
tff(fact_42_Abs__App__neq__Var__apps,axiom,
! [Ss1: list(dB),N2: nat,T2: dB,S3: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S3)),T2) != foldl(dB,dB,app,var(N2),Ss1) ).
tff(fact_43_Var__apps__neq__Abs__apps,axiom,
! [Ss1: list(dB),Ra: dB,Ts: list(dB),N2: nat] : foldl(dB,dB,app,var(N2),Ts) != foldl(dB,dB,app,abs(Ra),Ss1) ).
tff(fact_44_IT_OVar,axiom,
! [N2: nat,Rs1: list(dB)] :
( listsp(dB,it,Rs1)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(N2),Rs1))) ) ).
tff(fact_45_lifts__IT,axiom,
! [Ts: list(dB)] :
( listsp(dB,it,Ts)
=> listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),Ts)) ) ).
tff(fact_46_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_47_liftn__lift,axiom,
! [K: nat,T: dB,N: nat] : liftn(suc(N),T,K) = aa(nat,dB,aa(dB,fun(nat,dB),lift,liftn(N,T,K)),K) ).
tff(fact_48_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_49_beta,axiom,
! [T: dB,S: dB] : beta(aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S)),T),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S),T),zero_zero(nat))) ).
tff(fact_50_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,A1: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A1) )
<=> ( B3 = C2 ) ) ) ).
tff(fact_51_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,B3: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C2) )
<=> ( B3 = C2 ) ) ) ).
tff(fact_52_nat__add__right__cancel,axiom,
! [N2: 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),N2),K1) )
<=> ( M = N2 ) ) ).
tff(fact_53_nat__add__left__cancel,axiom,
! [N2: 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),N2) )
<=> ( M = N2 ) ) ).
tff(fact_54_listsp__conj__eq,axiom,
! [A: $tType,B2: fun(A,bool),A3: fun(A,bool),X3: list(A)] :
( listsp(A,combs(A,bool,bool,aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),A3),B2),X3)
<=> ( listsp(A,A3,X3)
& listsp(A,B2,X3) ) ) ).
tff(fact_55_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_56_add__is__0,axiom,
! [N2: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N2 = zero_zero(nat) ) ) ) ).
tff(fact_57_add__Suc,axiom,
! [N: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),suc(M1)),N) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N)) ).
tff(fact_58_add__Suc__right,axiom,
! [N: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),suc(N)) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N)) ).
tff(fact_59_appR,axiom,
! [U: dB,T: dB,S: dB] :
( beta(S,T)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,U),S),aa(dB,dB,aa(dB,fun(dB,dB),app,U),T)) ) ).
tff(fact_60_appL,axiom,
! [U: dB,T: dB,S: dB] :
( beta(S,T)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,S),U),aa(dB,dB,aa(dB,fun(dB,dB),app,T),U)) ) ).
tff(fact_61_beta__cases_I1_J,axiom,
! [T: dB,I1: nat] : ~ beta(var(I1),T) ).
tff(fact_62_abs,axiom,
! [T: dB,S: dB] :
( beta(S,T)
=> beta(abs(S),abs(T)) ) ).
tff(fact_63_liftn_Osimps_I2_J,axiom,
! [K: nat,T: dB,S: dB,N: nat] : liftn(N,aa(dB,dB,aa(dB,fun(dB,dB),app,S),T),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N,S,K)),liftn(N,T,K)) ).
tff(fact_64_dB_Osize_I5_J,axiom,
! [DB2: dB,DB1: dB] : size_size(dB,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),size_size(dB,DB1)),size_size(dB,DB2))),suc(zero_zero(nat))) ).
tff(fact_65_dB_Osize_I6_J,axiom,
! [DB: dB] : size_size(dB,abs(DB)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_size(dB,DB)),suc(zero_zero(nat))) ).
tff(fact_66_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A2) )
=> ( B1 = C1 ) ) ) ).
tff(fact_67_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B1: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_68_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_69_nat__add__assoc,axiom,
! [K: nat,N: 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),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) ).
tff(fact_70_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B1: 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),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C1)) ) ).
tff(fact_71_nat__add__left__commute,axiom,
! [Z: nat,Y1: nat,X2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y1),Z)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),Z)) ).
tff(fact_72_nat__add__commute,axiom,
! [N: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M1) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_74_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_75_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_76_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_77_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_78_add__eq__self__zero,axiom,
! [N: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N) = M1 )
=> ( N = zero_zero(nat) ) ) ).
tff(fact_79_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_80_plus__nat_Oadd__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).
tff(fact_81_add__Suc__shift,axiom,
! [N: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),suc(M1)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),suc(N)) ).
tff(fact_82_foldl__assoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [Zs: list(A),Y: A,X: A] : foldl(A,A,plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),Zs) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),foldl(A,A,plus_plus(A),Y,Zs)) ) ).
tff(fact_83_subst__preserves__beta,axiom,
! [I1: nat,T: dB,S: dB,R2: dB] :
( beta(R2,S)
=> beta(aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R2),T),I1),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S),T),I1)) ) ).
tff(fact_84_lift__preserves__beta,axiom,
! [I1: nat,S: dB,R2: dB] :
( beta(R2,S)
=> beta(aa(nat,dB,aa(dB,fun(nat,dB),lift,R2),I1),aa(nat,dB,aa(dB,fun(nat,dB),lift,S),I1)) ) ).
tff(fact_85_liftn__0,axiom,
! [K: nat,T: dB] : liftn(zero_zero(nat),T,K) = T ).
tff(fact_86_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_87_add__is__1,axiom,
! [N2: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = suc(zero_zero(nat)) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N2 = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N2 = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_88_one__is__add,axiom,
! [N2: nat,M: nat] :
( ( suc(zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N2 = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N2 = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_89_beta__cases_I2_J,axiom,
! [S: dB,R2: dB] :
( beta(abs(R2),S)
=> ~ ! [T1: dB] :
( ( S = abs(T1) )
=> ~ beta(R2,T1) ) ) ).
tff(fact_90_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_91_apps__preserves__beta,axiom,
! [Ss1: list(dB),S3: dB,Ra: dB] :
( beta(Ra,S3)
=> beta(foldl(dB,dB,app,Ra,Ss1),foldl(dB,dB,app,S3,Ss1)) ) ).
tff(fact_92_IT_Osimps,axiom,
! [A1: dB] :
( pp(aa(dB,bool,it,A1))
<=> ( ? [Rs: list(dB),N1: nat] :
( ( A1 = foldl(dB,dB,app,var(N1),Rs) )
& listsp(dB,it,Rs) )
| ? [R1: dB] :
( ( A1 = abs(R1) )
& pp(aa(dB,bool,it,R1)) )
| ? [R1: dB,S2: dB,Ss: list(dB)] :
( ( A1 = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R1)),S2),Ss) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R1),S2),zero_zero(nat)),Ss)))
& pp(aa(dB,bool,it,S2)) ) ) ) ).
tff(fact_93_substn__subst__n,axiom,
! [N: nat,S: dB,T: dB] : substn(T,S,N) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),liftn(N,S,zero_zero(nat))),N) ).
tff(fact_94_beta__cases_I3_J,axiom,
! [U: dB,T: dB,S: dB] :
( beta(aa(dB,dB,aa(dB,fun(dB,dB),app,S),T),U)
=> ( ! [S1: dB] :
( ( U = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T),zero_zero(nat)) )
=> ( S != abs(S1) ) )
=> ( ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,T1),T) )
=> ~ beta(S,T1) )
=> ~ ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,S),T1) )
=> ~ beta(T,T1) ) ) ) ) ).
tff(fact_95_substn_Osimps_I2_J,axiom,
! [K: nat,S: dB,U: dB,T: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T),U),S,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T,S,K)),substn(U,S,K)) ).
tff(fact_96_substn__subst__0,axiom,
! [S: dB,T: dB] : substn(T,S,zero_zero(nat)) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),S),zero_zero(nat)) ).
tff(fact_97_subst__Abs,axiom,
! [K: nat,S: dB,T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,abs(T)),S),K) = abs(aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),aa(nat,dB,aa(dB,fun(nat,dB),lift,S),zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat)))) ).
%----Arities (8)
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___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (9)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
%----Conjectures (5)
tff(conj_0,hypothesis,
pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,ra),s),zero_zero(nat)),ss))) ).
tff(conj_1,hypothesis,
! [I: nat,J: nat] : pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,ra),var(suc(I))),suc(J))),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,s),var(I)),J)),zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,var(I)),J),ss)))) ).
tff(conj_2,hypothesis,
pp(aa(dB,bool,it,s)) ).
tff(conj_3,hypothesis,
! [I: nat,J: nat] : pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,s),var(I)),J))) ).
tff(conj_4,conjecture,
pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,ra),var(suc(ia))),suc(ja))),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,s),var(ia)),ja)),zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,var(ia)),ja),ss)))) ).
%------------------------------------------------------------------------------