TPTP Problem File: LCL813_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL813_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 174
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_174 [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 : 165 ( 56 unt; 52 typ; 0 def)
% Number of atoms : 214 ( 75 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 139 ( 38 ~; 8 |; 12 &)
% ( 12 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 36 ( 19 >; 17 *; 0 +; 0 <<)
% Number of predicates : 5 ( 4 usr; 0 prp; 1-3 aty)
% Number of functors : 44 ( 44 usr; 24 con; 0-5 aty)
% Number of variables : 348 ( 314 !; 8 ?; 348 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:41
%------------------------------------------------------------------------------
%----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,
fun1: ( $tType * $tType ) > $tType ).
%----Explicit typings (46)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun1(fun1(B,C),fun1(fun1(A,B),fun1(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun1(A,fun1(B,C)) * B ) > fun1(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun1(A,A) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun1(A,fun1(B,C)) * fun1(A,B) ) > fun1(A,C) ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_InductTermi_OIT,type,
it: fun1(dB,bool) ).
tff(sy_c_Lambda_Obeta,type,
beta: ( dB * dB ) > $o ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
tff(sy_c_Lambda_OdB_OApp,type,
app: fun1(dB,fun1(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: fun1(dB,fun1(nat,dB)) ).
tff(sy_c_Lambda_Osubst,type,
subst: fun1(dB,fun1(dB,fun1(nat,dB))) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun1(B,fun1(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( fun1(A,fun1(B,B)) * list(A) * B ) > B ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T5: $tType,A: $tType] : ( ( T5 * fun1(A,fun1(list(A),T5)) * list(A) ) > T5 ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun1(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun1(A,B) * list(A) ) > list(B) ) ).
tff(sy_c_Type_Oshift,type,
shift:
!>[A: $tType] : ( ( fun1(nat,A) * nat * A ) > fun1(nat,A) ) ).
tff(sy_c_Type_Otype_OFun,type,
fun: fun1(type,fun1(type,type)) ).
tff(sy_c_Type_Otyping,type,
typing: fun1(nat,type) > fun1(dB,fun1(type,bool)) ).
tff(sy_c_Type_Otypings,type,
typings: ( fun1(nat,type) * list(dB) ) > fun1(list(type),bool) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun1(A,B) * A ) > B ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun1(fun1(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun1(bool,fun1(bool,bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_T_H_H____,type,
t2: type ).
tff(sy_v_T_H____,type,
t: type ).
tff(sy_v_T_Ha______,type,
t_a: type ).
tff(sy_v_T____,type,
t1: type ).
tff(sy_v_Ts____,type,
ts: list(type) ).
tff(sy_v_a____,type,
a: dB ).
tff(sy_v_as____,type,
as: list(dB) ).
tff(sy_v_e____,type,
e: fun1(nat,type) ).
tff(sy_v_ea______,type,
ea: fun1(nat,type) ).
tff(sy_v_i____,type,
i: nat ).
tff(sy_v_ia______,type,
ia: nat ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_rs____,type,
rs: list(dB) ).
tff(sy_v_t____,type,
t3: 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,t3)) ).
tff(fact_1_typing_OVar,axiom,
! [Ta1: type,X1: nat,Env: fun1(nat,type)] :
( ( aa(nat,type,Env,X1) = Ta1 )
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Env),var(X1)),Ta1)) ) ).
tff(fact_2_typing__elims_I1_J,axiom,
! [Ta1: type,Ib: nat,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),var(Ib)),Ta1))
=> ( aa(nat,type,Eb,Ib) = Ta1 ) ) ).
tff(fact_3_uT_H,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(e),u),aa(type,type,aa(type,fun1(type,type),fun,t2),foldr(type,type,fun,ts,t)))) ).
tff(fact_4_type_Osimps_I2_J,axiom,
! [Type21: type,Type11: type,Type2: type,Type1: type] :
( ( aa(type,type,aa(type,fun1(type,type),fun,Type1),Type2) = aa(type,type,aa(type,fun1(type,type),fun,Type11),Type21) )
<=> ( ( Type1 = Type11 )
& ( Type2 = Type21 ) ) ) ).
tff(fact_5_dB_Osimps_I1_J,axiom,
! [Nat2: nat,Nat1: nat] :
( ( var(Nat1) = var(Nat2) )
<=> ( Nat1 = Nat2 ) ) ).
tff(fact_6_T,axiom,
t1 = aa(type,type,aa(type,fun1(type,type),fun,t2),foldr(type,type,fun,ts,t)) ).
tff(fact_7_varT,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),var(n)),aa(type,type,aa(type,fun1(type,type),fun,t2),foldr(type,type,fun,ts,t)))) ).
tff(fact_8_uT,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(e),u),t1)) ).
tff(fact_9_Var_I3_J,axiom,
pp(aa(dB,bool,it,ua)) ).
tff(fact_10_shift__eq,axiom,
! [A: $tType,Ta1: A,Eb: fun1(nat,A),J1: nat,Ib: nat] :
( ( Ib = J1 )
=> ( aa(nat,A,shift(A,Eb,Ib,Ta1),J1) = Ta1 ) ) ).
tff(fact_11_Abs,axiom,
! [Ua: type,Ta: dB,Ta1: type,Env: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Env,zero_zero(nat),Ta1)),Ta),Ua))
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Env),abs(Ta)),aa(type,type,aa(type,fun1(type,type),fun,Ta1),Ua))) ) ).
tff(fact_12_uIT,axiom,
pp(aa(dB,bool,it,u)) ).
tff(fact_13_True,axiom,
n = i ).
tff(fact_14_dB_Osimps_I3_J,axiom,
! [DB4: dB,DB3: dB] :
( ( abs(DB3) = abs(DB4) )
<=> ( DB3 = DB4 ) ) ).
tff(fact_15_argT,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),a),t2)) ).
tff(fact_16_dB_Osimps_I6_J,axiom,
! [DB: dB,Nat: nat] : var(Nat) != abs(DB) ).
tff(fact_17_dB_Osimps_I7_J,axiom,
! [Nat: nat,DB: dB] : abs(DB) != var(Nat) ).
tff(fact_18_Var__IT,axiom,
! [N1: nat] : pp(aa(dB,bool,it,var(N1))) ).
tff(fact_19_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_20_typing__elims_I3_J,axiom,
! [Ta1: type,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),abs(Ta)),Ta1))
=> ~ ! [T3: type,U1: type] :
( ( Ta1 = aa(type,type,aa(type,fun1(type,type),fun,T3),U1) )
=> ~ pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Eb,zero_zero(nat),T3)),Ta),U1)) ) ) ).
tff(fact_21__096_B_Bthesis_O_A_I_B_BT_H_H_O_A_091_124_Ae_060i_058T_062_A_092_060turnstile_062_AVar_An_A_058_AT_H_H_A_092_060Rightarrow_062_ATs_A_061_062_062_AT_H_059_Ae_060i_058T_062_A_092_060turnstile_062_Aa_A_058_AT_H_H_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [T4: type] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),var(n)),aa(type,type,aa(type,fun1(type,type),fun,T4),foldr(type,type,fun,ts,t))))
=> ~ pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),a),T4)) ) ).
tff(fact_22_headT,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),aa(dB,dB,aa(dB,fun1(dB,dB),app,var(n)),a)),foldr(type,type,fun,ts,t))) ).
tff(fact_23_argsT,axiom,
pp(aa(list(type),bool,typings(shift(type,e,i,t1),as),ts)) ).
tff(fact_24_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_25_MI2,axiom,
! [Ub: dB,Ta1: type,Ib: nat,Eb: fun1(nat,type),Ta: dB,T21: type,T11: type] :
( ( t1 = aa(type,type,aa(type,fun1(type,type),fun,T11),T21) )
=> ( pp(aa(dB,bool,it,Ta))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Eb,Ib,T21)),Ta),Ta1))
=> ( pp(aa(dB,bool,it,Ub))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ub),T21))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,Ta),Ub),Ib))) ) ) ) ) ) ).
tff(fact_26_MI1,axiom,
! [Ub: dB,Ta1: type,Ib: nat,Eb: fun1(nat,type),Ta: dB,T21: type,T11: type] :
( ( t1 = aa(type,type,aa(type,fun1(type,type),fun,T11),T21) )
=> ( pp(aa(dB,bool,it,Ta))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Eb,Ib,T11)),Ta),Ta1))
=> ( pp(aa(dB,bool,it,Ub))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ub),T11))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,Ta),Ub),Ib))) ) ) ) ) ) ).
tff(fact_27_abs__typeE,axiom,
! [Ta1: type,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),abs(Ta)),Ta1))
=> ~ ! [U1: type,V: type] : ~ pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Eb,zero_zero(nat),U1)),Ta),V)) ) ).
tff(fact_28_nT,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),foldl(dB,dB,app,var(n),rs)),t)) ).
tff(fact_29_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB22: dB,DB12: dB] :
( ( aa(dB,dB,aa(dB,fun1(dB,dB),app,DB12),DB22) = aa(dB,dB,aa(dB,fun1(dB,dB),app,DB13),DB23) )
<=> ( ( DB12 = DB13 )
& ( DB22 = DB23 ) ) ) ).
tff(fact_30_subst__App,axiom,
! [K: nat,S: dB,U: dB,T: dB] : aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,aa(dB,dB,aa(dB,fun1(dB,dB),app,T),U)),S),K) = aa(dB,dB,aa(dB,fun1(dB,dB),app,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,T),S),K)),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,U),S),K)) ).
tff(fact_31__096IT_A_Iu_A_092_060degree_062_Aa_091u_Pi_093_J_096,axiom,
pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun1(dB,dB),app,u),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,a),u),i)))) ).
tff(fact_32_App,axiom,
! [Ta: dB,Ua: type,Ta1: type,S3: dB,Env: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Env),S3),aa(type,type,aa(type,fun1(type,type),fun,Ta1),Ua)))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Env),Ta),Ta1))
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Env),aa(dB,dB,aa(dB,fun1(dB,dB),app,S3),Ta)),Ua)) ) ) ).
tff(fact_33__096e_060i_058T_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_092_060degree_062_092_060degree_062_Aas_A_058_AT_H_096,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),foldl(dB,dB,app,aa(dB,dB,aa(dB,fun1(dB,dB),app,var(n)),a),as)),t)) ).
tff(fact_34_Beta,axiom,
! [Ss1: list(dB),S3: dB,R2: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,R2),S3),zero_zero(nat)),Ss1)))
=> ( pp(aa(dB,bool,it,S3))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun1(dB,dB),app,abs(R2)),S3),Ss1))) ) ) ).
tff(fact_35_Cons,axiom,
rs = cons(dB,a,as) ).
tff(fact_36__096_B_Bthesis_O_A_I_B_BTs_O_A_091_124_Ae_060i_058T_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_058_ATs_A_061_062_062_AT_H_059_Ae_060i_058T_062_A_124_124_N_Aas_A_058_ATs_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Ts1: list(type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,i,t1)),aa(dB,dB,aa(dB,fun1(dB,dB),app,var(n)),a)),foldr(type,type,fun,Ts1,t)))
=> ~ pp(aa(list(type),bool,typings(shift(type,e,i,t1),as),Ts1)) ) ).
tff(fact_37_var__app__type__eq,axiom,
! [Ua: type,Ta1: type,Ts: list(dB),Ib: nat,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,var(Ib),Ts)),Ta1))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,var(Ib),Ts)),Ua))
=> ( Ta1 = Ua ) ) ) ).
tff(fact_38_list__app__typeI,axiom,
! [Ts: list(dB),Ta1: type,Tsa: list(type),Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),foldr(type,type,fun,Tsa,Ta1)))
=> ( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,Ta,Ts)),Ta1)) ) ) ).
tff(fact_39_dB_Osimps_I4_J,axiom,
! [DB21: dB,DB11: dB,Nat: nat] : var(Nat) != aa(dB,dB,aa(dB,fun1(dB,dB),app,DB11),DB21) ).
tff(fact_40_dB_Osimps_I5_J,axiom,
! [Nat: nat,DB21: dB,DB11: dB] : aa(dB,dB,aa(dB,fun1(dB,dB),app,DB11),DB21) != var(Nat) ).
tff(fact_41_dB_Osimps_I8_J,axiom,
! [DB: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun1(dB,dB),app,DB1),DB2) != abs(DB) ).
tff(fact_42_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB: dB] : abs(DB) != aa(dB,dB,aa(dB,fun1(dB,dB),app,DB1),DB2) ).
tff(fact_43_subst__eq,axiom,
! [U: dB,K: nat] : aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,var(K)),U),K) = U ).
tff(fact_44_app__Var__IT,axiom,
! [I: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun1(dB,dB),app,T),var(I)))) ) ).
tff(fact_45_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,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,R3),var(I)),J))) ) ).
tff(fact_46_subst__lemma,axiom,
! [Ib: nat,Ua: type,Ub: dB,E: fun1(nat,type),Ta1: type,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),Ta1))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(E),Ub),Ua))
=> ( ( Eb = shift(type,E,Ib,Ua) )
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(E),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,Ta),Ub),Ib)),Ta1)) ) ) ) ).
tff(fact_47__096IT_A_I_Ilift_Au_A0_A_092_060degree_062_AVar_A0_J_091a_091u_Pi_093_P0_093_J_096,axiom,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,aa(dB,dB,aa(dB,fun1(dB,dB),app,aa(nat,dB,aa(dB,fun1(nat,dB),lift,u),zero_zero(nat))),var(zero_zero(nat)))),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,a),u),i)),zero_zero(nat)))) ).
tff(fact_48_typing__elims_I2_J,axiom,
! [Ta1: type,Ub: dB,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),aa(dB,dB,aa(dB,fun1(dB,dB),app,Ta),Ub)),Ta1))
=> ~ ! [T3: type] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),aa(type,type,aa(type,fun1(type,type),fun,T3),Ta1)))
=> ~ pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ub),T3)) ) ) ).
tff(fact_49_var__app__typesE,axiom,
! [Ta1: type,Ts: list(dB),Ib: nat,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,var(Ib),Ts)),Ta1))
=> ~ ! [Ts1: list(type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),var(Ib)),foldr(type,type,fun,Ts1,Ta1)))
=> ~ pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).
tff(fact_50_var__app__types,axiom,
! [Ua: type,Tsa: list(type),Ta1: type,Us: list(dB),Ts: list(dB),Ib: nat,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,foldl(dB,dB,app,var(Ib),Ts),Us)),Ta1))
=> ( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
=> ( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,var(Ib),Ts)),Ua))
=> ? [Us1: list(type)] :
( ( Ua = foldr(type,type,fun,Us1,Ta1) )
& pp(aa(list(type),bool,typings(Eb,Us),Us1)) ) ) ) ) ).
tff(fact_51_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss1: list(dB),S3: dB,Rsa: list(dB),R2: dB] :
( ( foldl(dB,dB,app,abs(R2),Rsa) = foldl(dB,dB,app,abs(S3),Ss1) )
<=> ( ( R2 = S3 )
& ( Rsa = Ss1 ) ) ) ).
tff(fact_52_Var__apps__eq__Var__apps__conv,axiom,
! [Ss1: list(dB),Na: nat,Rsa: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rsa) = foldl(dB,dB,app,var(Na),Ss1) )
<=> ( ( M = Na )
& ( Rsa = Ss1 ) ) ) ).
tff(fact_53_apps__eq__tail__conv,axiom,
! [S3: dB,Ts: list(dB),R2: dB] :
( ( foldl(dB,dB,app,R2,Ts) = foldl(dB,dB,app,S3,Ts) )
<=> ( R2 = S3 ) ) ).
tff(fact_54_lift_Osimps_I2_J,axiom,
! [K: nat,T: dB,S: dB] : aa(nat,dB,aa(dB,fun1(nat,dB),lift,aa(dB,dB,aa(dB,fun1(dB,dB),app,S),T)),K) = aa(dB,dB,aa(dB,fun1(dB,dB),app,aa(nat,dB,aa(dB,fun1(nat,dB),lift,S),K)),aa(nat,dB,aa(dB,fun1(nat,dB),lift,T),K)) ).
tff(fact_55_lift__IT,axiom,
! [I: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun1(nat,dB),lift,T),I))) ) ).
tff(fact_56_lift__type,axiom,
! [Ua: type,Ib: nat,Ta1: type,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),Ta1))
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,Eb,Ib,Ua)),aa(nat,dB,aa(dB,fun1(nat,dB),lift,Ta),Ib)),Ta1)) ) ).
tff(fact_57_subst__lift,axiom,
! [S: dB,K: nat,T: dB] : aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,aa(nat,dB,aa(dB,fun1(nat,dB),lift,T),K)),S),K) = T ).
tff(fact_58_Abs__App__neq__Var__apps,axiom,
! [Ss1: list(dB),Na: nat,Ta: dB,S3: dB] : aa(dB,dB,aa(dB,fun1(dB,dB),app,abs(S3)),Ta) != foldl(dB,dB,app,var(Na),Ss1) ).
tff(fact_59_Var__apps__neq__Abs__apps,axiom,
! [Ss1: list(dB),R2: dB,Ts: list(dB),Na: nat] : foldl(dB,dB,app,var(Na),Ts) != foldl(dB,dB,app,abs(R2),Ss1) ).
tff(fact_60_list__app__typeE,axiom,
! [Ta1: type,Ts: list(dB),Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,Ta,Ts)),Ta1))
=> ~ ! [Ts1: list(type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),foldr(type,type,fun,Ts1,Ta1)))
=> ~ pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).
tff(fact_61_list__app__typeD,axiom,
! [Ta1: type,Ts: list(dB),Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),foldl(dB,dB,app,Ta,Ts)),Ta1))
=> ? [Ts1: list(type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),foldr(type,type,fun,Ts1,Ta1)))
& pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).
tff(fact_62_list_Oinject,axiom,
! [A: $tType,List1: list(A),A2: A,List: list(A),Aa: A] :
( ( cons(A,Aa,List) = cons(A,A2,List1) )
<=> ( ( Aa = A2 )
& ( List = List1 ) ) ) ).
tff(fact_63_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(var(Nat)) = zero_zero(nat) ).
tff(fact_64_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : cons(A,X,Xs1) != Xs1 ).
tff(fact_65_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != cons(A,X,Xs1) ).
tff(fact_66_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,Aa: A,F: fun1(A,fun1(B,A))] : foldl(A,B,F,Aa,cons(B,X1,Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun1(B,A),F,Aa),X1),Xs) ).
tff(fact_67_foldr_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Aa: A,Xs: list(B),X1: B,F: fun1(B,fun1(A,A))] : foldr(B,A,F,cons(B,X1,Xs),Aa) = aa(A,A,aa(B,fun1(A,A),F,X1),foldr(B,A,F,Xs,Aa)) ).
tff(fact_68_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_69_Var_I2_J,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,ea,ia,t1)),foldl(dB,dB,app,var(n),rs)),t_a)) ).
tff(fact_70_beta,axiom,
! [T: dB,S: dB] : beta(aa(dB,dB,aa(dB,fun1(dB,dB),app,abs(S)),T),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,S),T),zero_zero(nat))) ).
tff(fact_71_Var_I4_J,axiom,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(ea),ua),t1)) ).
tff(fact_72_listsp__conj__eq,axiom,
! [A: $tType,B1: fun1(A,bool),A1: fun1(A,bool),X2: list(A)] :
( listsp(A,combs(A,bool,bool,aa(fun1(A,bool),fun1(A,fun1(bool,bool)),aa(fun1(bool,fun1(bool,bool)),fun1(fun1(A,bool),fun1(A,fun1(bool,bool))),combb(bool,fun1(bool,bool),A),fconj),A1),B1),X2)
<=> ( listsp(A,A1,X2)
& listsp(A,B1,X2) ) ) ).
tff(fact_73_appL,axiom,
! [U: dB,T: dB,S: dB] :
( beta(S,T)
=> beta(aa(dB,dB,aa(dB,fun1(dB,dB),app,S),U),aa(dB,dB,aa(dB,fun1(dB,dB),app,T),U)) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun1(A,B),F: fun1(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_75_appR,axiom,
! [U: dB,T: dB,S: dB] :
( beta(S,T)
=> beta(aa(dB,dB,aa(dB,fun1(dB,dB),app,U),S),aa(dB,dB,aa(dB,fun1(dB,dB),app,U),T)) ) ).
tff(fact_76_beta__cases_I1_J,axiom,
! [T: dB,I: nat] : ~ beta(var(I),T) ).
tff(fact_77_abs,axiom,
! [T: dB,S: dB] :
( beta(S,T)
=> beta(abs(S),abs(T)) ) ).
tff(fact_78_subst__preserves__beta,axiom,
! [I: nat,T: dB,S: dB,R3: dB] :
( beta(R3,S)
=> beta(aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,R3),T),I),aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,S),T),I)) ) ).
tff(fact_79_subject__reduction,axiom,
! [T2: dB,Ta1: type,Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ta),Ta1))
=> ( beta(Ta,T2)
=> pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),T2),Ta1)) ) ) ).
tff(fact_80_lift__preserves__beta,axiom,
! [I: nat,S: dB,R3: dB] :
( beta(R3,S)
=> beta(aa(nat,dB,aa(dB,fun1(nat,dB),lift,R3),I),aa(nat,dB,aa(dB,fun1(nat,dB),lift,S),I)) ) ).
tff(fact_81__096listsp_A_I_Ft_O_AEX_AT_H_O_Ae_060i_058T_062_A_092_060turnstile_062_At_A_058_AT_H_J_Aas_096,axiom,
listsp(dB,aa(fun1(dB,fun1(type,bool)),fun1(dB,bool),aa(fun1(fun1(type,bool),bool),fun1(fun1(dB,fun1(type,bool)),fun1(dB,bool)),combb(fun1(type,bool),bool,dB),fEx(type)),typing(shift(type,e,i,t1))),as) ).
tff(fact_82__096IT_A_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_096,axiom,
pp(aa(dB,bool,it,foldl(dB,dB,app,var(zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun1(nat,dB),subst,u),i),as))))) ).
tff(fact_83_map__ident,axiom,
! [A: $tType,X2: list(A)] : map(A,A,combi(A),X2) = X2 ).
tff(fact_84_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,F: fun1(B,A)] : map(B,A,F,cons(B,X1,Xs)) = cons(A,aa(B,A,F,X1),map(B,A,F,Xs)) ).
tff(fact_85__096listsp_AIT_A_Imap_A_I_Ft_O_Alift_At_A0_J_A_Imap_A_I_Ft_O_At_091u_Pi_093_J_Aas_J_J_096,axiom,
listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun1(nat,dB),subst,u),i),as))) ).
tff(fact_86_subst__map,axiom,
! [Ib: nat,Ub: dB,Ts: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,foldl(dB,dB,app,Ta,Ts)),Ub),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,Ta),Ub),Ib),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun1(nat,dB),subst,Ub),Ib),Ts)) ).
tff(fact_87_lift__map,axiom,
! [Ib: nat,Ts: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun1(nat,dB),lift,foldl(dB,dB,app,Ta,Ts)),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun1(nat,dB),lift,Ta),Ib),map(dB,dB,combc(dB,nat,dB,lift,Ib),Ts)) ).
tff(fact_88_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun1(C,B),Aa: A,G: fun1(A,fun1(B,A))] : foldl(A,B,G,Aa,map(C,B,F,Xs)) = foldl(A,C,combc(A,fun1(C,B),fun1(C,A),aa(fun1(A,fun1(B,A)),fun1(A,fun1(fun1(C,B),fun1(C,A))),aa(fun1(fun1(B,A),fun1(fun1(C,B),fun1(C,A))),fun1(fun1(A,fun1(B,A)),fun1(A,fun1(fun1(C,B),fun1(C,A)))),combb(fun1(B,A),fun1(fun1(C,B),fun1(C,A)),A),combb(B,A,C)),G),F),Aa,Xs) ).
tff(fact_89_lift__types,axiom,
! [Ua: type,Ib: nat,Tsa: list(type),Ts: list(dB),Eb: fun1(nat,type)] :
( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
=> pp(aa(list(type),bool,typings(shift(type,Eb,Ib,Ua),map(dB,dB,combc(dB,nat,dB,lift,Ib),Ts)),Tsa)) ) ).
tff(fact_90_lists__typings,axiom,
! [Tsa: list(type),Ts: list(dB),Eb: fun1(nat,type)] :
( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
=> listsp(dB,aa(fun1(dB,fun1(type,bool)),fun1(dB,bool),aa(fun1(fun1(type,bool),bool),fun1(fun1(dB,fun1(type,bool)),fun1(dB,bool)),combb(fun1(type,bool),bool,dB),fEx(type)),typing(Eb)),Ts) ) ).
tff(fact_91_substs__lemma,axiom,
! [Tsa: list(type),Ts: list(dB),Ib: nat,Ta1: type,Ub: dB,Eb: fun1(nat,type)] :
( pp(aa(type,bool,aa(dB,fun1(type,bool),typing(Eb),Ub),Ta1))
=> ( pp(aa(list(type),bool,typings(shift(type,Eb,Ib,Ta1),Ts),Tsa))
=> pp(aa(list(type),bool,typings(Eb,map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun1(nat,dB),subst,Ub),Ib),Ts)),Tsa)) ) ) ).
tff(fact_92_lifts__IT,axiom,
! [Ts: list(dB)] :
( listsp(dB,it,Ts)
=> listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),Ts)) ) ).
tff(fact_93_typings_Osimps_I2_J,axiom,
! [Tsa: list(type),Ts: list(dB),Ta: dB,Eb: fun1(nat,type)] :
( pp(aa(list(type),bool,typings(Eb,cons(dB,Ta,Ts)),Tsa))
<=> pp(list_case(bool,type,fFalse,combc(type,fun1(list(type),bool),fun1(list(type),bool),aa(fun1(type,fun1(bool,bool)),fun1(type,fun1(fun1(list(type),bool),fun1(list(type),bool))),aa(fun1(fun1(bool,bool),fun1(fun1(list(type),bool),fun1(list(type),bool))),fun1(fun1(type,fun1(bool,bool)),fun1(type,fun1(fun1(list(type),bool),fun1(list(type),bool)))),combb(fun1(bool,bool),fun1(fun1(list(type),bool),fun1(list(type),bool)),type),combb(bool,bool,list(type))),aa(fun1(type,bool),fun1(type,fun1(bool,bool)),aa(fun1(bool,fun1(bool,bool)),fun1(fun1(type,bool),fun1(type,fun1(bool,bool))),combb(bool,fun1(bool,bool),type),fconj),aa(dB,fun1(type,bool),typing(Eb),Ta))),typings(Eb,Ts)),Tsa)) ) ).
tff(fact_94_beta__cases_I2_J,axiom,
! [S: dB,R3: dB] :
( beta(abs(R3),S)
=> ~ ! [T1: dB] :
( ( S = abs(T1) )
=> ~ beta(R3,T1) ) ) ).
tff(fact_95_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun1(B,fun1(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,Aa,List)) = aa(list(B),A,aa(B,fun1(list(B),A),F2,Aa),List) ).
tff(fact_96_apps__preserves__beta,axiom,
! [Ss1: list(dB),S3: dB,R2: dB] :
( beta(R2,S3)
=> beta(foldl(dB,dB,app,R2,Ss1),foldl(dB,dB,app,S3,Ss1)) ) ).
tff(fact_97_IT_Osimps,axiom,
! [Aa: dB] :
( pp(aa(dB,bool,it,Aa))
<=> ( ? [Rs: list(dB),N: nat] :
( ( Aa = foldl(dB,dB,app,var(N),Rs) )
& listsp(dB,it,Rs) )
| ? [R1: dB] :
( ( Aa = abs(R1) )
& pp(aa(dB,bool,it,R1)) )
| ? [R1: dB,S2: dB,Ss: list(dB)] :
( ( Aa = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun1(dB,dB),app,abs(R1)),S2),Ss) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,R1),S2),zero_zero(nat)),Ss)))
& pp(aa(dB,bool,it,S2)) ) ) ) ).
tff(fact_98_beta__cases_I3_J,axiom,
! [U: dB,T: dB,S: dB] :
( beta(aa(dB,dB,aa(dB,fun1(dB,dB),app,S),T),U)
=> ( ! [S1: dB] :
( ( U = aa(nat,dB,aa(dB,fun1(nat,dB),aa(dB,fun1(dB,fun1(nat,dB)),subst,S1),T),zero_zero(nat)) )
=> ( S != abs(S1) ) )
=> ( ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun1(dB,dB),app,T1),T) )
=> ~ beta(S,T1) )
=> ~ ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun1(dB,dB),app,S),T1) )
=> ~ beta(T,T1) ) ) ) ) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (12)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fEx_1_1_U,axiom,
! [A: $tType,X: A,P: fun1(A,bool)] :
( ~ pp(aa(A,bool,P,X))
| pp(aa(fun1(A,bool),bool,fEx(A),P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun1(A,B),P: fun1(B,C)] : aa(A,C,aa(fun1(A,B),fun1(A,C),aa(fun1(B,C),fun1(fun1(A,B),fun1(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: fun1(A,fun1(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun1(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: fun1(A,B),P: fun1(A,fun1(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun1(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,fun1(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun1(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun1(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
pp(aa(type,bool,aa(dB,fun1(type,bool),typing(shift(type,e,zero_zero(nat),foldr(type,type,fun,ts,t))),var(zero_zero(nat))),foldr(type,type,fun,ts,t))) ).
%------------------------------------------------------------------------------