TPTP Problem File: LCL769_5.p
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%------------------------------------------------------------------------------
% File : LCL769_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 74
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_74 [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 : 155 ( 47 unt; 38 typ; 0 def)
% Number of atoms : 242 ( 181 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 176 ( 51 ~; 17 |; 34 &)
% ( 35 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 40 ( 23 >; 17 *; 0 +; 0 <<)
% Number of predicates : 5 ( 4 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 9 con; 0-5 aty)
% Number of variables : 429 ( 386 !; 12 ?; 429 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:43
%------------------------------------------------------------------------------
%----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 (33)
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(B,fun(A,C)) ) ).
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_InductTermi_OIT,type,
it: 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:
!>[T: $tType] : ( ( fun(nat,T) * fun(dB,fun(dB,T)) * fun(dB,T) * dB ) > T ) ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( 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_Olast,type,
last:
!>[A: $tType] : ( list(A) > 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:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
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_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( fun(A,list(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
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_pp,type,
pp: bool > $o ).
tff(sy_v_i,type,
i: nat ).
tff(sy_v_rs,type,
rs: list(dB) ).
%----Relevant facts (100)
tff(fact_0_Var__IT,axiom,
! [N2: nat] : pp(aa(dB,bool,it,var(N2))) ).
tff(fact_1_listsp_ONil,axiom,
! [A: $tType,A1: fun(A,bool)] : listsp(A,A1,nil(A)) ).
tff(fact_2_listsp__conj__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),X3: list(A)] :
( listsp(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,A1),B1),X3)
<=> ( listsp(A,A1,X3)
& listsp(A,B1,X3) ) ) ).
tff(fact_3_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_4_dB_Osimps_I2_J,axiom,
! [DB24: dB,DB14: 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,DB14),DB24) )
<=> ( ( DB11 = DB14 )
& ( DB21 = DB24 ) ) ) ).
tff(fact_5_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List: list(A),A2: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),List) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),List3) )
<=> ( ( A2 = A6 )
& ( List = List3 ) ) ) ).
tff(fact_6_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_7_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_8_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_9_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_10_listsp_Osimps,axiom,
! [A: $tType,A2: list(A),A1: fun(A,bool)] :
( listsp(A,A1,A2)
<=> ( ( A2 = nil(A) )
| ? [A4: A,L: list(A)] :
( ( A2 = 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_11_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_12_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y3: A,Ys4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) ).
tff(fact_13_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,A2: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A2,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,A2),X1),Xs) ).
tff(fact_14_foldl__Nil,axiom,
! [B: $tType,A: $tType,A2: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A2,nil(B)) = A2 ).
tff(fact_15_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_16_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_17_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_18_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),N1: nat,Rs: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs) = foldl(dB,dB,app,var(N1),Ss) )
<=> ( ( M = N1 )
& ( Rs = Ss ) ) ) ).
tff(fact_19_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_20_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A3: A,List1: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List1) ) ).
tff(fact_21_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_22_foldl__fun__comm,axiom,
! [B: $tType,A: $tType,X1: A,Xs: list(A),S: B,F: fun(B,fun(A,B))] :
( ! [X2: A,Y2: A,S1: B] : aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S1),X2)),Y2) = aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S1),Y2)),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_23_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ).
tff(fact_24_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_25_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A2: 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),A2),List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A2),List) ).
tff(fact_26_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A),X: A] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys1)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),splice(A,Xs1,Ys1))) ).
tff(fact_27_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys1: list(A)] : splice(A,nil(A),Ys1) = Ys1 ).
tff(fact_28_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_29_App__eq__foldl__conv,axiom,
! [Ts1: list(dB),Ta: dB,S: dB,R1: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,R1),S) = foldl(dB,dB,app,Ta,Ts1) )
<=> ( ( ( Ts1 = nil(dB) )
=> ( aa(dB,dB,aa(dB,fun(dB,dB),app,R1),S) = Ta ) )
& ( ( Ts1 != nil(dB) )
=> ? [Ss1: list(dB)] :
( ( Ts1 = aa(list(dB),list(dB),aa(list(dB),fun(list(dB),list(dB)),append(dB),Ss1),aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),S),nil(dB))) )
& ( R1 = foldl(dB,dB,app,Ta,Ss1) ) ) ) ) ) ).
tff(fact_30_app__last,axiom,
! [U1: dB,Ts1: list(dB),Ta: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,foldl(dB,dB,app,Ta,Ts1)),U1) = foldl(dB,dB,app,Ta,aa(list(dB),list(dB),aa(list(dB),fun(list(dB),list(dB)),append(dB),Ts1),aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),U1),nil(dB)))) ).
tff(fact_31_Abs__eq__apps__conv,axiom,
! [Ss: list(dB),S: dB,R1: dB] :
( ( abs(R1) = foldl(dB,dB,app,S,Ss) )
<=> ( ( abs(R1) = S )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_32_apps__eq__Abs__conv,axiom,
! [R1: dB,Ss: list(dB),S: dB] :
( ( foldl(dB,dB,app,S,Ss) = abs(R1) )
<=> ( ( S = abs(R1) )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_33_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_34_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_35_list__nonempty__induct,axiom,
! [A: $tType,P1: fun(list(A),bool),Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ! [X2: A] : pp(aa(list(A),bool,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P1,Xs2))
=> pp(aa(list(A),bool,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2))) ) )
=> pp(aa(list(A),bool,P1,Xs)) ) ) ) ).
tff(fact_36_append__same__eq,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xs) )
<=> ( Ys = Zs ) ) ).
tff(fact_37_same__append__eq,axiom,
! [A: $tType,Zs: list(A),Ys: list(A),Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs) )
<=> ( Ys = Zs ) ) ).
tff(fact_38_append__assoc,axiom,
! [A: $tType,Zs1: list(A),Ys1: list(A),Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)),Zs1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys1),Zs1)) ).
tff(fact_39_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB4: dB] :
( ( abs(DB4) = abs(DB5) )
<=> ( DB4 = DB5 ) ) ).
tff(fact_40_append__Cons,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A),X: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Ys1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) ).
tff(fact_41_append__self__conv2,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
<=> ( Xs = nil(A) ) ) ).
tff(fact_42_append__self__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
<=> ( Ys = nil(A) ) ) ).
tff(fact_43_append__is__Nil__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
tff(fact_44_self__append__conv2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
<=> ( Xs = nil(A) ) ) ).
tff(fact_45_self__append__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
<=> ( Ys = nil(A) ) ) ).
tff(fact_46_Nil__is__append__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
tff(fact_47_foldl__append,axiom,
! [A: $tType,B: $tType,Ys: list(B),Xs: list(B),A2: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = foldl(A,B,F,foldl(A,B,F,A2,Xs),Ys) ).
tff(fact_48_append__in__listsp__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),A1: fun(A,bool)] :
( listsp(A,A1,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( listsp(A,A1,Xs)
& listsp(A,A1,Ys) ) ) ).
tff(fact_49_Lambda,axiom,
! [R2: dB] :
( pp(aa(dB,bool,it,R2))
=> pp(aa(dB,bool,it,abs(R2))) ) ).
tff(fact_50_append1__eq__conv,axiom,
! [A: $tType,Y1: A,Ys: list(A),X1: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),nil(A))) )
<=> ( ( Xs = Ys )
& ( X1 = Y1 ) ) ) ).
tff(fact_51_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs: list(dB),R1: dB] :
( ( foldl(dB,dB,app,abs(R1),Rs) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R1 = S )
& ( Rs = Ss ) ) ) ).
tff(fact_52_append__eq__appendI,axiom,
! [A: $tType,Us1: list(A),Ys1: list(A),Zs1: list(A),Xs11: list(A),Xs1: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Xs11) = Zs1 )
=> ( ( Ys1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs11),Us1) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs1),Us1) ) ) ) ).
tff(fact_53_dB_Osimps_I12_J,axiom,
! [A: $tType,DB4: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB4)) = aa(dB,A,F3,DB4) ).
tff(fact_54_append__eq__append__conv2,axiom,
! [A: $tType,Ts1: list(A),Zs: list(A),Ys: list(A),Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts1) )
<=> ? [Us: list(A)] :
( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys) = Ts1 ) )
| ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = Zs )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ts1) ) ) ) ) ).
tff(fact_55_Cons__eq__appendI,axiom,
! [A: $tType,Zs1: list(A),Xs1: list(A),Ys1: list(A),Xs11: list(A),X: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs11) = Ys1 )
=> ( ( Xs1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs11),Zs1) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys1),Zs1) ) ) ) ).
tff(fact_56_eq__Nil__appendI,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( Xs1 = Ys1 )
=> ( Xs1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys1) ) ) ).
tff(fact_57_append__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),nil(A)) = Xs1 ).
tff(fact_58_append__Nil,axiom,
! [A: $tType,Ys1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys1) = Ys1 ).
tff(fact_59_dB_Osimps_I8_J,axiom,
! [DB3: dB,DB22: dB,DB12: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) != abs(DB3) ).
tff(fact_60_dB_Osimps_I9_J,axiom,
! [DB22: dB,DB12: dB,DB3: dB] : abs(DB3) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) ).
tff(fact_61_dB_Osimps_I6_J,axiom,
! [DB3: dB,Nat1: nat] : var(Nat1) != abs(DB3) ).
tff(fact_62_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB3: dB] : abs(DB3) != var(Nat1) ).
tff(fact_63_Cons__eq__append__conv,axiom,
! [A: $tType,Zs: list(A),Ys: list(A),Xs: list(A),X1: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
<=> ( ( ( Ys = nil(A) )
& ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs) = Zs ) )
| ? [Ys3: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Ys3) = Ys )
& ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) ) ) ) ) ).
tff(fact_64_append__eq__Cons__conv,axiom,
! [A: $tType,Xs: list(A),X1: A,Zs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs) )
<=> ( ( ( Ys = nil(A) )
& ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs) ) )
| ? [Ys3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Ys3) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) = Xs ) ) ) ) ).
tff(fact_65_Var__apps__neq__Abs__apps,axiom,
! [Ss: list(dB),R1: dB,Ts1: list(dB),N1: nat] : foldl(dB,dB,app,var(N1),Ts1) != foldl(dB,dB,app,abs(R1),Ss) ).
tff(fact_66_Abs__App__neq__Var__apps,axiom,
! [Ss: 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),Ss) ).
tff(fact_67_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_68_rev__cases,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 != nil(A) )
=> ~ ! [Ys2: list(A),Y2: A] : Xs1 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))) ) ).
tff(fact_69_rev__induct,axiom,
! [A: $tType,Xs: list(A),P1: fun(list(A),bool)] :
( pp(aa(list(A),bool,P1,nil(A)))
=> ( ! [X2: A,Xs2: list(A)] :
( pp(aa(list(A),bool,P1,Xs2))
=> pp(aa(list(A),bool,P1,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))) )
=> pp(aa(list(A),bool,P1,Xs)) ) ) ).
tff(fact_70_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,combb(bool,fun(bool,bool),A,fdisj,combb(bool,bool,A,fNot,P1)),fequal(A,X1)),Xs) )
& ( ~ pp(aa(A,bool,P1,X1))
=> list_ex1(A,P1,Xs) ) ) ) ).
tff(fact_71_rotate1__def,axiom,
! [A: $tType,Xs: list(A)] : rotate1(A,Xs) = list_case(list(A),A,nil(A),combb(list(A),fun(list(A),list(A)),A,combc(list(A),list(A),list(A),append(A)),aa(list(A),fun(A,list(A)),combc(A,list(A),list(A),cons(A)),nil(A))),Xs) ).
tff(fact_72_rev__exhaust2,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 != nil(A) )
=> ~ ! [Ys2: list(A),Y2: A] : Xs1 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))) ) ).
tff(fact_73_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_74_list__all__simps_I2_J,axiom,
! [A: $tType,P1: fun(A,bool)] : list_all(A,P1,nil(A)) ).
tff(fact_75_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_76_list__all__append,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),P1: fun(A,bool)] :
( list_all(A,P1,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( list_all(A,P1,Xs)
& list_all(A,P1,Ys) ) ) ).
tff(fact_77_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( rotate1(A,Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
tff(fact_78_rotate__simps,axiom,
! [A: $tType,B: $tType,Xs1: list(B),X: B] :
( ( rotate1(A,nil(A)) = nil(A) )
& ( rotate1(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs1)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ) ) ).
tff(fact_79_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,F: fun(B,list(A))] : maps(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(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F,X1)),maps(B,A,F,Xs)) ).
tff(fact_80_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_81_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
tff(fact_82_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_83_last__snoc,axiom,
! [A: $tType,X: A,Xs1: list(A)] : last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).
tff(fact_84_last_Osimps,axiom,
! [A: $tType,X: A,Xs1: list(A)] :
( ( ( Xs1 = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = X ) )
& ( ( Xs1 != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = last(A,Xs1) ) ) ) ).
tff(fact_85_last__ConsR,axiom,
! [A: $tType,X: A,Xs1: list(A)] :
( ( Xs1 != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = last(A,Xs1) ) ) ).
tff(fact_86_last__ConsL,axiom,
! [A: $tType,X: A,Xs1: list(A)] :
( ( Xs1 = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = X ) ) ).
tff(fact_87_last__append,axiom,
! [A: $tType,Xs1: list(A),Ys1: list(A)] :
( ( ( Ys1 = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = last(A,Xs1) ) )
& ( ( Ys1 != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = last(A,Ys1) ) ) ) ).
tff(fact_88_last__appendR,axiom,
! [A: $tType,Xs1: list(A),Ys1: list(A)] :
( ( Ys1 != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = last(A,Ys1) ) ) ).
tff(fact_89_last__appendL,axiom,
! [A: $tType,Xs1: list(A),Ys1: list(A)] :
( ( Ys1 = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = last(A,Xs1) ) ) ).
tff(fact_90_snoc__eq__iff__butlast,axiom,
! [A: $tType,Ys: list(A),X1: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))) = Ys )
<=> ( ( Ys != nil(A) )
& ( butlast(A,Ys) = Xs )
& ( last(A,Ys) = X1 ) ) ) ).
tff(fact_91_append__butlast__last__id,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 != nil(A) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),butlast(A,Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,Xs1)),nil(A))) = Xs1 ) ) ).
tff(fact_92_butlast_Osimps_I1_J,axiom,
! [A: $tType] : butlast(A,nil(A)) = nil(A) ).
tff(fact_93_butlast_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs1: list(A)] :
( ( ( Xs1 = nil(A) )
=> ( butlast(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = nil(A) ) )
& ( ( Xs1 != nil(A) )
=> ( butlast(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),butlast(A,Xs1)) ) ) ) ).
tff(fact_94_butlast__append,axiom,
! [A: $tType,Xs1: list(A),Ys1: list(A)] :
( ( ( Ys1 = nil(A) )
=> ( butlast(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = butlast(A,Xs1) ) )
& ( ( Ys1 != nil(A) )
=> ( butlast(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Ys1)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),butlast(A,Ys1)) ) ) ) ).
tff(fact_95_butlast__snoc,axiom,
! [A: $tType,X: A,Xs1: list(A)] : butlast(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = Xs1 ).
tff(fact_96_rev__eq__Cons__iff,axiom,
! [A: $tType,Ys: list(A),Y1: A,Xs: list(A)] :
( ( rev(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) )
<=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),nil(A))) ) ) ).
tff(fact_97_rev_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),X: A] : rev(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
tff(fact_98_rev__is__rev__conv,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( rev(A,Xs) = rev(A,Ys) )
<=> ( Xs = Ys ) ) ).
tff(fact_99_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( rev(A,Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
%----Helper facts (15)
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,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,aa(B,fun(A,C),combc(A,B,C,P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
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,combb(dB,bool,dB,it,aa(dB,fun(dB,dB),combc(dB,dB,dB,app),var(i))),rs) ) ).
tff(conj_1,conjecture,
listsp(dB,it,aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),var(i)),nil(dB))) ).
%------------------------------------------------------------------------------