TPTP Problem File: LCL770_5.p
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%------------------------------------------------------------------------------
% File : LCL770_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 75
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_75 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v6.4.0
% Syntax : Number of formulae : 166 ( 60 unt; 48 typ; 0 def)
% Number of atoms : 228 ( 127 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 161 ( 51 ~; 17 |; 26 &)
% ( 23 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 53 ( 27 >; 26 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-4 aty)
% Number of functors : 38 ( 38 usr; 12 con; 0-5 aty)
% Number of variables : 431 ( 376 !; 19 ?; 431 :)
% ( 36 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:09
%------------------------------------------------------------------------------
%----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 (43)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[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_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: fun(dB,fun(dB,bool)) ).
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__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_ListOrder_Ostep1,type,
step1:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T2: $tType,A: $tType] : ( ( T2 * fun(A,fun(list(A),T2)) * list(A) ) > T2 ) ).
tff(sy_c_List_Olist_Olist__rec,type,
list_rec:
!>[T2: $tType,A: $tType] : ( ( T2 * fun(A,fun(list(A),fun(T2,T2))) * list(A) ) > T2 ) ).
tff(sy_c_List_Olist_Olist__size,type,
list_size:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
tff(sy_c_List_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
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) > fun(list(A),list(B)) ) ).
tff(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list(A) * fun(nat,bool) ) > list(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_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_i,type,
i: nat ).
tff(sy_v_rs,type,
rs: list(dB) ).
%----Relevant facts (99)
tff(fact_0_Var__IT,axiom,
! [N3: nat] : pp(aa(dB,bool,it,var(N3))) ).
tff(fact_1_listsp__conj__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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),A1),B1),X3)
<=> ( listsp(A,A1,X3)
& listsp(A,B1,X3) ) ) ).
tff(fact_2_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_3_dB_Osimps_I2_J,axiom,
! [DB24: dB,DB14: 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,DB14),DB24) )
<=> ( ( DB12 = DB14 )
& ( DB22 = DB24 ) ) ) ).
tff(fact_4_dB_Osimps_I4_J,axiom,
! [DB23: dB,DB13: dB,Nat1: nat] : var(Nat1) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) ).
tff(fact_5_dB_Osimps_I5_J,axiom,
! [Nat1: nat,DB23: dB,DB13: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) != var(Nat1) ).
tff(fact_6_IT_OVar,axiom,
! [N1: nat,Rs: list(dB)] :
( listsp(dB,it,Rs)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(N1),Rs))) ) ).
tff(fact_7_subst__Var__IT,axiom,
! [J: nat,I: nat,R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R3),var(I)),J))) ) ).
tff(fact_8_lift__IT,axiom,
! [I: nat,T1: dB] :
( pp(aa(dB,bool,it,T1))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,T1),I))) ) ).
tff(fact_9_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat2: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_10_Var__apps__eq__Var__apps__conv,axiom,
! [Ss1: list(dB),N1: nat,Rs: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs) = foldl(dB,dB,app,var(N1),Ss1) )
<=> ( ( M = N1 )
& ( Rs = Ss1 ) ) ) ).
tff(fact_11_dB_Osimps_I11_J,axiom,
! [A: $tType,DB22: dB,DB12: 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,DB12),DB22)) = aa(dB,A,aa(dB,fun(dB,A),F2,DB12),DB22) ).
tff(fact_12_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U1: dB,T1: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T1,S1,K)),substn(U1,S1,K)) ).
tff(fact_13_apps__eq__tail__conv,axiom,
! [S: dB,Ts1: list(dB),R1: dB] :
( ( foldl(dB,dB,app,R1,Ts1) = foldl(dB,dB,app,S,Ts1) )
<=> ( R1 = S ) ) ).
tff(fact_14_subst__App,axiom,
! [K: nat,S1: dB,U1: dB,T1: 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,T1),U1)),S1),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,T1),S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U1),S1),K)) ).
tff(fact_15_lift_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1)),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),lift,S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),lift,T1),K)) ).
tff(fact_16_subst__lift,axiom,
! [S1: dB,K: nat,T1: 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,T1),K)),S1),K) = T1 ).
tff(fact_17_subst__eq,axiom,
! [U1: dB,K: nat] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,var(K)),U1),K) = U1 ).
tff(fact_18_foldl__fun__comm,axiom,
! [B: $tType,A: $tType,X1: A,Xs: list(A),S: B,F: fun(B,fun(A,B))] :
( ! [X2: A,Y1: A,S2: B] : aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),X2)),Y1) = aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),Y1)),X2)
=> ( aa(A,B,aa(B,fun(A,B),F,foldl(B,A,F,S,Xs)),X1) = foldl(B,A,F,aa(A,B,aa(B,fun(A,B),F,S),X1),Xs) ) ) ).
tff(fact_19_Var__eq__apps__conv,axiom,
! [Ss1: list(dB),S: dB,M: nat] :
( ( var(M) = foldl(dB,dB,app,S,Ss1) )
<=> ( ( var(M) = S )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_20_lift__map,axiom,
! [Ia: nat,Ts1: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,foldl(dB,dB,app,Ta,Ts1)),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),lift,Ta),Ia),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,lift,Ia)),Ts1)) ).
tff(fact_21_subst__map,axiom,
! [Ia: nat,U2: dB,Ts1: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,Ta,Ts1)),U2),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),U2),Ia),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,U2),Ia)),Ts1)) ).
tff(fact_22_Var__apps__neq__Abs__apps,axiom,
! [Ss1: list(dB),R1: dB,Ts1: list(dB),N1: nat] : foldl(dB,dB,app,var(N1),Ts1) != foldl(dB,dB,app,abs(R1),Ss1) ).
tff(fact_23_Abs__App__neq__Var__apps,axiom,
! [Ss1: list(dB),N1: nat,Ta: dB,S: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S)),Ta) != foldl(dB,dB,app,var(N1),Ss1) ).
tff(fact_24_liftn_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB,N3: nat] : liftn(N3,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N3,S1,K)),liftn(N3,T1,K)) ).
tff(fact_25_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss1: list(dB),S: dB,Rs: list(dB),R1: dB] :
( ( foldl(dB,dB,app,abs(R1),Rs) = foldl(dB,dB,app,abs(S),Ss1) )
<=> ( ( R1 = S )
& ( Rs = Ss1 ) ) ) ).
tff(fact_26_substn__subst__0,axiom,
! [S1: dB,T1: dB] : substn(T1,S1,zero_zero(nat)) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T1),S1),zero_zero(nat)) ).
tff(fact_27_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB3: dB] :
( ( abs(DB3) = abs(DB5) )
<=> ( DB3 = DB5 ) ) ).
tff(fact_28_map__ident,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),map(A,A,combi(A)),X3) = X3 ).
tff(fact_29_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( nil(A) = aa(list(B),list(A),map(B,A,F),Xs) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_30_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),nil(B)) = nil(A) ).
tff(fact_31_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),map(B,A,F),Xs) = nil(A) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_32_listsp_ONil,axiom,
! [A: $tType,A1: fun(A,bool)] : listsp(A,A1,nil(A)) ).
tff(fact_33_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_34_apps__eq__Abs__conv,axiom,
! [R1: dB,Ss1: list(dB),S: dB] :
( ( foldl(dB,dB,app,S,Ss1) = abs(R1) )
<=> ( ( S = abs(R1) )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_35_Abs__eq__apps__conv,axiom,
! [Ss1: list(dB),S: dB,R1: dB] :
( ( abs(R1) = foldl(dB,dB,app,S,Ss1) )
<=> ( ( abs(R1) = S )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_36_substn__subst__n,axiom,
! [N3: nat,S1: dB,T1: dB] : substn(T1,S1,N3) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T1),liftn(N3,S1,zero_zero(nat))),N3) ).
tff(fact_37_Beta,axiom,
! [Ss1: list(dB),S: dB,R1: 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,R1),S),zero_zero(nat)),Ss1)))
=> ( 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(R1)),S),Ss1))) ) ) ).
tff(fact_38_liftn__0,axiom,
! [K: nat,T1: dB] : liftn(zero_zero(nat),T1,K) = T1 ).
tff(fact_39_foldl__Nil,axiom,
! [B: $tType,A: $tType,A3: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A3,nil(B)) = A3 ).
tff(fact_40_dB_Osimps_I9_J,axiom,
! [DB21: dB,DB11: dB,DB4: dB] : abs(DB4) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) ).
tff(fact_41_dB_Osimps_I8_J,axiom,
! [DB4: dB,DB21: dB,DB11: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) != abs(DB4) ).
tff(fact_42_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB4: dB] : abs(DB4) != var(Nat1) ).
tff(fact_43_dB_Osimps_I6_J,axiom,
! [DB4: dB,Nat1: nat] : var(Nat1) != abs(DB4) ).
tff(fact_44_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun(C,B),A3: A,G: fun(A,fun(B,A))] : foldl(A,B,G,A3,aa(list(C),list(B),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),A3,Xs) ).
tff(fact_45_lifts__IT,axiom,
! [Ts1: list(dB)] :
( listsp(dB,it,Ts1)
=> listsp(dB,it,aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat))),Ts1)) ) ).
tff(fact_46_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_47_IT_Osimps,axiom,
! [A3: dB] :
( pp(aa(dB,bool,it,A3))
<=> ( ? [Rs2: list(dB),N2: nat] :
( ( A3 = foldl(dB,dB,app,var(N2),Rs2) )
& listsp(dB,it,Rs2) )
| ? [R4: dB] :
( ( A3 = abs(R4) )
& pp(aa(dB,bool,it,R4)) )
| ? [R4: dB,S3: dB,Ss2: list(dB)] :
( ( A3 = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R4)),S3),Ss2) )
& 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,R4),S3),zero_zero(nat)),Ss2)))
& pp(aa(dB,bool,it,S3)) ) ) ) ).
tff(fact_48_dB_Osize_I1_J,axiom,
! [Nat1: nat] : dB_size(var(Nat1)) = zero_zero(nat) ).
tff(fact_49_beta,axiom,
! [T1: dB,S1: dB] : pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T1)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T1),zero_zero(nat)))) ).
tff(fact_50_dB_Osize_I4_J,axiom,
! [Nat1: nat] : size_size(dB,var(Nat1)) = zero_zero(nat) ).
tff(fact_51_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_52_appR,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,U1),S1)),aa(dB,dB,aa(dB,fun(dB,dB),app,U1),T1))) ) ).
tff(fact_53_appL,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U1)),aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1))) ) ).
tff(fact_54_beta__cases_I1_J,axiom,
! [T1: dB,I: nat] : ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,var(I)),T1)) ).
tff(fact_55_abs,axiom,
! [T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(S1)),abs(T1))) ) ).
tff(fact_56_subst__preserves__beta,axiom,
! [I: nat,T1: dB,S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),S1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R3),T1),I)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T1),I))) ) ).
tff(fact_57_lift__preserves__beta,axiom,
! [I: nat,S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),S1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(nat,dB,aa(dB,fun(nat,dB),lift,R3),I)),aa(nat,dB,aa(dB,fun(nat,dB),lift,S1),I))) ) ).
tff(fact_58_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_59_beta__cases_I2_J,axiom,
! [S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(R3)),S1))
=> ~ ! [T: dB] :
( ( S1 = abs(T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),T)) ) ) ).
tff(fact_60_apps__preserves__beta,axiom,
! [Ss1: list(dB),S: dB,R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R1),S))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Ss1)),foldl(dB,dB,app,S,Ss1))) ) ).
tff(fact_61_beta__cases_I3_J,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1)),U1))
=> ( ! [S2: dB] :
( ( U1 = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S2),T1),zero_zero(nat)) )
=> ( S1 != abs(S2) ) )
=> ( ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,T),T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T)) )
=> ~ ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,T1),T)) ) ) ) ) ).
tff(fact_62_list_Osize_I1_J,axiom,
! [A: $tType,Fa: fun(A,nat)] : list_size(A,Fa,nil(A)) = zero_zero(nat) ).
tff(fact_63_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_64_apps__preserves__betas,axiom,
! [R1: dB,Ss1: list(dB),Rs: list(dB)] :
( step1(dB,beta,Rs,Ss1)
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Rs)),foldl(dB,dB,app,R1,Ss1))) ) ).
tff(fact_65_dB_Oexhaust,axiom,
! [Y: dB] :
( ! [Nat: nat] : Y != var(Nat)
=> ( ! [DB1: dB,DB2: dB] : Y != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2)
=> ~ ! [DB: dB] : Y != abs(DB) ) ) ).
tff(fact_66_head__Var__reduction,axiom,
! [V: dB,Rs: list(dB),N1: nat] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,var(N1),Rs)),V))
=> ? [Ss: list(dB)] :
( step1(dB,beta,Rs,Ss)
& ( V = foldl(dB,dB,app,var(N1),Ss) ) ) ) ).
tff(fact_67_not__Nil__step1,axiom,
! [A: $tType,Xs: list(A),R1: fun(A,fun(A,bool))] : ~ step1(A,R1,nil(A),Xs) ).
tff(fact_68_not__step1__Nil,axiom,
! [A: $tType,Xs: list(A),R1: fun(A,fun(A,bool))] : ~ step1(A,R1,Xs,nil(A)) ).
tff(fact_69_apps__betasE,axiom,
! [S: dB,Rs: list(dB),R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Rs)),S))
=> ( ! [R2: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R1),R2))
=> ( S != foldl(dB,dB,app,R2,Rs) ) )
=> ( ! [Rs1: list(dB)] :
( step1(dB,beta,Rs,Rs1)
=> ( S != foldl(dB,dB,app,R1,Rs1) ) )
=> ~ ! [T: dB] :
( ( R1 = abs(T) )
=> ! [U: dB,Us: list(dB)] :
( ( Rs = aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),U),Us) )
=> ( S != foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),U),zero_zero(nat)),Us) ) ) ) ) ) ) ).
tff(fact_70_ex__head__tail,axiom,
! [Ta: dB] :
? [Ts: list(dB),H: dB] :
( ( Ta = foldl(dB,dB,app,H,Ts) )
& ( ? [N: nat] : H = var(N)
| ? [U: dB] : H = abs(U) ) ) ).
tff(fact_71_list_Orecs_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_72_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List1: list(A),A3: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),List3) )
<=> ( ( A3 = A6 )
& ( List1 = List3 ) ) ) ).
tff(fact_73_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,X1)),aa(list(B),list(A),map(B,A,F),Xs)) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] :
( member(A,X1,A1)
<=> pp(aa(A,bool,A1,X1)) ) ).
tff(fact_76_Cons__step1__Cons,axiom,
! [A: $tType,Xs: list(A),X1: A,Ys: list(A),Y3: A,R1: fun(A,fun(A,bool))] :
( step1(A,R1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),R1,Y3),X1))
& ( Xs = Ys ) )
| ( ( X1 = Y3 )
& step1(A,R1,Ys,Xs) ) ) ) ).
tff(fact_77_list_Orecs_I2_J,axiom,
! [A: $tType,B: $tType,List1: list(B),A3: B,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),List1)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F2,A3),List1),list_rec(A,B,F1,F2,List1)) ).
tff(fact_78_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) != Xs1 ).
tff(fact_79_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) ).
tff(fact_80_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,A3: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun(B,A),F,A3),X1),Xs) ).
tff(fact_81_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A5: A] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) ).
tff(fact_82_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A5: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) != nil(A) ).
tff(fact_83_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List1: list(B),A3: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),List1)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A3),List1) ).
tff(fact_84_listsp_Osimps,axiom,
! [A: $tType,A3: list(A),A1: fun(A,bool)] :
( listsp(A,A1,A3)
<=> ( ( A3 = nil(A) )
| ? [A4: A,L: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),L) )
& pp(aa(A,bool,A1,A4))
& listsp(A,A1,L) ) ) ) ).
tff(fact_85_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys: list(A),Y3: A,Xs: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),map(B,A,F),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys) )
<=> ? [Z: B,Zs1: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs1) )
& ( aa(B,A,F,Z) = Y3 )
& ( aa(list(B),list(A),map(B,A,F),Zs1) = Ys ) ) ) ).
tff(fact_86_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list(B),F: fun(B,A),Xs: list(A),X1: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs) = aa(list(B),list(A),map(B,A,F),Ys) )
<=> ? [Z: B,Zs1: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs1) )
& ( X1 = aa(B,A,F,Z) )
& ( Xs = aa(list(B),list(A),map(B,A,F),Zs1) ) ) ) ).
tff(fact_87_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y2: A,Ys1: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys1) ) ).
tff(fact_88_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A2: A,List: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),List) ) ).
tff(fact_89_Cons__step1E,axiom,
! [A: $tType,Xs: list(A),X1: A,Ys: list(A),R1: fun(A,fun(A,bool))] :
( step1(A,R1,Ys,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs))
=> ( ! [Y1: A] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Xs) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),R1,Y1),X1)) )
=> ~ ! [Zs: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Zs) )
=> ~ step1(A,R1,Zs,Xs) ) ) ) ).
tff(fact_90_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).
tff(fact_91_list__ex1__simps_I2_J,axiom,
! [A: $tType,Xs: list(A),X1: A,P1: fun(A,bool)] :
( list_ex1(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs))
<=> ( ( pp(aa(A,bool,P1,X1))
=> list_all(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),fdisj),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fNot),P1)),fequal(A,X1)),Xs) )
& ( ~ pp(aa(A,bool,P1,X1))
=> list_ex1(A,P1,Xs) ) ) ) ).
tff(fact_92_list__all__simps_I1_J,axiom,
! [A: $tType,Xs: list(A),X1: A,P1: fun(A,bool)] :
( list_all(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs))
<=> ( pp(aa(A,bool,P1,X1))
& list_all(A,P1,Xs) ) ) ).
tff(fact_93_list__all__simps_I2_J,axiom,
! [A: $tType,P1: fun(A,bool)] : list_all(A,P1,nil(A)) ).
tff(fact_94_sublist__singleton,axiom,
! [A: $tType,X1: A,A1: fun(nat,bool)] :
( ( member(nat,zero_zero(nat),A1)
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A)),A1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A)) ) )
& ( ~ member(nat,zero_zero(nat),A1)
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A)),A1) = nil(A) ) ) ) ).
tff(fact_95_concat__map__singleton,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),combc(B,list(A),list(A),aa(fun(B,A),fun(B,fun(list(A),list(A))),aa(fun(A,fun(list(A),list(A))),fun(fun(B,A),fun(B,fun(list(A),list(A)))),combb(A,fun(list(A),list(A)),B),cons(A)),F),nil(A))),Xs)) = aa(list(B),list(A),map(B,A,F),Xs) ).
tff(fact_96_concat_Osimps_I1_J,axiom,
! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).
tff(fact_97_sublist__nil,axiom,
! [A: $tType,A1: fun(nat,bool)] : sublist(A,nil(A),A1) = nil(A) ).
tff(fact_98_map__concat,axiom,
! [A: $tType,B: $tType,Xs: list(list(B)),F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F)),Xs)) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (16)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
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) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,fequal(A,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,fequal(A,X),Y)) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
( listsp(dB,it,rs)
& listsp(dB,aa(fun(dB,dB),fun(dB,bool),aa(fun(dB,bool),fun(fun(dB,dB),fun(dB,bool)),combb(dB,bool,dB),it),combc(dB,dB,dB,app,var(i))),rs) ) ).
tff(conj_1,conjecture,
pp(aa(dB,bool,it,var(i))) ).
%------------------------------------------------------------------------------