TPTP Problem File: LCL843_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL843_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 252
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_252 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 147 ( 53 unt; 44 typ; 0 def)
% Number of atoms : 199 ( 111 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 139 ( 43 ~; 3 |; 15 &)
% ( 15 <=>; 63 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 58 ( 27 >; 31 *; 0 +; 0 <<)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-4 aty)
% Number of functors : 35 ( 35 usr; 10 con; 0-5 aty)
% Number of variables : 371 ( 346 !; 7 ?; 371 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:48
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_Type_Otype,type,
type: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (38)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
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:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(dB,fun(dB,T4)) * fun(dB,T4) * dB ) > T4 ) ).
tff(sy_c_Lambda_OdB_OdB__rec,type,
dB_rec:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(dB,fun(dB,fun(T4,fun(T4,T4)))) * fun(dB,fun(T4,T4)) * dB ) > T4 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * nat ) > dB ).
tff(sy_c_ListOrder_Ostep1,type,
step1:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * 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:
!>[T4: $tType,A: $tType] : ( ( T4 * fun(A,fun(list(A),T4)) * list(A) ) > T4 ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Type_Oshift,type,
shift:
!>[A: $tType] : ( ( fun(nat,A) * nat * A ) > fun(nat,A) ) ).
tff(sy_c_Type_Otype_OAtom,type,
atom: nat > type ).
tff(sy_c_Type_Otype_OFun,type,
fun1: ( type * type ) > type ).
tff(sy_c_Type_Otype_Otype__case,type,
type_case:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(type,fun(type,T4)) * type ) > T4 ) ).
tff(sy_c_Type_Otype_Otype__size,type,
type_size: type > nat ).
tff(sy_c_Type_Otyping,type,
typing: ( fun(nat,type) * dB * type ) > $o ).
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_pp,type,
pp: bool > $o ).
tff(sy_v_T,type,
t: type ).
tff(sy_v_Ta______,type,
ta: type ).
tff(sy_v_e,type,
e: fun(nat,type) ).
tff(sy_v_env______,type,
env: nat > type ).
tff(sy_v_t,type,
t1: dB ).
tff(sy_v_x______,type,
x: nat ).
%----Relevant facts (99)
tff(fact_0_Var,axiom,
env(x) = ta ).
tff(fact_1_Var__IT,axiom,
! [N2: nat] : pp(aa(dB,bool,it,var(N2))) ).
tff(fact_2_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_3_subst__Var__IT,axiom,
! [J1: nat,I1: nat,R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,subst(R3,var(I1),J1))) ) ).
tff(fact_4_app__Var__IT,axiom,
! [I1: nat,T1: dB] :
( pp(aa(dB,bool,it,T1))
=> pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun(dB,dB),app,T1),var(I1)))) ) ).
tff(fact_5_lift__IT,axiom,
! [I1: nat,T1: dB] :
( pp(aa(dB,bool,it,T1))
=> pp(aa(dB,bool,it,lift(T1,I1))) ) ).
tff(fact_6_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_7_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat2: nat,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,var(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_8_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_9_beta__cases_I1_J,axiom,
! [T1: dB,I1: nat] : ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,var(I1)),T1)) ).
tff(fact_10_typing_OVar,axiom,
! [Tb: type,Xa: nat,Enva: fun(nat,type)] :
( ( aa(nat,type,Enva,Xa) = Tb )
=> typing(Enva,var(Xa),Tb) ) ).
tff(fact_11_typing__elims_I1_J,axiom,
! [Tb: type,I: nat,Ea: fun(nat,type)] :
( typing(Ea,var(I),Tb)
=> ( aa(nat,type,Ea,I) = Tb ) ) ).
tff(fact_12_subst__eq,axiom,
! [U2: dB,K: nat] : subst(var(K),U2,K) = U2 ).
tff(fact_13_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB3: dB] : abs(DB3) != var(Nat1) ).
tff(fact_14_assms,axiom,
typing(e,t1,t) ).
tff(fact_15_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_16_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB4: dB] :
( ( abs(DB4) = abs(DB5) )
<=> ( DB4 = DB5 ) ) ).
tff(fact_17_appL,axiom,
! [U2: dB,T1: dB,S3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S3),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S3),U2)),aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U2))) ) ).
tff(fact_18_appR,axiom,
! [U2: dB,T1: dB,S3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S3),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,U2),S3)),aa(dB,dB,aa(dB,fun(dB,dB),app,U2),T1))) ) ).
tff(fact_19_subst__App,axiom,
! [K: nat,S3: dB,U2: dB,T1: dB] : subst(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U2),S3,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,subst(T1,S3,K)),subst(U2,S3,K)) ).
tff(fact_20_abs,axiom,
! [T1: dB,S3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S3),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(S3)),abs(T1))) ) ).
tff(fact_21_lift_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S3: dB] : lift(aa(dB,dB,aa(dB,fun(dB,dB),app,S3),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,lift(S3,K)),lift(T1,K)) ).
tff(fact_22_dB_Osimps_I9_J,axiom,
! [DB23: dB,DB13: dB,DB3: dB] : abs(DB3) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) ).
tff(fact_23_dB_Osimps_I8_J,axiom,
! [DB3: dB,DB23: dB,DB13: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) != abs(DB3) ).
tff(fact_24_subst__lift,axiom,
! [S3: dB,K: nat,T1: dB] : subst(lift(T1,K),S3,K) = T1 ).
tff(fact_25_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_26_dB_Orecs_I3_J,axiom,
! [A: $tType,DB4: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,abs(DB4)) = aa(A,A,aa(dB,fun(A,A),F3,DB4),dB_rec(A,F1,F2,F3,DB4)) ).
tff(fact_27_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_28_dB_Orecs_I2_J,axiom,
! [A: $tType,DB22: dB,DB12: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22)) = aa(A,A,aa(A,fun(A,A),aa(dB,fun(A,fun(A,A)),aa(dB,fun(dB,fun(A,fun(A,A))),F2,DB12),DB22),dB_rec(A,F1,F2,F3,DB12)),dB_rec(A,F1,F2,F3,DB22)) ).
tff(fact_29_lift__preserves__beta,axiom,
! [I1: nat,S3: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),S3))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,lift(R3,I1)),lift(S3,I1))) ) ).
tff(fact_30_subst__preserves__beta,axiom,
! [I1: nat,T1: dB,S3: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),S3))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,subst(R3,T1,I1)),subst(S3,T1,I1))) ) ).
tff(fact_31_subject__reduction,axiom,
! [T3: dB,Tb: type,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,Ta,Tb)
=> ( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,Ta),T3))
=> typing(Ea,T3,Tb) ) ) ).
tff(fact_32_dB_Osimps_I5_J,axiom,
! [Nat1: nat,DB21: dB,DB11: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) != var(Nat1) ).
tff(fact_33_dB_Osimps_I4_J,axiom,
! [DB21: dB,DB11: dB,Nat1: nat] : var(Nat1) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) ).
tff(fact_34_dB_Osimps_I6_J,axiom,
! [DB3: dB,Nat1: nat] : var(Nat1) != abs(DB3) ).
tff(fact_35_beta__cases_I2_J,axiom,
! [S3: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(R3)),S3))
=> ~ ! [T: dB] :
( ( S3 = abs(T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),T)) ) ) ).
tff(fact_36_beta,axiom,
! [T1: dB,S3: dB] : pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S3)),T1)),subst(S3,T1,zero_zero(nat)))) ).
tff(fact_37_subst__type__IT,axiom,
! [U4: dB,Tb: type,U1: type,I: nat,Ea: fun(nat,type),Ta: dB] :
( pp(aa(dB,bool,it,Ta))
=> ( typing(shift(type,Ea,I,U1),Ta,Tb)
=> ( pp(aa(dB,bool,it,U4))
=> ( typing(Ea,U4,U1)
=> pp(aa(dB,bool,it,subst(Ta,U4,I))) ) ) ) ) ).
tff(fact_38_lift__type,axiom,
! [U1: type,I: nat,Tb: type,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,Ta,Tb)
=> typing(shift(type,Ea,I,U1),lift(Ta,I),Tb) ) ).
tff(fact_39_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_40_App,axiom,
! [Ta: dB,U1: type,Tb: type,S: dB,Enva: fun(nat,type)] :
( typing(Enva,S,fun1(Tb,U1))
=> ( typing(Enva,Ta,Tb)
=> typing(Enva,aa(dB,dB,aa(dB,fun(dB,dB),app,S),Ta),U1) ) ) ).
tff(fact_41_substn_Osimps_I2_J,axiom,
! [K: nat,S3: dB,U2: dB,T1: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U2),S3,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T1,S3,K)),substn(U2,S3,K)) ).
tff(fact_42_subst__lemma,axiom,
! [I: nat,U1: type,U4: dB,E: fun(nat,type),Tb: type,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,Ta,Tb)
=> ( typing(E,U4,U1)
=> ( ( Ea = shift(type,E,I,U1) )
=> typing(E,subst(Ta,U4,I),Tb) ) ) ) ).
tff(fact_43_liftn_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S3: dB,N2: nat] : liftn(N2,aa(dB,dB,aa(dB,fun(dB,dB),app,S3),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N2,S3,K)),liftn(N2,T1,K)) ).
tff(fact_44_type_Osimps_I2_J,axiom,
! [Type23: type,Type13: type,Type22: type,Type12: type] :
( ( fun1(Type12,Type22) = fun1(Type13,Type23) )
<=> ( ( Type12 = Type13 )
& ( Type22 = Type23 ) ) ) ).
tff(fact_45_substn__subst__n,axiom,
! [N2: nat,S3: dB,T1: dB] : substn(T1,S3,N2) = subst(T1,liftn(N2,S3,zero_zero(nat)),N2) ).
tff(fact_46_Abs,axiom,
! [U1: type,Ta: dB,Tb: type,Enva: fun(nat,type)] :
( typing(shift(type,Enva,zero_zero(nat),Tb),Ta,U1)
=> typing(Enva,abs(Ta),fun1(Tb,U1)) ) ).
tff(fact_47_liftn__0,axiom,
! [K: nat,T1: dB] : liftn(zero_zero(nat),T1,K) = T1 ).
tff(fact_48_shift__eq,axiom,
! [A: $tType,Tb: A,Ea: fun(nat,A),J: nat,I: nat] :
( ( I = J )
=> ( aa(nat,A,shift(A,Ea,I,Tb),J) = Tb ) ) ).
tff(fact_49_substn__subst__0,axiom,
! [S3: dB,T1: dB] : substn(T1,S3,zero_zero(nat)) = subst(T1,S3,zero_zero(nat)) ).
tff(fact_50_typing__elims_I3_J,axiom,
! [Tb: type,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,abs(Ta),Tb)
=> ~ ! [T2: type,U3: type] :
( ( Tb = fun1(T2,U3) )
=> ~ typing(shift(type,Ea,zero_zero(nat),T2),Ta,U3) ) ) ).
tff(fact_51_typing__elims_I2_J,axiom,
! [Tb: type,U4: dB,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,aa(dB,dB,aa(dB,fun(dB,dB),app,Ta),U4),Tb)
=> ~ ! [T2: type] :
( typing(Ea,Ta,fun1(T2,Tb))
=> ~ typing(Ea,U4,T2) ) ) ).
tff(fact_52_abs__typeE,axiom,
! [Tb: type,Ta: dB,Ea: fun(nat,type)] :
( typing(Ea,abs(Ta),Tb)
=> ~ ! [U3: type,V2: type] : ~ typing(shift(type,Ea,zero_zero(nat),U3),Ta,V2) ) ).
tff(fact_53_beta__cases_I3_J,axiom,
! [U2: dB,T1: dB,S3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S3),T1)),U2))
=> ( ! [S1: dB] :
( ( U2 = subst(S1,T1,zero_zero(nat)) )
=> ( S3 != abs(S1) ) )
=> ( ! [T: dB] :
( ( U2 = aa(dB,dB,aa(dB,fun(dB,dB),app,T),T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S3),T)) )
=> ~ ! [T: dB] :
( ( U2 = aa(dB,dB,aa(dB,fun(dB,dB),app,S3),T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,T1),T)) ) ) ) ) ).
tff(fact_54_dB_Osize_I1_J,axiom,
! [Nat1: nat] : dB_size(var(Nat1)) = zero_zero(nat) ).
tff(fact_55_Beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R,S,zero_zero(nat)),Ss)))
=> ( pp(aa(dB,bool,it,S))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R)),S),Ss))) ) ) ).
tff(fact_56_type_Osimps_I6_J,axiom,
! [A: $tType,Type22: type,Type12: type,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,fun1(Type12,Type22)) = aa(type,A,aa(type,fun(type,A),F2,Type12),Type22) ).
tff(fact_57_var__app__type__eq,axiom,
! [U1: type,Tb: type,Ts: list(dB),I: nat,Ea: fun(nat,type)] :
( typing(Ea,foldl(dB,dB,app,var(I),Ts),Tb)
=> ( typing(Ea,foldl(dB,dB,app,var(I),Ts),U1)
=> ( Tb = U1 ) ) ) ).
tff(fact_58_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_59_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs: list(dB),R: dB] :
( ( foldl(dB,dB,app,abs(R),Rs) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R = S )
& ( Rs = Ss ) ) ) ).
tff(fact_60_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),N: nat,Rs: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs) = foldl(dB,dB,app,var(N),Ss) )
<=> ( ( M = N )
& ( Rs = Ss ) ) ) ).
tff(fact_61_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),R: dB] :
( ( foldl(dB,dB,app,R,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( R = S ) ) ).
tff(fact_62_Abs__App__neq__Var__apps,axiom,
! [Ss: list(dB),N: nat,Ta: dB,S: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S)),Ta) != foldl(dB,dB,app,var(N),Ss) ).
tff(fact_63_Var__apps__neq__Abs__apps,axiom,
! [Ss: list(dB),R: dB,Ts: list(dB),N: nat] : foldl(dB,dB,app,var(N),Ts) != foldl(dB,dB,app,abs(R),Ss) ).
tff(fact_64_apps__preserves__beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R),S))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Ss)),foldl(dB,dB,app,S,Ss))) ) ).
tff(fact_65_IT_OVar,axiom,
! [N: nat,Rs: list(dB)] :
( listsp(dB,it,Rs)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(N),Rs))) ) ).
tff(fact_66_dB_Osize_I4_J,axiom,
! [Nat1: nat] : size_size(dB,var(Nat1)) = zero_zero(nat) ).
tff(fact_67_IT_Osimps,axiom,
! [A3: dB] :
( pp(aa(dB,bool,it,A3))
<=> ( ? [Rs2: list(dB),N1: nat] :
( ( A3 = foldl(dB,dB,app,var(N1),Rs2) )
& listsp(dB,it,Rs2) )
| ? [R2: dB] :
( ( A3 = abs(R2) )
& pp(aa(dB,bool,it,R2)) )
| ? [R2: dB,S2: dB,Ss2: list(dB)] :
( ( A3 = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R2)),S2),Ss2) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R2,S2,zero_zero(nat)),Ss2)))
& pp(aa(dB,bool,it,S2)) ) ) ) ).
tff(fact_68_apps__preserves__betas,axiom,
! [R: dB,Ss: list(dB),Rs: list(dB)] :
( step1(dB,beta,Rs,Ss)
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Rs)),foldl(dB,dB,app,R,Ss))) ) ).
tff(fact_69_type_Osimps_I5_J,axiom,
! [A: $tType,Nat2: nat,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,atom(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_70_type_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( atom(Nat2) = atom(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_71_type_Osimps_I4_J,axiom,
! [Nat1: nat,Type21: type,Type11: type] : fun1(Type11,Type21) != atom(Nat1) ).
tff(fact_72_type_Osimps_I3_J,axiom,
! [Type21: type,Type11: type,Nat1: nat] : atom(Nat1) != fun1(Type11,Type21) ).
tff(fact_73_head__Var__reduction,axiom,
! [V1: dB,Rs: list(dB),N: nat] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,var(N),Rs)),V1))
=> ? [Ss1: list(dB)] :
( step1(dB,beta,Rs,Ss1)
& ( V1 = foldl(dB,dB,app,var(N),Ss1) ) ) ) ).
tff(fact_74_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_75_type_Osize_I1_J,axiom,
! [Nat1: nat] : type_size(atom(Nat1)) = zero_zero(nat) ).
tff(fact_76_type_Osize_I3_J,axiom,
! [Nat1: nat] : size_size(type,atom(Nat1)) = zero_zero(nat) ).
tff(fact_77_type_Oexhaust,axiom,
! [Y: type] :
( ! [Nat: nat] : Y != atom(Nat)
=> ~ ! [Type1: type,Type2: type] : Y != fun1(Type1,Type2) ) ).
tff(fact_78_apps__betasE,axiom,
! [S: dB,Rs: list(dB),R: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Rs)),S))
=> ( ! [R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R),R1))
=> ( S != foldl(dB,dB,app,R1,Rs) ) )
=> ( ! [Rs1: list(dB)] :
( step1(dB,beta,Rs,Rs1)
=> ( S != foldl(dB,dB,app,R,Rs1) ) )
=> ~ ! [T: dB] :
( ( R = abs(T) )
=> ! [U: dB,Us: list(dB)] :
( ( Rs = cons(dB,U,Us) )
=> ( S != foldl(dB,dB,app,subst(T,U,zero_zero(nat)),Us) ) ) ) ) ) ) ).
tff(fact_79_Cons__step1__Cons,axiom,
! [A: $tType,Xs: list(A),Xa: A,Ys1: list(A),Y2: A,R: fun(A,fun(A,bool))] :
( step1(A,R,cons(A,Y2,Ys1),cons(A,Xa,Xs))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,Y2),Xa))
& ( Xs = Ys1 ) )
| ( ( Xa = Y2 )
& step1(A,R,Ys1,Xs) ) ) ) ).
tff(fact_80_list_Oinject,axiom,
! [A: $tType,List2: list(A),A4: A,List1: list(A),A3: A] :
( ( cons(A,A3,List1) = cons(A,A4,List2) )
<=> ( ( A3 = A4 )
& ( List1 = List2 ) ) ) ).
tff(fact_81_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : cons(A,X,Xs1) != Xs1 ).
tff(fact_82_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != cons(A,X,Xs1) ).
tff(fact_83_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),Xa: B,A3: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A3,cons(B,Xa,Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun(B,A),F,A3),Xa),Xs) ).
tff(fact_84_Cons__step1E,axiom,
! [A: $tType,Xs: list(A),Xa: A,Ys1: list(A),R: fun(A,fun(A,bool))] :
( step1(A,R,Ys1,cons(A,Xa,Xs))
=> ( ! [Y1: A] :
( ( Ys1 = cons(A,Y1,Xs) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Xa)) )
=> ~ ! [Zs: list(A)] :
( ( Ys1 = cons(A,Xa,Zs) )
=> ~ step1(A,R,Zs,Xs) ) ) ) ).
tff(fact_85_foldl__fun__comm,axiom,
! [B: $tType,A: $tType,Xa: A,Xs: list(A),S: B,F: fun(B,fun(A,B))] :
( ! [X1: A,Y1: A,S1: B] : aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S1),X1)),Y1) = aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S1),Y1)),X1)
=> ( aa(A,B,aa(B,fun(A,B),F,foldl(B,A,F,S,Xs)),Xa) = foldl(B,A,F,aa(A,B,aa(B,fun(A,B),F,S),Xa),Xs) ) ) ).
tff(fact_86_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,cons(B,A3,List1)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A3),List1) ).
tff(fact_87_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y: A,Xs1: list(A),X: A] : splice(A,cons(A,X,Xs1),cons(A,Y,Ys)) = cons(A,X,cons(A,Y,splice(A,Xs1,Ys))) ).
tff(fact_88_Abs__eq__apps__conv,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( ( abs(R) = foldl(dB,dB,app,S,Ss) )
<=> ( ( abs(R) = S )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_89_apps__eq__Abs__conv,axiom,
! [R: dB,Ss: list(dB),S: dB] :
( ( foldl(dB,dB,app,S,Ss) = abs(R) )
<=> ( ( S = abs(R) )
& ( Ss = nil(dB) ) ) ) ).
tff(fact_90_listsp_ONil,axiom,
! [A: $tType,A2: fun(A,bool)] : listsp(A,A2,nil(A)) ).
tff(fact_91_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_92_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).
tff(fact_93_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_94_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_95_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,cons(A,V,Va),nil(A)) = cons(A,V,Va) ).
tff(fact_96_list_Osimps_I2_J,axiom,
! [A: $tType,List: list(A),A1: A] : nil(A) != cons(A,A1,List) ).
tff(fact_97_list_Osimps_I3_J,axiom,
! [A: $tType,List: list(A),A1: A] : cons(A,A1,List) != nil(A) ).
tff(fact_98_not__Nil__step1,axiom,
! [A: $tType,Xs: list(A),R: fun(A,fun(A,bool))] : ~ step1(A,R,nil(A),Xs) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
pp(aa(dB,bool,it,var(x))) ).
%------------------------------------------------------------------------------