TPTP Problem File: LCL819_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : LCL819_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 187
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_187 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 160 ( 64 unt; 49 typ; 0 def)
% Number of atoms : 203 ( 121 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 130 ( 38 ~; 8 |; 22 &)
% ( 21 <=>; 41 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 14 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 41 ( 23 >; 18 *; 0 +; 0 <<)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-4 aty)
% Number of functors : 41 ( 41 usr; 16 con; 0-5 aty)
% Number of variables : 386 ( 338 !; 15 ?; 386 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:11
%------------------------------------------------------------------------------
%----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 (44)
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_COMBK,type,
combk:
!>[A: $tType,B: $tType] : fun(A,fun(B,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__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)) ) > fun(list(A),T2) ) ).
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_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list(A) * fun(nat,bool) ) > list(A) ) ).
tff(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(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_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,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_a____,type,
a: dB ).
tff(sy_v_as____,type,
as: list(dB) ).
tff(sy_v_i____,type,
i: nat ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_rs____,type,
rs: list(dB) ).
tff(sy_v_t____,type,
t: dB ).
tff(sy_v_u____,type,
u: dB ).
tff(sy_v_ua______,type,
ua: dB ).
%----Relevant facts (99)
tff(fact_0__096IT_At_096,axiom,
pp(aa(dB,bool,it,t)) ).
tff(fact_1_Var_I3_J,axiom,
pp(aa(dB,bool,it,ua)) ).
tff(fact_2_True,axiom,
n = i ).
tff(fact_3_Cons,axiom,
rs = aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),a),as) ).
tff(fact_4_uIT,axiom,
pp(aa(dB,bool,it,u)) ).
tff(fact_5_Var__IT,axiom,
! [N: nat] : pp(aa(dB,bool,it,var(N))) ).
tff(fact_6_app__Var__IT,axiom,
! [I: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun(dB,dB),app,T),var(I)))) ) ).
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,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),I))) ) ).
tff(fact_9_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),Na: nat,Rsa: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rsa) = foldl(dB,dB,app,var(Na),Ss) )
<=> ( ( M = Na )
& ( Rsa = Ss ) ) ) ).
tff(fact_10_subst__App,axiom,
! [K: nat,S1: 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)),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,T),S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U),S1),K)) ).
tff(fact_11_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),R1: dB] :
( ( foldl(dB,dB,app,R1,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( R1 = S ) ) ).
tff(fact_12_dB_Osimps_I1_J,axiom,
! [Nat2: nat,Nat1: nat] :
( ( var(Nat1) = var(Nat2) )
<=> ( Nat1 = Nat2 ) ) ).
tff(fact_13_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_14_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_15_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_16_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_17__096IT_A_I_IVar_A0_A_092_060degree_062_092_060degree_062_Amap_A_I_Ft_O_Alift_At_A0_J_A_Imap_A_I_Ft_O_At_091u_Pi_093_J_Aas_J_J_091u_A_092_060degree_062_Aa_091u_Pi_093_P0_093_J_096,axiom,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,var(zero_zero(nat)),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat))),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,u),i)),as)))),aa(dB,dB,aa(dB,fun(dB,dB),app,u),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,a),u),i))),zero_zero(nat)))) ).
tff(fact_18_IT_OVar,axiom,
! [Na: nat,Rsa: list(dB)] :
( listsp(dB,it,Rsa)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(Na),Rsa))) ) ).
tff(fact_19_Beta,axiom,
! [Ss: 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)),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(R1)),S),Ss))) ) ) ).
tff(fact_20_dB_Osimps_I3_J,axiom,
! [DB4: dB,DB3: dB] :
( ( abs(DB3) = abs(DB4) )
<=> ( DB3 = DB4 ) ) ).
tff(fact_21_lift_Osimps_I2_J,axiom,
! [K: nat,T: dB,S1: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T)),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,T),K)) ).
tff(fact_22_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_23_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rsa: list(dB),R1: dB] :
( ( foldl(dB,dB,app,abs(R1),Rsa) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R1 = S )
& ( Rsa = Ss ) ) ) ).
tff(fact_24_subst__map,axiom,
! [Ib: nat,Ub: dB,Ts: 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,Ts)),Ub),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),Ub),Ib),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,Ub),Ib)),Ts)) ).
tff(fact_25_lift__map,axiom,
! [Ib: nat,Ts: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,foldl(dB,dB,app,Ta,Ts)),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),lift,Ta),Ib),aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,lift,Ib)),Ts)) ).
tff(fact_26_lifts__IT,axiom,
! [Ts: list(dB)] :
( listsp(dB,it,Ts)
=> listsp(dB,it,aa(list(dB),list(dB),map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat))),Ts)) ) ).
tff(fact_27_dB_Osimps_I8_J,axiom,
! [DB: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB) ).
tff(fact_28_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB: dB] : abs(DB) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_29_dB_Osimps_I6_J,axiom,
! [DB: dB,Nat: nat] : var(Nat) != abs(DB) ).
tff(fact_30_dB_Osimps_I7_J,axiom,
! [Nat: nat,DB: dB] : abs(DB) != var(Nat) ).
tff(fact_31_subst__lift,axiom,
! [S1: 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)),S1),K) = T ).
tff(fact_32_Var__apps__neq__Abs__apps,axiom,
! [Ss: list(dB),R1: dB,Ts: list(dB),Na: nat] : foldl(dB,dB,app,var(Na),Ts) != foldl(dB,dB,app,abs(R1),Ss) ).
tff(fact_33_Abs__App__neq__Var__apps,axiom,
! [Ss: list(dB),Na: nat,Ta: dB,S: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S)),Ta) != foldl(dB,dB,app,var(Na),Ss) ).
tff(fact_34_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X: 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),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,X)),aa(list(B),list(A),map(B,A,F),Xs)) ).
tff(fact_35_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_36_map__ident,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),map(A,A,combi(A)),X3) = X3 ).
tff(fact_37_IT_Osimps,axiom,
! [Aa: dB] :
( pp(aa(dB,bool,it,Aa))
<=> ( ? [Rs1: list(dB),N1: nat] :
( ( Aa = foldl(dB,dB,app,var(N1),Rs1) )
& listsp(dB,it,Rs1) )
| ? [R4: dB] :
( ( Aa = abs(R4) )
& pp(aa(dB,bool,it,R4)) )
| ? [R4: dB,S3: dB,Ss2: list(dB)] :
( ( Aa = 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_38_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List2: list(A),Aa: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa),List2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List3) )
<=> ( ( Aa = A5 )
& ( List2 = List3 ) ) ) ).
tff(fact_39_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun(C,B),Aa: A,G: fun(A,fun(B,A))] : foldl(A,B,G,Aa,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),Aa,Xs) ).
tff(fact_40_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(var(Nat)) = zero_zero(nat) ).
tff(fact_41_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_42_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1) != Xs1 ).
tff(fact_43_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1) ).
tff(fact_44_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),X: B,Aa: A,F: fun(A,fun(B,A))] : foldl(A,B,F,Aa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun(B,A),F,Aa),X),Xs) ).
tff(fact_45_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys2: 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),Ys2) )
<=> ? [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) = Ys2 ) ) ) ).
tff(fact_46_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys2: list(B),F: fun(B,A),Xs: list(A),X: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),map(B,A,F),Ys2) )
<=> ? [Z: B,Zs1: list(B)] :
( ( Ys2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs1) )
& ( X = aa(B,A,F,Z) )
& ( Xs = aa(list(B),list(A),map(B,A,F),Zs1) ) ) ) ).
tff(fact_47_beta,axiom,
! [T: dB,S1: dB] : pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T),zero_zero(nat)))) ).
tff(fact_48_dB_Osize_I4_J,axiom,
! [Nat: nat] : size_size(dB,var(Nat)) = zero_zero(nat) ).
tff(fact_49_foldl__fun__comm,axiom,
! [B: $tType,A: $tType,X: A,Xs: list(A),S: B,F: fun(B,fun(A,B))] :
( ! [X2: A,Y2: A,S2: B] : aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),X2)),Y2) = aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),Y2)),X2)
=> ( aa(A,B,aa(B,fun(A,B),F,foldl(B,A,F,S,Xs)),X) = foldl(B,A,F,aa(A,B,aa(B,fun(A,B),F,S),X),Xs) ) ) ).
tff(fact_50_appR,axiom,
! [U: dB,T: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,U),S1)),aa(dB,dB,aa(dB,fun(dB,dB),app,U),T))) ) ).
tff(fact_51_appL,axiom,
! [U: dB,T: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U)),aa(dB,dB,aa(dB,fun(dB,dB),app,T),U))) ) ).
tff(fact_52_beta__cases_I1_J,axiom,
! [T: dB,I: nat] : ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,var(I)),T)) ).
tff(fact_53_abs,axiom,
! [T: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(S1)),abs(T))) ) ).
tff(fact_54_subst__preserves__beta,axiom,
! [I: nat,T: 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),T),I)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T),I))) ) ).
tff(fact_55_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_56_beta__cases_I2_J,axiom,
! [S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(R3)),S1))
=> ~ ! [T1: dB] :
( ( S1 = abs(T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),T1)) ) ) ).
tff(fact_57_apps__preserves__beta,axiom,
! [Ss: 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,Ss)),foldl(dB,dB,app,S,Ss))) ) ).
tff(fact_58_beta__cases_I3_J,axiom,
! [U: dB,T: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T)),U))
=> ( ! [S2: dB] :
( ( U = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S2),T),zero_zero(nat)) )
=> ( S1 != abs(S2) ) )
=> ( ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,T1),T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1)) )
=> ~ ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,T),T1)) ) ) ) ) ).
tff(fact_59_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List2: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : aa(list(B),A,list_case(A,B,F1,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa),List2)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List2) ).
tff(fact_60_apps__betasE,axiom,
! [S: dB,Rsa: list(dB),R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Rsa)),S))
=> ( ! [R2: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R1),R2))
=> ( S != foldl(dB,dB,app,R2,Rsa) ) )
=> ( ! [Rs: list(dB)] :
( step1(dB,beta,Rsa,Rs)
=> ( S != foldl(dB,dB,app,R1,Rs) ) )
=> ~ ! [T1: dB] :
( ( R1 = abs(T1) )
=> ! [U1: dB,Us: list(dB)] :
( ( Rsa = aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),U1),Us) )
=> ( S != foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T1),U1),zero_zero(nat)),Us) ) ) ) ) ) ) ).
tff(fact_61_substn__subst__0,axiom,
! [S1: dB,T: dB] : substn(T,S1,zero_zero(nat)) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),S1),zero_zero(nat)) ).
tff(fact_62_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A),X1: A] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),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),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),splice(A,Xs1,Ys1))) ).
tff(fact_63_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U: dB,T: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T),U),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T,S1,K)),substn(U,S1,K)) ).
tff(fact_64_apps__preserves__betas,axiom,
! [R1: dB,Ss: list(dB),Rsa: list(dB)] :
( step1(dB,beta,Rsa,Ss)
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Rsa)),foldl(dB,dB,app,R1,Ss))) ) ).
tff(fact_65_head__Var__reduction,axiom,
! [V1: dB,Rsa: list(dB),Na: nat] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,var(Na),Rsa)),V1))
=> ? [Ss1: list(dB)] :
( step1(dB,beta,Rsa,Ss1)
& ( V1 = foldl(dB,dB,app,var(Na),Ss1) ) ) ) ).
tff(fact_66_Cons__step1__Cons,axiom,
! [A: $tType,Xs: list(A),X: A,Ys2: 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),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),R1,Y3),X))
& ( Xs = Ys2 ) )
| ( ( X = Y3 )
& step1(A,R1,Ys2,Xs) ) ) ) ).
tff(fact_67_substn__subst__n,axiom,
! [N: nat,S1: dB,T: dB] : substn(T,S1,N) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),liftn(N,S1,zero_zero(nat))),N) ).
tff(fact_68_liftn_Osimps_I2_J,axiom,
! [K: nat,T: dB,S1: dB,N: nat] : liftn(N,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N,S1,K)),liftn(N,T,K)) ).
tff(fact_69_liftn__0,axiom,
! [K: nat,T: dB] : liftn(zero_zero(nat),T,K) = T ).
tff(fact_70_Cons__step1E,axiom,
! [A: $tType,Xs: list(A),X: A,Ys2: list(A),R1: fun(A,fun(A,bool))] :
( step1(A,R1,Ys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
=> ( ! [Y2: A] :
( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),R1,Y2),X)) )
=> ~ ! [Zs: list(A)] :
( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs) )
=> ~ step1(A,R1,Zs,Xs) ) ) ) ).
tff(fact_71_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_72_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_73_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_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),X: A] :
( member(A,X,A1)
<=> pp(aa(A,bool,A1,X)) ) ).
tff(fact_76_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_77_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_78_listsp_ONil,axiom,
! [A: $tType,A1: fun(A,bool)] : listsp(A,A1,nil(A)) ).
tff(fact_79_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_80_not__Nil__step1,axiom,
! [A: $tType,Xs: list(A),R1: fun(A,fun(A,bool))] : ~ step1(A,R1,nil(A),Xs) ).
tff(fact_81_not__step1__Nil,axiom,
! [A: $tType,Xs: list(A),R1: fun(A,fun(A,bool))] : ~ step1(A,R1,Xs,nil(A)) ).
tff(fact_82_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : aa(list(B),A,list_case(A,B,F1,F2),nil(B)) = F1 ).
tff(fact_83_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A4: A] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List1) ).
tff(fact_84_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A4: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List1) != nil(A) ).
tff(fact_85_foldl__Nil,axiom,
! [B: $tType,A: $tType,Aa: A,F: fun(A,fun(B,A))] : foldl(A,B,F,Aa,nil(B)) = Aa ).
tff(fact_86_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys1: list(A)] : splice(A,nil(A),Ys1) = Ys1 ).
tff(fact_87_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_88_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_89_listsp_Osimps,axiom,
! [A: $tType,Aa: list(A),A1: fun(A,bool)] :
( listsp(A,A1,Aa)
<=> ( ( Aa = nil(A) )
| ? [A3: A,L: list(A)] :
( ( Aa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),L) )
& pp(aa(A,bool,A1,A3))
& listsp(A,A1,L) ) ) ) ).
tff(fact_90_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y1: A,Ys: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) ) ).
tff(fact_91_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_92_insert__Nil,axiom,
! [A: $tType,X1: A] : insert(A,X1,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A)) ).
tff(fact_93_sublist__singleton,axiom,
! [A: $tType,X: 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),X),nil(A)),A1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
& ( ~ member(nat,zero_zero(nat),A1)
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A1) = nil(A) ) ) ) ).
tff(fact_94_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_95_concat_Osimps_I1_J,axiom,
! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).
tff(fact_96_sublist__nil,axiom,
! [A: $tType,A1: fun(nat,bool)] : sublist(A,nil(A),A1) = nil(A) ).
tff(fact_97_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)) ).
tff(fact_98_transpose_Osimps_I3_J,axiom,
! [A: $tType,Xss: list(list(A)),Xs: list(A),X: A] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),list_case(list(A),A,nil(A),aa(fun(A,list(A)),fun(A,fun(list(A),list(A))),aa(fun(list(A),fun(list(A),list(A))),fun(fun(A,list(A)),fun(A,fun(list(A),list(A)))),combb(list(A),fun(list(A),list(A)),A),combk(list(A),list(A))),combc(A,list(A),list(A),cons(A),nil(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),list_case(list(list(A)),A,nil(list(A)),aa(fun(list(A),list(list(A))),fun(A,fun(list(A),list(list(A)))),combk(fun(list(A),list(list(A))),A),combc(list(A),list(list(A)),list(list(A)),cons(list(A)),nil(list(A)))))),Xss))))) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (10)
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_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,aa(A,fun(B,A),combk(A,B),P),Q) = 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 (1)
tff(conj_0,conjecture,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,var(n),rs)),u),i))) ).
%------------------------------------------------------------------------------