%------------------------------------------------------------------------------
% File     : LCL803_5 : TPTP v9.0.0. Released v6.0.0.
% Domain   : Logic Calculi
% Problem  : Strong normalization of typed lambda calculus line 156
% Version  : Especial.
% English  : 

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla13]
% Names    : sn_156 [Bla13]

% Status   : Theorem
% Rating   : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v7.1.0, 0.33 v6.4.0
% Syntax   : Number of formulae    :  178 (  55 unt;  60 typ;   0 def)
%            Number of atoms       :  231 (  81 equ)
%            Maximal formula atoms :    8 (   1 avg)
%            Number of connectives :  161 (  48   ~;  15   |;  12   &)
%                                         (  13 <=>;  73  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   5 avg)
%            Maximal term depth    :   28 (   2 avg)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  :   35 (  19   >;  16   *;   0   +;   0  <<)
%            Number of predicates  :    5 (   4 usr;   0 prp; 1-4 aty)
%            Number of functors    :   52 (  52 usr;  26 con; 0-5 aty)
%            Number of variables   :  362 ( 322   !;   8   ?; 362   :)
%                                         (  32  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-12-13 16:21:17
%------------------------------------------------------------------------------
%----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 (54)
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(fun(A,fun(B,C)),fun(B,fun(A,C))) ).

tff(sy_c_COMBS,type,
    combs: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(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: ( dB * nat ) > dB ).

tff(sy_c_Lambda_Osubst,type,
    subst: fun(dB,fun(dB,fun(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_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_Olist__case,type,
    list_case: 
      !>[T5: $tType,A: $tType] : ( ( T5 * fun(A,fun(list(A),T5)) * list(A) ) > T5 ) ).

tff(sy_c_List_Olistset,type,
    listset: 
      !>[A: $tType] : ( list(fun(A,bool)) > fun(list(A),bool) ) ).

tff(sy_c_List_Olistsp,type,
    listsp: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( fun(A,bool) * fun(list(A),bool) ) > fun(list(A),bool) ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : ( A > nat ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).

tff(sy_c_Type_Oshift,type,
    shift: 
      !>[A: $tType] : fun(fun(nat,A),fun(nat,fun(A,fun(nat,A)))) ).

tff(sy_c_Type_Otype_OFun,type,
    fun1: fun(type,fun(type,type)) ).

tff(sy_c_Type_Otype_Otype__case,type,
    type_case: 
      !>[T5: $tType] : ( ( fun(nat,T5) * fun(type,fun(type,T5)) * type ) > T5 ) ).

tff(sy_c_Type_Otyping,type,
    typing: fun(fun(nat,type),fun(dB,fun(type,bool))) ).

tff(sy_c_Type_Otypings,type,
    typings: ( fun(nat,type) * list(dB) ) > fun(list(type),bool) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

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_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : fun(A,fun(fun(A,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,
    t1: type ).

tff(sy_v_T_Ha______,type,
    t_a: type ).

tff(sy_v_T____,type,
    t: 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: 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,
    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_Var_I3_J,axiom,
    pp(aa(dB,bool,it,ua)) ).

tff(fact_2_uIT,axiom,
    pp(aa(dB,bool,it,u)) ).

tff(fact_3_True,axiom,
    n = i ).

tff(fact_4__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(fun(dB,fun(type,bool)),fun(dB,bool),aa(fun(fun(type,bool),bool),fun(fun(dB,fun(type,bool)),fun(dB,bool)),combb(fun(type,bool),bool,dB),fEx(type)),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t))),as) ).

tff(fact_5_Var_I4_J,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,ea),ua),t)) ).

tff(fact_6__096listsp_A_I_Ft_O_AALL_Ae_AT_H_Au_Ai_O_Ae_060i_058T_062_A_092_060turnstile_062_At_A_058_AT_H_A_N_N_062_AIT_Au_A_N_N_062_Ae_A_092_060turnstile_062_Au_A_058_AT_A_N_N_062_AIT_A_It_091u_Pi_093_J_J_Aas_096,axiom,
    listsp(dB,aa(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool),aa(fun(fun(fun(nat,type),bool),bool),fun(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool)),combb(fun(fun(nat,type),bool),bool,dB),fAll(fun(nat,type))),aa(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool)),aa(fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),fun(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool))),combb(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool),dB),aa(fun(fun(type,bool),bool),fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),combb(fun(type,bool),bool,fun(nat,type)),fAll(type))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool)))),combb(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool)),dB),aa(fun(fun(type,fun(dB,bool)),fun(type,bool)),fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),combb(fun(type,fun(dB,bool)),fun(type,bool),fun(nat,type)),aa(fun(fun(dB,bool),bool),fun(fun(type,fun(dB,bool)),fun(type,bool)),combb(fun(dB,bool),bool,type),fAll(dB)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool))))),combb(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool))),dB),aa(fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),combb(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool)),fun(nat,type)),aa(fun(fun(dB,fun(nat,bool)),fun(dB,bool)),fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),combb(fun(dB,fun(nat,bool)),fun(dB,bool),type),aa(fun(fun(nat,bool),bool),fun(fun(dB,fun(nat,bool)),fun(dB,bool)),combb(fun(nat,bool),bool,dB),fAll(nat))))),aa(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),combs(dB,fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),dB),combs(fun(nat,type),fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),dB),aa(fun(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),combb(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))),fun(nat,type)),combc(type,fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),dB),aa(fun(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),combb(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(nat,type)),aa(fun(fun(fun(nat,bool),fun(nat,bool)),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),combb(fun(fun(nat,bool),fun(nat,bool)),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),type),combb(fun(nat,bool),fun(nat,bool),dB)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),aa(fun(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))))),combb(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),dB),aa(fun(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),combb(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(nat,type)),aa(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),type),combs(nat,bool,bool)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),aa(fun(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))))),combb(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),dB),aa(fun(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool)))),fun(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),combb(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool))),fun(nat,type)),aa(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),type),aa(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fimplies)))),aa(fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),aa(fun(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool))))),combb(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool))),dB),aa(fun(fun(nat,fun(type,bool)),fun(type,fun(nat,bool))),fun(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool)))),combb(fun(nat,fun(type,bool)),fun(type,fun(nat,bool)),fun(nat,type)),combc(nat,type,bool))),aa(fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),combc(fun(nat,type),dB,fun(nat,fun(type,bool))),aa(fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool)))),aa(fun(fun(nat,fun(dB,fun(type,bool))),fun(dB,fun(nat,fun(type,bool)))),fun(fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool))))),combb(fun(nat,fun(dB,fun(type,bool))),fun(dB,fun(nat,fun(type,bool))),fun(nat,type)),combc(nat,dB,fun(type,bool))),aa(fun(fun(nat,type),fun(nat,fun(nat,type))),fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),aa(fun(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool)))),fun(fun(fun(nat,type),fun(nat,fun(nat,type))),fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool))))),combb(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool))),fun(nat,type)),aa(fun(fun(nat,type),fun(dB,fun(type,bool))),fun(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool)))),combb(fun(nat,type),fun(dB,fun(type,bool)),nat),typing)),aa(type,fun(fun(nat,type),fun(nat,fun(nat,type))),aa(fun(fun(nat,type),fun(type,fun(nat,fun(nat,type)))),fun(type,fun(fun(nat,type),fun(nat,fun(nat,type)))),combc(fun(nat,type),type,fun(nat,fun(nat,type))),aa(fun(fun(nat,type),fun(nat,fun(type,fun(nat,type)))),fun(fun(nat,type),fun(type,fun(nat,fun(n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).

tff(fact_7_uT,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,e),u),t)) ).

tff(fact_8_Var_Ohyps,axiom,
    listsp(dB,aa(fun(dB,bool),fun(dB,bool),aa(fun(dB,fun(bool,bool)),fun(fun(dB,bool),fun(dB,bool)),combs(dB,bool,bool),aa(fun(dB,bool),fun(dB,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(dB,bool),fun(dB,fun(bool,bool))),combb(bool,fun(bool,bool),dB),fconj),it)),aa(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool),aa(fun(fun(fun(nat,type),bool),bool),fun(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool)),combb(fun(fun(nat,type),bool),bool,dB),fAll(fun(nat,type))),aa(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool)),aa(fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),fun(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool))),combb(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool),dB),aa(fun(fun(type,bool),bool),fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),combb(fun(type,bool),bool,fun(nat,type)),fAll(type))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool)))),combb(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool)),dB),aa(fun(fun(type,fun(dB,bool)),fun(type,bool)),fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),combb(fun(type,fun(dB,bool)),fun(type,bool),fun(nat,type)),aa(fun(fun(dB,bool),bool),fun(fun(type,fun(dB,bool)),fun(type,bool)),combb(fun(dB,bool),bool,type),fAll(dB)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool))))),combb(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool))),dB),aa(fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),combb(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool)),fun(nat,type)),aa(fun(fun(dB,fun(nat,bool)),fun(dB,bool)),fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),combb(fun(dB,fun(nat,bool)),fun(dB,bool),type),aa(fun(fun(nat,bool),bool),fun(fun(dB,fun(nat,bool)),fun(dB,bool)),combb(fun(nat,bool),bool,dB),fAll(nat))))),aa(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),combs(dB,fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),dB),combs(fun(nat,type),fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),dB),aa(fun(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),combb(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))),fun(nat,type)),combc(type,fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),dB),aa(fun(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),combb(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(nat,type)),aa(fun(fun(fun(nat,bool),fun(nat,bool)),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),combb(fun(fun(nat,bool),fun(nat,bool)),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),type),combb(fun(nat,bool),fun(nat,bool),dB)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),aa(fun(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))))),combb(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool)))),dB),aa(fun(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),combb(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool))),fun(nat,type)),aa(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(type,fun(nat,fun(bool,bool))),fun(type,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),type),combs(nat,bool,bool)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),aa(fun(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))))),combb(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool)))),dB),aa(fun(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool)))),fun(fun(fun(nat,type),fun(type,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(nat,fun(bool,bool))))),combb(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool))),fun(nat,type)),aa(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(type,fun(nat,bool)),fun(type,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),type),aa(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fimplies)))),aa(fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool)))),aa(fun(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool)))),fun(fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(nat,bool))))),combb(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool))),dB),aa(fun(fun(nat,fun(type,bool)),fun(type,fun(nat,bool))),fun(fun(fun(nat,type),fun(nat,fun(type,bool))),fun(fun(nat,type),fun(type,fun(nat,bool)))),combb(fun(nat,fun(type,bool)),fun(type,fun(nat,bool)),fun(nat,type)),combc(nat,type,bool))),aa(fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool)))),fun(dB,fun(fun(nat,type),fun(nat,fun(type,bool)))),combc(fun(nat,type),dB,fun(nat,fun(type,bool))),aa(fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool)))),aa(fun(fun(nat,fun(dB,fun(type,bool))),fun(dB,fun(nat,fun(type,bool)))),fun(fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),fun(fun(nat,type),fun(dB,fun(nat,fun(type,bool))))),combb(fun(nat,fun(dB,fun(type,bool))),fun(dB,fun(nat,fun(type,bool))),fun(nat,type)),combc(nat,dB,fun(type,bool))),aa(fun(fun(nat,type),fun(nat,fun(nat,type))),fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool)))),aa(fun(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool)))),fun(fun(fun(nat,type),fun(nat,fun(nat,type))),fun(fun(nat,type),fun(nat,fun(dB,fun(type,bool))))),combb(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool))),fun(nat,type)),aa(fun(fun(nat,type),fun(dB,fun(type,bool))),fun(fun(nat,fun(nat,type)),fun(nat,fun(dB,fun(type,bool)))),combb(fun(nat,type),fun(dB,fun(type,bool)),nat),typing)),aa(type,fun(fun(nat,type),fun(nat,fun(nat,type))),aa(fun(fun(nat,type),fun(type,fun(nat,fun(nat,type)))),fun(type,fun(fun(nat,type),fun(nat,fun(nat,type)))),combc(fun(nat,type),type,fun(nat,fun(nat,type))),aa(fun(fun(nat,type),fun(nat,fun(type,fun(nat,type)))),fun(fun(nat,type),fun(type,fun(nat,fun(nat,type)))),aa(fun(fun(nat,fun(type,fun(nat,type))),fun(type,fun(nat,fun(nat,type)))),fun(fun(fun(nat,type),fun(nat,fun(type,fun(nat,type)))),fun(fun(nat,type),fun(type,fun(nat,fun(nat,type))))),combb(fun(nat,fun(type,fun(nat,type))),fun(type,fun(nat,fun(nat,type))),fun(nat,type)),combc(nat,type,fun(nat,type))),shift(type))),t))))))))))),aa(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),aa(fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool))))),combb(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(dB,fun(nat,bool))),dB),aa(fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(dB,fun(nat,bool)))),combb(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)),fun(nat,type)),aa(fun(dB,fun(fun(nat,bool),fun(nat,bool))),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),combs(dB,fun(nat,bool),fun(nat,bool)),aa(fun(dB,fun(bool,bool)),fun(dB,fun(fun(nat,bool),fun(nat,bool))),aa(fun(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool))),fun(fun(dB,fun(bool,bool)),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),dB),combb(bool,bool,nat)),aa(fun(dB,bool),fun(dB,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(dB,bool),fun(dB,fun(bool,bool))),combb(bool,fun(bool,bool),dB),fimplies),it))))),aa(fun(dB,fun(dB,fun(nat,bool))),fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),aa(fun(fun(dB,fun(nat,bool)),fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(dB,fun(nat,bool))),fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool))))),combb(fun(dB,fun(nat,bool)),fun(fun(nat,type),fun(dB,fun(nat,bool))),dB),aa(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(fun(nat,type),fun(dB,fun(nat,bool)))),combc(fun(nat,type),fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),aa(fun(fun(nat,type),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),aa(fun(fun(dB,fun(fun(nat,bool),fun(nat,bool))),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(fun(nat,type),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),combb(fun(dB,fun(fun(nat,bool),fun(nat,bool))),fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))),fun(nat,type)),combs(dB,fun(nat,bool),fun(nat,bool))),aa(fun(fun(nat,type),fun(dB,fun(bool,bool))),fun(fun(nat,type),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),aa(fun(fun(dB,fun(bool,bool)),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(fun(nat,type),fun(dB,fun(bool,bool))),fun(fun(nat,type),fun(dB,fun(fun(nat,bool),fun(nat,bool))))),combb(fun(dB,fun(bool,bool)),fun(dB,fun(fun(nat,bool),fun(nat,bool))),fun(nat,type)),aa(fun(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool))),fun(fun(dB,fun(bool,bool)),fun(dB,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),dB),combb(bool,bool,nat))),aa(fun(fun(nat,type),fun(dB,bool)),fun(fun(nat,type),fun(dB,fun(bool,bool))),aa(fun(fun(dB,bool),fun(dB,fun(bool,bool))),fun(fun(fun(nat,type),fun(dB,bool)),fun(fun(nat,type),fun(dB,fun(bool,bool)))),combb(fun(dB,bool),fun(dB,fun(bool,bool)),fun(nat,type)),aa(fun(bool,fun(bool,bool)),fun(fun(dB,bool),fun(dB,fun(bool,bool))),combb(bool,fun(bool,bool),dB),fimplies)),aa(type,fun(fun(nat,type),fun(dB,bool)),aa(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(type,fun(fun(nat,type),fun(dB,bool))),combc(fun(nat,type),type,fun(dB,bool)),aa(fun(fun(nat,type),fun(dB,fun(type,bool))),fun(fun(nat,type),fun(type,fun(dB,bool))),aa(fun(fun(dB,fun(type,bool)),fun(type,fun(dB,bool))),fun(fun(fun(nat,type),fun(dB,fun(type,bool))),fun(fun(nat,type),fun(type,fun(dB,bool)))),combb(fun(dB,fun(type,bool)),fun(type,fun(dB,bool)),fun(nat,type)),combc(dB,type,bool)),typing)),t)))))),aa(fun(dB,fun(dB,fun(nat,dB))),fun(dB,fun(dB,fun(nat,bool))),aa(fun(fun(dB,fun(nat,dB)),fun(dB,fun(nat,bool))),fun(fun(dB,fun(dB,fun(nat,dB))),fun(dB,fun(dB,fun(nat,bool)))),combb(fun(dB,fun(nat,dB)),fun(dB,fun(nat,bool)),dB),aa(fun(fun(nat,dB),fun(nat,bool)),fun(fun(dB,fun(nat,dB)),fun(dB,fun(nat,bool))),combb(fun(nat,dB),fun(nat,bool),dB),aa(fun(dB,bool),fun(fun(nat,dB),fun(nat,bool)),combb(dB,bool,nat),it))),subst))))))))),rs) ).

tff(fact_9_argT,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),a),t2)) ).

tff(fact_10_argsT,axiom,
    pp(aa(list(type),bool,typings(aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t),as),ts)) ).

tff(fact_11_MI1,axiom,
    ! [Ub: dB,Ta1: type,Ib: nat,Eb: fun(nat,type),Ta: dB,T21: type,T11: type] :
      ( ( t = aa(type,type,aa(type,fun(type,type),fun1,T11),T21) )
     => ( pp(aa(dB,bool,it,Ta))
       => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Eb),Ib),T11)),Ta),Ta1))
         => ( pp(aa(dB,bool,it,Ub))
           => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ub),T11))
             => pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),Ub),Ib))) ) ) ) ) ) ).

tff(fact_12_MI2,axiom,
    ! [Ub: dB,Ta1: type,Ib: nat,Eb: fun(nat,type),Ta: dB,T21: type,T11: type] :
      ( ( t = aa(type,type,aa(type,fun(type,type),fun1,T11),T21) )
     => ( pp(aa(dB,bool,it,Ta))
       => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Eb),Ib),T21)),Ta),Ta1))
         => ( pp(aa(dB,bool,it,Ub))
           => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ub),T21))
             => pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),Ub),Ib))) ) ) ) ) ) ).

tff(fact_13_listsp__conj__eq,axiom,
    ! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),X3: list(A)] :
      ( listsp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),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_14_subst__lemma,axiom,
    ! [Ib: nat,Ua: type,Ub: dB,E: fun(nat,type),Ta1: type,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),Ta1))
     => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,E),Ub),Ua))
       => ( ( Eb = aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),E),Ib),Ua) )
         => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,E),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),Ub),Ib)),Ta1)) ) ) ) ).

tff(fact_15_shift__eq,axiom,
    ! [A: $tType,Ta1: A,Eb: fun(nat,A),J1: nat,Ib: nat] :
      ( ( Ib = J1 )
     => ( aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aa(fun(nat,A),fun(nat,fun(A,fun(nat,A))),shift(A),Eb),Ib),Ta1),J1) = Ta1 ) ) ).

tff(fact_16_lists__typings,axiom,
    ! [Tsa: list(type),Ts: list(dB),Eb: fun(nat,type)] :
      ( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
     => listsp(dB,aa(fun(dB,fun(type,bool)),fun(dB,bool),aa(fun(fun(type,bool),bool),fun(fun(dB,fun(type,bool)),fun(dB,bool)),combb(fun(type,bool),bool,dB),fEx(type)),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb)),Ts) ) ).

tff(fact_17_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_18_lift__type,axiom,
    ! [Ua: type,Ib: nat,Ta1: type,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),Ta1))
     => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Eb),Ib),Ua)),lift(Ta,Ib)),Ta1)) ) ).

tff(fact_19__096IT_A_Iu_A_092_060degree_062_Aa_091u_Pi_093_J_096,axiom,
    pp(aa(dB,bool,it,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)))) ).

tff(fact_20_lift__IT,axiom,
    ! [I: nat,T1: dB] :
      ( pp(aa(dB,bool,it,T1))
     => pp(aa(dB,bool,it,lift(T1,I))) ) ).

tff(fact_21_type_Osimps_I2_J,axiom,
    ! [Type21: type,Type11: type,Type2: type,Type1: type] :
      ( ( aa(type,type,aa(type,fun(type,type),fun1,Type1),Type2) = aa(type,type,aa(type,fun(type,type),fun1,Type11),Type21) )
    <=> ( ( Type1 = Type11 )
        & ( Type2 = Type21 ) ) ) ).

tff(fact_22_typing__elims_I1_J,axiom,
    ! [Ta1: type,Ib: nat,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),var(Ib)),Ta1))
     => ( aa(nat,type,Eb,Ib) = Ta1 ) ) ).

tff(fact_23_typing_OVar,axiom,
    ! [Ta1: type,X1: nat,Env: fun(nat,type)] :
      ( ( aa(nat,type,Env,X1) = Ta1 )
     => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Env),var(X1)),Ta1)) ) ).

tff(fact_24_App,axiom,
    ! [Ta: dB,Ua: type,Ta1: type,S: dB,Env: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Env),S),aa(type,type,aa(type,fun(type,type),fun1,Ta1),Ua)))
     => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Env),Ta),Ta1))
       => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Env),aa(dB,dB,aa(dB,fun(dB,dB),app,S),Ta)),Ua)) ) ) ).

tff(fact_25_Cons,axiom,
    rs = aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),a),as) ).

tff(fact_26_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_27_Var__IT,axiom,
    ! [N1: nat] : pp(aa(dB,bool,it,var(N1))) ).

tff(fact_28_headT,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),aa(dB,dB,aa(dB,fun(dB,dB),app,var(n)),a)),foldr(type,type,fun1,ts,t1))) ).

tff(fact_29_varT,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),var(n)),aa(type,type,aa(type,fun(type,type),fun1,t2),foldr(type,type,fun1,ts,t1)))) ).

tff(fact_30__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,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(dB,dB,aa(dB,fun(dB,dB),app,lift(u,zero_zero(nat))),var(zero_zero(nat)))),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,a),u),i)),zero_zero(nat)))) ).

tff(fact_31_typing__elims_I2_J,axiom,
    ! [Ta1: type,Ub: dB,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),aa(dB,dB,aa(dB,fun(dB,dB),app,Ta),Ub)),Ta1))
     => ~ ! [T3: type] :
            ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),aa(type,type,aa(type,fun(type,type),fun1,T3),Ta1)))
           => ~ pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ub),T3)) ) ) ).

tff(fact_32_Var_I2_J,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),ea),ia),t)),foldl(dB,dB,app,var(n),rs)),t_a)) ).

tff(fact_33_nT,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),foldl(dB,dB,app,var(n),rs)),t1)) ).

tff(fact_34_lift_Osimps_I2_J,axiom,
    ! [K: nat,T1: dB,S1: dB] : ( lift(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,lift(S1,K)),lift(T1,K)) ) ).

tff(fact_35_list_Oinject,axiom,
    ! [A: $tType,List1: list(A),A2: A,List: list(A),Aa: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa),List) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),List1) )
    <=> ( ( Aa = A2 )
        & ( List = List1 ) ) ) ).

tff(fact_36_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_37_dB_Osimps_I1_J,axiom,
    ! [Nat2: nat,Nat1: nat] :
      ( ( var(Nat1) = var(Nat2) )
    <=> ( Nat1 = Nat2 ) ) ).

tff(fact_38_subst__App,axiom,
    ! [K: nat,S1: dB,U1: dB,T1: 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,T1),U1)),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,T1),S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U1),S1),K)) ) ).

tff(fact_39_T,axiom,
    t = aa(type,type,aa(type,fun(type,type),fun1,t2),foldr(type,type,fun1,ts,t1)) ).

tff(fact_40_uT_H,axiom,
    pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,e),u),aa(type,type,aa(type,fun(type,type),fun1,t2),foldr(type,type,fun1,ts,t1)))) ).

tff(fact_41__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,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,var(n)),a),as)),t1)) ).

tff(fact_42__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,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),var(n)),aa(type,type,aa(type,fun(type,type),fun1,T4),foldr(type,type,fun1,ts,t1))))
       => ~ pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),a),T4)) ) ).

tff(fact_43__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,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t)),aa(dB,dB,aa(dB,fun(dB,dB),app,var(n)),a)),foldr(type,type,fun1,Ts1,t1)))
       => ~ pp(aa(list(type),bool,typings(aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),e),i),t),as),Ts1)) ) ).

tff(fact_44_not__Cons__self,axiom,
    ! [A: $tType,X: A,Xs1: list(A)] : ( Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) ) ).

tff(fact_45_not__Cons__self2,axiom,
    ! [A: $tType,Xs1: list(A),X: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) != Xs1 ) ).

tff(fact_46_foldl__Cons,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),X1: 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),X1),Xs)) = foldl(A,B,F,aa(B,A,aa(A,fun(B,A),F,Aa),X1),Xs) ) ).

tff(fact_47_foldr_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,Aa: A,Xs: list(B),X1: B,F: fun(B,fun(A,A))] : ( foldr(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),Xs),Aa) = aa(A,A,aa(B,fun(A,A),F,X1),foldr(B,A,F,Xs,Aa)) ) ).

tff(fact_48_list__app__typeI,axiom,
    ! [Ts: list(dB),Ta1: type,Tsa: list(type),Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),foldr(type,type,fun1,Tsa,Ta1)))
     => ( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
       => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,Ta,Ts)),Ta1)) ) ) ).

tff(fact_49_var__app__type__eq,axiom,
    ! [Ua: type,Ta1: type,Ts: list(dB),Ib: nat,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,var(Ib),Ts)),Ta1))
     => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,var(Ib),Ts)),Ua))
       => ( Ta1 = Ua ) ) ) ).

tff(fact_50_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_51_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_52_subst__eq,axiom,
    ! [U1: dB,K: nat] : ( aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,var(K)),U1),K) = U1 ) ).

tff(fact_53_subst__lift,axiom,
    ! [S1: dB,K: nat,T1: dB] : ( aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,lift(T1,K)),S1),K) = T1 ) ).

tff(fact_54_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_55_var__app__types,axiom,
    ! [Ua: type,Tsa: list(type),Ta1: type,Us1: list(dB),Ts: list(dB),Ib: nat,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,foldl(dB,dB,app,var(Ib),Ts),Us1)),Ta1))
     => ( pp(aa(list(type),bool,typings(Eb,Ts),Tsa))
       => ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,var(Ib),Ts)),Ua))
         => ? [Us2: list(type)] :
              ( ( Ua = foldr(type,type,fun1,Us2,Ta1) )
              & pp(aa(list(type),bool,typings(Eb,Us1),Us2)) ) ) ) ) ).

tff(fact_56_var__app__typesE,axiom,
    ! [Ta1: type,Ts: list(dB),Ib: nat,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,var(Ib),Ts)),Ta1))
     => ~ ! [Ts1: list(type)] :
            ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),var(Ib)),foldr(type,type,fun1,Ts1,Ta1)))
           => ~ pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).

tff(fact_57_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_58_list__app__typeE,axiom,
    ! [Ta1: type,Ts: list(dB),Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,Ta,Ts)),Ta1))
     => ~ ! [Ts1: list(type)] :
            ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),foldr(type,type,fun1,Ts1,Ta1)))
           => ~ pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).

tff(fact_59_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_60_list__app__typeD,axiom,
    ! [Ta1: type,Ts: list(dB),Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),foldl(dB,dB,app,Ta,Ts)),Ta1))
     => ? [Ts1: list(type)] :
          ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),foldr(type,type,fun1,Ts1,Ta1)))
          & pp(aa(list(type),bool,typings(Eb,Ts),Ts1)) ) ) ).

tff(fact_61_typings_Osimps_I2_J,axiom,
    ! [Tsa: list(type),Ts: list(dB),Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(list(type),bool,typings(Eb,aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),Ta),Ts)),Tsa))
    <=> pp(list_case(bool,type,fFalse,aa(fun(list(type),bool),fun(type,fun(list(type),bool)),aa(fun(type,fun(fun(list(type),bool),fun(list(type),bool))),fun(fun(list(type),bool),fun(type,fun(list(type),bool))),combc(type,fun(list(type),bool),fun(list(type),bool)),aa(fun(type,fun(bool,bool)),fun(type,fun(fun(list(type),bool),fun(list(type),bool))),aa(fun(fun(bool,bool),fun(fun(list(type),bool),fun(list(type),bool))),fun(fun(type,fun(bool,bool)),fun(type,fun(fun(list(type),bool),fun(list(type),bool)))),combb(fun(bool,bool),fun(fun(list(type),bool),fun(list(type),bool)),type),combb(bool,bool,list(type))),aa(fun(type,bool),fun(type,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(type,bool),fun(type,fun(bool,bool))),combb(bool,fun(bool,bool),type),fconj),aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta)))),typings(Eb,Ts)),Tsa)) ) ).

tff(fact_62_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X1: A] :
          ( ( zero_zero(A) = X1 )
        <=> ( X1 = zero_zero(A) ) ) ) ).

tff(fact_63_list_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : ( list_case(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa),List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List) ) ).

tff(fact_64_dB_Osize_I1_J,axiom,
    ! [Nat: nat] : ( dB_size(var(Nat)) = zero_zero(nat) ) ).

tff(fact_65_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_66_set__Cons__def,axiom,
    ! [A: $tType,XS: fun(list(A),bool),A1: fun(A,bool)] : ( set_Cons(A,A1,XS) = collect(list(A),aa(fun(list(A),fun(A,bool)),fun(list(A),bool),aa(fun(fun(A,bool),bool),fun(fun(list(A),fun(A,bool)),fun(list(A),bool)),combb(fun(A,bool),bool,list(A)),fEx(A)),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool)),aa(fun(fun(A,fun(list(A),bool)),fun(A,bool)),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool))),combb(fun(A,fun(list(A),bool)),fun(A,bool),list(A)),aa(fun(fun(list(A),bool),bool),fun(fun(A,fun(list(A),bool)),fun(A,bool)),combb(fun(list(A),bool),bool,A),fEx(list(A)))),aa(fun(A,fun(list(A),bool)),fun(list(A),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),fun(fun(A,fun(list(A),bool)),fun(list(A),fun(A,fun(list(A),bool)))),combc(list(A),fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),aa(fun(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))))),combb(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))),list(A)),combs(A,fun(list(A),bool),fun(list(A),bool))),aa(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),aa(fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool))))),combb(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),list(A)),aa(fun(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool)),A),combs(list(A),bool,bool))),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),aa(fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool))))),combb(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool))),list(A)),aa(fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),combb(fun(list(A),bool),fun(list(A),fun(bool,bool)),A),aa(fun(bool,fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),combb(bool,fun(bool,bool),list(A)),fconj))),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),fun(fun(A,fun(list(A),list(A))),fun(list(A),fun(A,fun(list(A),bool)))),combc(list(A),fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),aa(fun(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))))),combb(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))),list(A)),combb(fun(list(A),list(A)),fun(list(A),bool),A)),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),aa(fun(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool)))),combb(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool)),list(A)),combb(list(A),bool,list(A))),fequal(list(A))))),cons(A)))))),aa(fun(list(A),bool),fun(A,fun(list(A),bool)),aa(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(list(A),bool),fun(A,fun(list(A),bool))),combc(A,fun(list(A),bool),fun(list(A),bool)),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),aa(fun(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool)),A),combb(bool,bool,list(A))),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),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A1)))),aa(fun(list(A),bool),fun(list(A),bool),aa(fun(list(A),fun(fun(list(A),bool),bool)),fun(fun(list(A),bool),fun(list(A),bool)),combc(list(A),fun(list(A),bool),bool),member(list(A))),XS)))))) ) ).

tff(fact_67_dB_Osimps_I3_J,axiom,
    ! [DB4: dB,DB3: dB] :
      ( ( abs(DB3) = abs(DB4) )
    <=> ( DB3 = DB4 ) ) ).

tff(fact_68_Lambda,axiom,
    ! [R3: dB] :
      ( pp(aa(dB,bool,it,R3))
     => pp(aa(dB,bool,it,abs(R3))) ) ).

tff(fact_69_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_70_Abs,axiom,
    ! [Ua: type,Ta: dB,Ta1: type,Env: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Env),zero_zero(nat)),Ta1)),Ta),Ua))
     => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Env),abs(Ta)),aa(type,type,aa(type,fun(type,type),fun1,Ta1),Ua))) ) ).

tff(fact_71_dB_Osimps_I7_J,axiom,
    ! [Nat: nat,DB: dB] : ( abs(DB) != var(Nat) ) ).

tff(fact_72_dB_Osimps_I6_J,axiom,
    ! [DB: dB,Nat: nat] : ( var(Nat) != abs(DB) ) ).

tff(fact_73_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_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),X1: A] :
      ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A1))
    <=> pp(aa(A,bool,A1,X1)) ) ).

tff(fact_76_Collect__def,axiom,
    ! [A: $tType,P1: fun(A,bool)] : ( collect(A,P1) = P1 ) ).

tff(fact_77_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_78_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_79_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_80_typing__elims_I3_J,axiom,
    ! [Ta1: type,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),abs(Ta)),Ta1))
     => ~ ! [T3: type,U2: type] :
            ( ( Ta1 = aa(type,type,aa(type,fun(type,type),fun1,T3),U2) )
           => ~ pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Eb),zero_zero(nat)),T3)),Ta),U2)) ) ) ).

tff(fact_81_abs__typeE,axiom,
    ! [Ta1: type,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),abs(Ta)),Ta1))
     => ~ ! [U2: type,V: type] : ~ pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,aa(type,fun(nat,type),aa(nat,fun(type,fun(nat,type)),aa(fun(nat,type),fun(nat,fun(type,fun(nat,type))),shift(type),Eb),zero_zero(nat)),U2)),Ta),V)) ) ).

tff(fact_82_listset_Osimps_I2_J,axiom,
    ! [A: $tType,As: list(fun(A,bool)),A1: fun(A,bool)] : ( listset(A,aa(list(fun(A,bool)),list(fun(A,bool)),aa(fun(A,bool),fun(list(fun(A,bool)),list(fun(A,bool))),cons(fun(A,bool)),A1),As)) = set_Cons(A,A1,listset(A,As)) ) ).

tff(fact_83_IT_Osimps,axiom,
    ! [Aa: dB] :
      ( pp(aa(dB,bool,it,Aa))
    <=> ( ? [Rs1: list(dB),N: nat] :
            ( ( Aa = foldl(dB,dB,app,var(N),Rs1) )
            & listsp(dB,it,Rs1) )
        | ? [R4: dB] :
            ( ( Aa = abs(R4) )
            & pp(aa(dB,bool,it,R4)) )
        | ? [R4: dB,S3: dB,Ss1: list(dB)] :
            ( ( Aa = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R4)),S3),Ss1) )
            & 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)),Ss1)))
            & pp(aa(dB,bool,it,S3)) ) ) ) ).

tff(fact_84_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)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T1),zero_zero(nat)))) ).

tff(fact_85_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_86_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_87_beta__cases_I1_J,axiom,
    ! [T1: dB,I: nat] : ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,var(I)),T1)) ).

tff(fact_88_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_89_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,lift(R3,I)),lift(S1,I))) ) ).

tff(fact_90_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,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R3),T1),I)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,S1),T1),I))) ) ).

tff(fact_91_subject__reduction,axiom,
    ! [T2: dB,Ta1: type,Ta: dB,Eb: fun(nat,type)] :
      ( pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),Ta),Ta1))
     => ( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,Ta),T2))
       => pp(aa(type,bool,aa(dB,fun(type,bool),aa(fun(nat,type),fun(dB,fun(type,bool)),typing,Eb),T2),Ta1)) ) ) ).

tff(fact_92_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_93_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_94_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 = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),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_95_dB_Osize_I4_J,axiom,
    ! [Nat: nat] : ( size_size(dB,var(Nat)) = zero_zero(nat) ) ).

tff(fact_96_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 = aa(list(dB),list(dB),aa(dB,fun(list(dB),list(dB)),cons(dB),U),Us) )
                   => ( S != foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),U),zero_zero(nat)),Us) ) ) ) ) ) ) ).

tff(fact_97_type_Osimps_I6_J,axiom,
    ! [A: $tType,Type2: type,Type1: type,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : ( type_case(A,F1,F2,aa(type,type,aa(type,fun(type,type),fun1,Type1),Type2)) = aa(type,A,aa(type,fun(type,A),F2,Type1),Type2) ) ).

tff(fact_98_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))) ) ).

%----Arities (1)
tff(arity_Nat_Onat___Groups_Ozero,axiom,
    zero(nat) ).

%----Helper facts (17)
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: fun(A,bool)] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ~ pp(aa(fun(A,bool),bool,fAll(A),P))
      | pp(aa(A,bool,P,X)) ) ).

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,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(A,C)),combc(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : ( aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(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_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,Y: A,X: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(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,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

%----Conjectures (1)
tff(conj_0,conjecture,
    listsp(dB,aa(fun(dB,bool),fun(dB,bool),aa(fun(dB,fun(bool,bool)),fun(fun(dB,bool),fun(dB,bool)),combs(dB,bool,bool),aa(fun(dB,bool),fun(dB,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(dB,bool),fun(dB,fun(bool,bool))),combb(bool,fun(bool,bool),dB),fconj),aa(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool),aa(fun(fun(fun(nat,type),bool),bool),fun(fun(dB,fun(fun(nat,type),bool)),fun(dB,bool)),combb(fun(fun(nat,type),bool),bool,dB),fAll(fun(nat,type))),aa(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool)),aa(fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),fun(fun(dB,fun(fun(nat,type),fun(type,bool))),fun(dB,fun(fun(nat,type),bool))),combb(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool),dB),aa(fun(fun(type,bool),bool),fun(fun(fun(nat,type),fun(type,bool)),fun(fun(nat,type),bool)),combb(fun(type,bool),bool,fun(nat,type)),fAll(type))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(dB,fun(fun(nat,type),fun(type,bool)))),combb(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool)),dB),aa(fun(fun(type,fun(dB,bool)),fun(type,bool)),fun(fun(fun(nat,type),fun(type,fun(dB,bool))),fun(fun(nat,type),fun(type,bool))),combb(fun(type,fun(dB,bool)),fun(type,bool),fun(nat,type)),aa(fun(fun(dB,bool),bool),fun(fun(type,fun(dB,bool)),fun(type,bool)),combb(fun(dB,bool),bool,type),fAll(dB)))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool)))),aa(fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,bool))))),combb(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool))),dB),aa(fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),fun(fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))),fun(fun(nat,type),fun(type,fun(dB,bool)))),combb(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool)),fun(nat,type)),aa(fun(fun(dB,fun(nat,bool)),fun(dB,bool)),fun(fun(type,fun(dB,fun(nat,bool))),fun(type,fun(dB,bool))),combb(fun(dB,fun(nat,bool)),fun(dB,bool),type),aa(fun(fun(nat,bool),bool),fun(fun(dB,fun(nat,bool)),fun(dB,bool)),combb(fun(nat,bool),bool,dB),fAll(nat))))),aa(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(dB,fun(nat,bool)))),fun(dB,fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),combs(dB,fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(dB,fun(nat,bool))),fun(fun(nat,type),fun(type,fun(dB,fun(nat,bool))))),dB),combs(fun(nat,type),fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),aa(fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),fun(fun(dB,fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))))),fun(dB,fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))))),combb(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),dB),aa(fun(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool))))),fun(fun(fun(nat,type),fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),fun(fun(nat,type),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))))),combb(fun(type,fun(fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool)))),fun(fun(dB,fun(nat,bool)),fun(type,fun(dB,fun(nat,bool)))),fun(nat,type)),combc(type,fun(dB,fun(nat,bool)),fun(dB,fun(nat,bool))))),aa(fun(dB,fun(fun(nat,type),fun(type,fun(fun(nat,bool),fun(nat,bool))))),fun(dB,fun(fun(n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).

%------------------------------------------------------------------------------