TPTP Problem File: LCL782_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL782_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 111
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_111 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 171 ( 53 unt; 56 typ; 0 def)
% Number of atoms : 236 ( 110 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 178 ( 57 ~; 16 |; 19 &)
% ( 19 <=>; 67 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 58 ( 29 >; 29 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-4 aty)
% Number of functors : 45 ( 45 usr; 20 con; 0-5 aty)
% Number of variables : 388 ( 351 !; 11 ?; 388 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:35
%------------------------------------------------------------------------------
%----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 (50)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,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_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_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_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
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_Olist_Olist__size,type,
list_size:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
tff(sy_c_List_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_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_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_T_H____,type,
t1: type ).
tff(sy_v_T_Ha______,type,
t_a: type ).
tff(sy_v_T____,type,
t: type ).
tff(sy_v_e____,type,
e: fun(nat,type) ).
tff(sy_v_ea______,type,
ea: fun(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,
t2: 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,t2)) ).
tff(fact_1_Var_I3_J,axiom,
pp(aa(dB,bool,it,ua)) ).
tff(fact_2_uIT,axiom,
pp(aa(dB,bool,it,u)) ).
tff(fact_3_Var__IT,axiom,
! [N: nat] : pp(aa(dB,bool,it,var(N))) ).
tff(fact_4_app__Var__IT,axiom,
! [I: 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(I)))) ) ).
tff(fact_5_subst__Var__IT,axiom,
! [J1: nat,I: nat,R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,subst(R3,var(I),J1))) ) ).
tff(fact_6_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_7_subst__App,axiom,
! [K: nat,S1: dB,U1: dB,T1: dB] : subst(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,subst(T1,S1,K)),subst(U1,S1,K)) ).
tff(fact_8_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_9_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_10_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_11_subst__eq,axiom,
! [U1: dB,K: nat] : subst(var(K),U1,K) = U1 ).
tff(fact_12_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_13_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_14_nT,axiom,
typing(shift(type,e,i,t),foldl(dB,dB,app,var(n),rs),t1) ).
tff(fact_15_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_16_uT,axiom,
typing(e,u,t) ).
tff(fact_17_Var__eq__apps__conv,axiom,
! [Ss1: list(dB),S: dB,M: nat] :
( ( var(M) = foldl(dB,dB,app,S,Ss1) )
<=> ( ( var(M) = S )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_18_MI2,axiom,
! [Ub: dB,Ta1: type,Ib: nat,Eb: fun(nat,type),Ta: dB,T21: type,T11: type] :
( ( t = fun1(T11,T21) )
=> ( pp(aa(dB,bool,it,Ta))
=> ( typing(shift(type,Eb,Ib,T21),Ta,Ta1)
=> ( pp(aa(dB,bool,it,Ub))
=> ( typing(Eb,Ub,T21)
=> pp(aa(dB,bool,it,subst(Ta,Ub,Ib))) ) ) ) ) ) ).
tff(fact_19_MI1,axiom,
! [Ub: dB,Ta1: type,Ib: nat,Eb: fun(nat,type),Ta: dB,T21: type,T11: type] :
( ( t = fun1(T11,T21) )
=> ( pp(aa(dB,bool,it,Ta))
=> ( typing(shift(type,Eb,Ib,T11),Ta,Ta1)
=> ( pp(aa(dB,bool,it,Ub))
=> ( typing(Eb,Ub,T11)
=> pp(aa(dB,bool,it,subst(Ta,Ub,Ib))) ) ) ) ) ) ).
tff(fact_20_typing__elims_I1_J,axiom,
! [Ta1: type,Ib: nat,Eb: fun(nat,type)] :
( typing(Eb,var(Ib),Ta1)
=> ( aa(nat,type,Eb,Ib) = Ta1 ) ) ).
tff(fact_21_typing_OVar,axiom,
! [Ta1: type,X1: nat,Env: fun(nat,type)] :
( ( aa(nat,type,Env,X1) = Ta1 )
=> typing(Env,var(X1),Ta1) ) ).
tff(fact_22_listsp_ONil,axiom,
! [A: $tType,A2: fun(A,bool)] : listsp(A,A2,nil(A)) ).
tff(fact_23_var__app__type__eq,axiom,
! [Ua: type,Ta1: type,Ts: list(dB),Ib: nat,Eb: fun(nat,type)] :
( typing(Eb,foldl(dB,dB,app,var(Ib),Ts),Ta1)
=> ( typing(Eb,foldl(dB,dB,app,var(Ib),Ts),Ua)
=> ( Ta1 = Ua ) ) ) ).
tff(fact_24_subst__lemma,axiom,
! [Ib: nat,Ua: type,Ub: dB,E: fun(nat,type),Ta1: type,Ta: dB,Eb: fun(nat,type)] :
( typing(Eb,Ta,Ta1)
=> ( typing(E,Ub,Ua)
=> ( ( Eb = shift(type,E,Ib,Ua) )
=> typing(E,subst(Ta,Ub,Ib),Ta1) ) ) ) ).
tff(fact_25_foldl__Nil,axiom,
! [B: $tType,A: $tType,A3: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A3,nil(B)) = A3 ).
tff(fact_26_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_27_App,axiom,
! [Ta: dB,Ua: type,Ta1: type,S: dB,Env: fun(nat,type)] :
( typing(Env,S,fun1(Ta1,Ua))
=> ( typing(Env,Ta,Ta1)
=> typing(Env,aa(dB,dB,aa(dB,fun(dB,dB),app,S),Ta),Ua) ) ) ).
tff(fact_28_shift__eq,axiom,
! [A: $tType,Ta1: A,Eb: fun(nat,A),J: nat,Ib: nat] :
( ( Ib = J )
=> ( aa(nat,A,shift(A,Eb,Ib,Ta1),J) = Ta1 ) ) ).
tff(fact_29_typing__elims_I2_J,axiom,
! [Ta1: type,Ub: dB,Ta: dB,Eb: fun(nat,type)] :
( typing(Eb,aa(dB,dB,aa(dB,fun(dB,dB),app,Ta),Ub),Ta1)
=> ~ ! [T3: type] :
( typing(Eb,Ta,fun1(T3,Ta1))
=> ~ typing(Eb,Ub,T3) ) ) ).
tff(fact_30_Var_I2_J,axiom,
typing(shift(type,ea,ia,t),foldl(dB,dB,app,var(n),rs),t_a) ).
tff(fact_31_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_32_foldl__fun__comm,axiom,
! [B: $tType,A: $tType,X1: A,Xs: list(A),S: B,F: fun(B,fun(A,B))] :
( ! [X2: A,Y1: A,S2: B] : aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),X2)),Y1) = aa(A,B,aa(B,fun(A,B),F,aa(A,B,aa(B,fun(A,B),F,S2),Y1)),X2)
=> ( aa(A,B,aa(B,fun(A,B),F,foldl(B,A,F,S,Xs)),X1) = foldl(B,A,F,aa(A,B,aa(B,fun(A,B),F,S),X1),Xs) ) ) ).
tff(fact_33_Abs__eq__apps__conv,axiom,
! [Ss1: list(dB),S: dB,R1: dB] :
( ( abs(R1) = foldl(dB,dB,app,S,Ss1) )
<=> ( ( abs(R1) = S )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_34_apps__eq__Abs__conv,axiom,
! [R1: dB,Ss1: list(dB),S: dB] :
( ( foldl(dB,dB,app,S,Ss1) = abs(R1) )
<=> ( ( S = abs(R1) )
& ( Ss1 = nil(dB) ) ) ) ).
tff(fact_35_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_36_Var_I4_J,axiom,
typing(ea,ua,t) ).
tff(fact_37_dB_Osimps_I3_J,axiom,
! [DB4: dB,DB3: dB] :
( ( abs(DB3) = abs(DB4) )
<=> ( DB3 = DB4 ) ) ).
tff(fact_38_listsp__conj__eq,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool),X3: list(A)] :
( listsp(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,A2),B1),X3)
<=> ( listsp(A,A2,X3)
& listsp(A,B1,X3) ) ) ).
tff(fact_39_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_40_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss1: list(dB),S: dB,Rsa: list(dB),R1: dB] :
( ( foldl(dB,dB,app,abs(R1),Rsa) = foldl(dB,dB,app,abs(S),Ss1) )
<=> ( ( R1 = S )
& ( Rsa = Ss1 ) ) ) ).
tff(fact_41_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_42_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_43_dB_Osimps_I6_J,axiom,
! [DB: dB,Nat1: nat] : var(Nat1) != abs(DB) ).
tff(fact_44_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB: dB] : abs(DB) != var(Nat1) ).
tff(fact_45_Var__apps__neq__Abs__apps,axiom,
! [Ss1: list(dB),R1: dB,Ts: list(dB),Na: nat] : foldl(dB,dB,app,var(Na),Ts) != foldl(dB,dB,app,abs(R1),Ss1) ).
tff(fact_46_Abs__App__neq__Var__apps,axiom,
! [Ss1: 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),Ss1) ).
tff(fact_47_Beta,axiom,
! [Ss1: list(dB),S: dB,R1: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R1,S,zero_zero(nat)),Ss1)))
=> ( pp(aa(dB,bool,it,S))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R1)),S),Ss1))) ) ) ).
tff(fact_48_Abs,axiom,
! [Ua: type,Ta: dB,Ta1: type,Env: fun(nat,type)] :
( typing(shift(type,Env,zero_zero(nat),Ta1),Ta,Ua)
=> typing(Env,abs(Ta),fun1(Ta1,Ua)) ) ).
tff(fact_49_typing__elims_I3_J,axiom,
! [Ta1: type,Ta: dB,Eb: fun(nat,type)] :
( typing(Eb,abs(Ta),Ta1)
=> ~ ! [T3: type,U2: type] :
( ( Ta1 = fun1(T3,U2) )
=> ~ typing(shift(type,Eb,zero_zero(nat),T3),Ta,U2) ) ) ).
tff(fact_50_IT_Osimps,axiom,
! [A3: dB] :
( pp(aa(dB,bool,it,A3))
<=> ( ? [Rs1: list(dB),N1: nat] :
( ( A3 = foldl(dB,dB,app,var(N1),Rs1) )
& listsp(dB,it,Rs1) )
| ? [R4: dB] :
( ( A3 = abs(R4) )
& pp(aa(dB,bool,it,R4)) )
| ? [R4: dB,S3: dB,Ss2: list(dB)] :
( ( A3 = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R4)),S3),Ss2) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,subst(R4,S3,zero_zero(nat)),Ss2)))
& pp(aa(dB,bool,it,S3)) ) ) ) ).
tff(fact_51_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_52_abs__typeE,axiom,
! [Ta1: type,Ta: dB,Eb: fun(nat,type)] :
( typing(Eb,abs(Ta),Ta1)
=> ~ ! [U2: type,V1: type] : ~ typing(shift(type,Eb,zero_zero(nat),U2),Ta,V1) ) ).
tff(fact_53_dB_Osize_I1_J,axiom,
! [Nat1: nat] : dB_size(var(Nat1)) = zero_zero(nat) ).
tff(fact_54_beta,axiom,
! [T1: dB,S1: dB] : pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T1)),subst(S1,T1,zero_zero(nat)))) ).
tff(fact_55_appR,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,U1),S1)),aa(dB,dB,aa(dB,fun(dB,dB),app,U1),T1))) ) ).
tff(fact_56_appL,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U1)),aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1))) ) ).
tff(fact_57_beta__cases_I1_J,axiom,
! [T1: dB,I: nat] : ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,var(I)),T1)) ).
tff(fact_58_abs,axiom,
! [T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(S1)),abs(T1))) ) ).
tff(fact_59_subject__reduction,axiom,
! [T2: dB,Ta1: type,Ta: dB,Eb: fun(nat,type)] :
( typing(Eb,Ta,Ta1)
=> ( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,Ta),T2))
=> typing(Eb,T2,Ta1) ) ) ).
tff(fact_60_subst__preserves__beta,axiom,
! [I: nat,T1: dB,S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),S1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,subst(R3,T1,I)),subst(S1,T1,I))) ) ).
tff(fact_61_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_62_beta__cases_I2_J,axiom,
! [S1: dB,R3: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(R3)),S1))
=> ~ ! [T: dB] :
( ( S1 = abs(T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R3),T)) ) ) ).
tff(fact_63_apps__preserves__beta,axiom,
! [Ss1: list(dB),S: dB,R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R1),S))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Ss1)),foldl(dB,dB,app,S,Ss1))) ) ).
tff(fact_64_beta__cases_I3_J,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1)),U1))
=> ( ! [S2: dB] :
( ( U1 = subst(S2,T1,zero_zero(nat)) )
=> ( S1 != abs(S2) ) )
=> ( ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,T),T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T)) )
=> ~ ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,T1),T)) ) ) ) ) ).
tff(fact_65_dB_Osize_I4_J,axiom,
! [Nat1: nat] : size_size(dB,var(Nat1)) = zero_zero(nat) ).
tff(fact_66_substn__subst__0,axiom,
! [S1: dB,T1: dB] : substn(T1,S1,zero_zero(nat)) = subst(T1,S1,zero_zero(nat)) ).
tff(fact_67_list_Osize_I1_J,axiom,
! [A: $tType,Fa: fun(A,nat)] : list_size(A,Fa,nil(A)) = zero_zero(nat) ).
tff(fact_68_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U1: dB,T1: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T1,S1,K)),substn(U1,S1,K)) ).
tff(fact_69_substn__subst__n,axiom,
! [N: nat,S1: dB,T1: dB] : substn(T1,S1,N) = subst(T1,liftn(N,S1,zero_zero(nat)),N) ).
tff(fact_70_apps__preserves__betas,axiom,
! [R1: dB,Ss1: list(dB),Rsa: list(dB)] :
( step1(dB,beta,Rsa,Ss1)
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R1,Rsa)),foldl(dB,dB,app,R1,Ss1))) ) ).
tff(fact_71_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_72_type_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( atom(Nat2) = atom(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_73_liftn_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB,N: nat] : liftn(N,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N,S1,K)),liftn(N,T1,K)) ).
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_liftn__0,axiom,
! [K: nat,T1: dB] : liftn(zero_zero(nat),T1,K) = T1 ).
tff(fact_76_type_Osimps_I4_J,axiom,
! [Nat1: nat,Type21: type,Type11: type] : fun1(Type11,Type21) != atom(Nat1) ).
tff(fact_77_type_Osimps_I3_J,axiom,
! [Type21: type,Type11: type,Nat1: nat] : atom(Nat1) != fun1(Type11,Type21) ).
tff(fact_78_head__Var__reduction,axiom,
! [V: dB,Rsa: list(dB),Na: nat] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,var(Na),Rsa)),V))
=> ? [Ss: list(dB)] :
( step1(dB,beta,Rsa,Ss)
& ( V = foldl(dB,dB,app,var(Na),Ss) ) ) ) ).
tff(fact_79_type_Osize_I1_J,axiom,
! [Nat1: nat] : type_size(atom(Nat1)) = zero_zero(nat) ).
tff(fact_80_type_Osize_I3_J,axiom,
! [Nat1: nat] : size_size(type,atom(Nat1)) = zero_zero(nat) ).
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_not__Nil__step1,axiom,
! [A: $tType,Xs: list(A),R1: fun(A,fun(A,bool))] : ~ step1(A,R1,nil(A),Xs) ).
tff(fact_83_type_Oexhaust,axiom,
! [Y: type] :
( ! [Nat: nat] : Y != atom(Nat)
=> ~ ! [Type1: type,Type2: type] : Y != fun1(Type1,Type2) ) ).
tff(fact_84_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) ) )
=> ~ ! [T: dB] :
( ( R1 = abs(T) )
=> ! [U: dB,Us: list(dB)] :
( ( Rsa = cons(dB,U,Us) )
=> ( S != foldl(dB,dB,app,subst(T,U,zero_zero(nat)),Us) ) ) ) ) ) ) ).
tff(fact_85_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List1: list(A),A3: A] :
( ( cons(A,A3,List1) = cons(A,A6,List3) )
<=> ( ( A3 = A6 )
& ( List1 = List3 ) ) ) ).
tff(fact_86_Cons__step1__Cons,axiom,
! [A: $tType,Xs: list(A),X1: A,Ys: list(A),Y3: A,R1: fun(A,fun(A,bool))] :
( step1(A,R1,cons(A,Y3,Ys),cons(A,X1,Xs))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),R1,Y3),X1))
& ( Xs = Ys ) )
| ( ( X1 = Y3 )
& step1(A,R1,Ys,Xs) ) ) ) ).
tff(fact_87_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A5: A] : nil(A) != cons(A,A5,List2) ).
tff(fact_88_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A5: A] : cons(A,A5,List2) != nil(A) ).
tff(fact_89_foldl__Cons,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,A3: A,F: fun(A,fun(B,A))] : foldl(A,B,F,A3,cons(B,X1,Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun(B,A),F,A3),X1),Xs) ).
tff(fact_90_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_91_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : cons(A,X,Xs1) != Xs1 ).
tff(fact_92_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != cons(A,X,Xs1) ).
tff(fact_93_listsp_Osimps,axiom,
! [A: $tType,A3: list(A),A2: fun(A,bool)] :
( listsp(A,A2,A3)
<=> ( ( A3 = nil(A) )
| ? [A4: A,L: list(A)] :
( ( A3 = cons(A,A4,L) )
& pp(aa(A,bool,A2,A4))
& listsp(A,A2,L) ) ) ) ).
tff(fact_94_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y2: A,Ys1: list(A)] : Xs = cons(A,Y2,Ys1) ) ).
tff(fact_95_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A1: A,List: list(A)] : Y != cons(A,A1,List) ) ).
tff(fact_96_Cons__step1E,axiom,
! [A: $tType,Xs: list(A),X1: A,Ys: list(A),R1: fun(A,fun(A,bool))] :
( step1(A,R1,Ys,cons(A,X1,Xs))
=> ( ! [Y1: A] :
( ( Ys = cons(A,Y1,Xs) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),R1,Y1),X1)) )
=> ~ ! [Zs: list(A)] :
( ( Ys = cons(A,X1,Zs) )
=> ~ step1(A,R1,Zs,Xs) ) ) ) ).
tff(fact_97_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = cons(A,X,nil(A)) ).
tff(fact_98_list__ex1__simps_I2_J,axiom,
! [A: $tType,Xs: list(A),X1: A,P1: fun(A,bool)] :
( list_ex1(A,P1,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) ) ) ) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (14)
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_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 (1)
tff(conj_0,conjecture,
pp(aa(dB,bool,it,subst(foldl(dB,dB,app,var(n),rs),u,i))) ).
%------------------------------------------------------------------------------