ITP001 Axioms: ITP078_7.ax


%------------------------------------------------------------------------------
% File     : ITP078_7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : update.ax [Gau19]
%          : HL4078_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  169 (  58 unt;  82 typ;   0 def)
%            Number of atoms       :  149 ( 101 equ)
%            Maximal formula atoms :   10 (   0 avg)
%            Number of connectives :   72 (  10   ~;   3   |;  15   &)
%                                         (  22 <=>;  22  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   6 avg)
%            Maximal term depth    :   10 (   2 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   76 (  42   >;  34   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   80 (  80 usr;   6 con; 0-6 aty)
%            Number of variables   :  494 ( 391   !;   1   ?; 494   :)
%                                         ( 102  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: $tType > $tType ).

tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

tff(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).

tff(tyop_2Eoption_2Eoption,type,
    tyop_2Eoption_2Eoption: $tType > $tType ).

tff(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).

tff(app_2E2,type,
    app_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(p,type,
    p: tyop_2Emin_2Ebool > $o ).

tff(combin_i_2E0,type,
    combin_i_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).

tff(combin_k_2E0,type,
    combin_k_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).

tff(combin_s_2E0,type,
    combin_s_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).

tff(c_2Ebool_2E_21_2E0,type,
    c_2Ebool_2E_21_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_21_2E1,type,
    c_2Ebool_2E_21_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Epair_2E_2C_2E0,type,
    c_2Epair_2E_2C_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).

tff(c_2Epair_2E_2C_2E2,type,
    c_2Epair_2E_2C_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).

tff(c_2Ebool_2E_2F_5C_2E0,type,
    c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_2F_5C_2E2,type,
    c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Emin_2E_3D_2E0,type,
    c_2Emin_2E_3D_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_2E2,type,
    c_2Emin_2E_3D_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_3D_3D_3E_2E0,type,
    c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_3D_3E_2E2,type,
    c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Ebool_2E_3F_2E0,type,
    c_2Ebool_2E_3F_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_3F_2E1,type,
    c_2Ebool_2E_3F_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_40_2E0,type,
    c_2Emin_2E_40_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).

tff(c_2Emin_2E_40_2E1,type,
    c_2Emin_2E_40_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).

tff(c_2Elist_2EALL__DISTINCT_2E0,type,
    c_2Elist_2EALL__DISTINCT_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool) ).

tff(c_2Elist_2EALL__DISTINCT_2E1,type,
    c_2Elist_2EALL__DISTINCT_2E1: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > tyop_2Emin_2Ebool ) ).

tff(c_2Elist_2EAPPEND_2E0,type,
    c_2Elist_2EAPPEND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Elist_2EAPPEND_2E2,type,
    c_2Elist_2EAPPEND_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Ebool_2ECOND_2E0,type,
    c_2Ebool_2ECOND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ebool_2ECOND_2E3,type,
    c_2Ebool_2ECOND_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).

tff(c_2Elist_2ECONS_2E0,type,
    c_2Elist_2ECONS_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Elist_2ECONS_2E2,type,
    c_2Elist_2ECONS_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Elist_2EFILTER_2E0,type,
    c_2Elist_2EFILTER_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Elist_2EFILTER_2E2,type,
    c_2Elist_2EFILTER_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Eupdate_2EFIND_2E0,type,
    c_2Eupdate_2EFIND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27a))) ).

tff(c_2Eupdate_2EFIND_2E2,type,
    c_2Eupdate_2EFIND_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Eoption_2Eoption(A_27a) ) ).

tff(c_2Epair_2EFST_2E0,type,
    c_2Epair_2EFST_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).

tff(c_2Epair_2EFST_2E1,type,
    c_2Epair_2EFST_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).

tff(c_2Ecombin_2EI_2E0,type,
    c_2Ecombin_2EI_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).

tff(c_2Ecombin_2EI_2E1,type,
    c_2Ecombin_2EI_2E1: 
      !>[A_27a: $tType] : ( A_27a > A_27a ) ).

tff(c_2Eupdate_2ELIST__UPDATE_2E0,type,
    c_2Eupdate_2ELIST__UPDATE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))) ).

tff(c_2Eupdate_2ELIST__UPDATE_2E1,type,
    c_2Eupdate_2ELIST__UPDATE_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ) ).

tff(c_2Eupdate_2ELIST__UPDATE_2E2,type,
    c_2Eupdate_2ELIST__UPDATE_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)) * tyop_2Emin_2Efun(A_27a,A_27b) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).

tff(c_2Eupdate_2ELIST__UPDATE_2E3,type,
    c_2Eupdate_2ELIST__UPDATE_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)) * tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(c_2Elist_2EMAP_2E0,type,
    c_2Elist_2EMAP_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b))) ).

tff(c_2Elist_2EMAP_2E2,type,
    c_2Elist_2EMAP_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27b) ) ).

tff(c_2Elist_2ENIL_2E0,type,
    c_2Elist_2ENIL_2E0: 
      !>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).

tff(c_2Eoption_2ENONE_2E0,type,
    c_2Eoption_2ENONE_2E0: 
      !>[A_27a: $tType] : tyop_2Eoption_2Eoption(A_27a) ).

tff(c_2Eupdate_2EOVERRIDE_2E0,type,
    c_2Eupdate_2EOVERRIDE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))) ).

tff(c_2Eupdate_2EOVERRIDE_2E1,type,
    c_2Eupdate_2EOVERRIDE_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)) > tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)) ) ).

tff(c_2Esorting_2EPERM_2E0,type,
    c_2Esorting_2EPERM_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Esorting_2EPERM_2E2,type,
    c_2Esorting_2EPERM_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Esorting_2EQSORT_2E0,type,
    c_2Esorting_2EQSORT_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Esorting_2EQSORT_2E2,type,
    c_2Esorting_2EQSORT_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Epair_2ESND_2E0,type,
    c_2Epair_2ESND_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).

tff(c_2Epair_2ESND_2E1,type,
    c_2Epair_2ESND_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).

tff(c_2Elist_2ESNOC_2E0,type,
    c_2Elist_2ESNOC_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Elist_2ESNOC_2E2,type,
    c_2Elist_2ESNOC_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Eoption_2ESOME_2E0,type,
    c_2Eoption_2ESOME_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)) ).

tff(c_2Eoption_2ESOME_2E1,type,
    c_2Eoption_2ESOME_2E1: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Eoption_2Eoption(A_27a) ) ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

tff(c_2Ecombin_2EUPDATE_2E0,type,
    c_2Ecombin_2EUPDATE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)))) ).

tff(c_2Ecombin_2EUPDATE_2E2,type,
    c_2Ecombin_2EUPDATE_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ) ).

tff(c_2Ecombin_2EUPDATE_2E3,type,
    c_2Ecombin_2EUPDATE_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).

tff(c_2Ecombin_2EUPDATE_2E4,type,
    c_2Ecombin_2EUPDATE_2E4: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(c_2Erelation_2EWF_2E0,type,
    c_2Erelation_2EWF_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).

tff(c_2Erelation_2EWF_2E1,type,
    c_2Erelation_2EWF_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).

tff(c_2Erelation_2EWFREC_2E0,type,
    c_2Erelation_2EWFREC_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,A_27b))) ).

tff(c_2Erelation_2EWFREC_2E2,type,
    c_2Erelation_2EWFREC_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).

tff(c_2Ebool_2E_5C_2F_2E0,type,
    c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_5C_2F_2E2,type,
    c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Elist_2Elist__CASE_2E0,type,
    c_2Elist_2Elist__CASE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)),A_27b))) ).

tff(c_2Elist_2Elist__CASE_2E3,type,
    c_2Elist_2Elist__CASE_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)) ) > A_27b ) ).

tff(c_2Ecombin_2Eo_2E0,type,
    c_2Ecombin_2Eo_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).

tff(c_2Ecombin_2Eo_2E2,type,
    c_2Ecombin_2Eo_2E2: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).

tff(c_2Eoption_2Eoption__CASE_2E0,type,
    c_2Eoption_2Eoption__CASE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b))) ).

tff(c_2Eoption_2Eoption__CASE_2E3,type,
    c_2Eoption_2Eoption__CASE_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) ) > A_27b ) ).

tff(c_2Epair_2Epair__CASE_2E0,type,
    c_2Epair_2Epair__CASE_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),A_27a)) ).

tff(c_2Epair_2Epair__CASE_2E2,type,
    c_2Epair_2Epair__CASE_2E2: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod(A_27b,A_27c) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)) ) > A_27a ) ).

tff(c_2Ebool_2E_7E_2E0,type,
    c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_7E_2E1,type,
    c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).

tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
      ( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
     => ( V0f_2E0 = V1g_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Eboolext,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(V0_2E0)
      <=> p(V1_2E0) )
     => ( V0_2E0 = V1_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Etruth,axiom,
    p(c_2Ebool_2ET_2E0) ).

tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
    ~ p(c_2Ebool_2EF_2E0) ).

tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0: tyop_2Emin_2Ebool] :
      ( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
      | ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Es__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).

tff(logicdef_2E_2F_5C,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        & p(V1_2E0) ) ) ).

tff(logicdef_2E_5C_2F,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        | p(V1_2E0) ) ) ).

tff(logicdef_2E_7E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_7E_2E1(V0_2E0))
    <=> ~ p(V0_2E0) ) ).

tff(logicdef_2E_3D_3D_3E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
       => p(V1_2E0) ) ) ).

tff(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
      ( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
    <=> ( V0_2E0 = V1_2E0 ) ) ).

tff(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
    <=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
    <=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Epair_2E_2C_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c,X1_2E0: A_27d] : ( c_2Epair_2E_2C_2E2(A_27c,A_27d,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Epair_2E_2C_2E0(A_27c,A_27d),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27e_20mono_2EA_27f,axiom,
    ! [A_27e: $tType,A_27f: $tType,X0_2E0: A_27e,X1_2E0: A_27f] : ( c_2Epair_2E_2C_2E2(A_27e,A_27f,X0_2E0,X1_2E0) = app_2E2(A_27f,tyop_2Epair_2Eprod(A_27e,A_27f),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Epair_2Eprod(A_27e,A_27f)),c_2Epair_2E_2C_2E0(A_27e,A_27f),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27g_20mono_2EA_27h,axiom,
    ! [A_27g: $tType,A_27h: $tType,X0_2E0: A_27g,X1_2E0: A_27h] : ( c_2Epair_2E_2C_2E2(A_27g,A_27h,X0_2E0,X1_2E0) = app_2E2(A_27h,tyop_2Epair_2Eprod(A_27g,A_27h),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(A_27g,A_27h)),c_2Epair_2E_2C_2E0(A_27g,A_27h),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        & p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
      ( ( X0_2E0 = X1_2E0 )
    <=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
       => p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29_2Ctyop_2Emin_2Efun_28tyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29_2Ctyop_2Emin_2Ebool_29_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),c_2Emin_2E_40_2E0(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool))),X0_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2EALL__DISTINCT_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EALL__DISTINCT_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Elist_2EAPPEND_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2EAPPEND_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27e_2CA_27f_29,axiom,
    ! [A_27e: $tType,A_27f: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f))] : ( c_2Elist_2EAPPEND_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f))),c_2Elist_2EAPPEND_2E0(tyop_2Epair_2Eprod(A_27e,A_27f)),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27b,X2_2E0: A_27b] : ( c_2Ebool_2ECOND_2E3(A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),c_2Ebool_2ECOND_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Eoption_2Eoption(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ECONS_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2ECONS_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2ECONS_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27c,A_27d),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))] : ( c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))),c_2Elist_2ECONS_2E0(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27e_2CA_27f_29,axiom,
    ! [A_27e: $tType,A_27f: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27e,A_27f),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f))] : ( c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),app_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f))),c_2Elist_2ECONS_2E0(tyop_2Epair_2Eprod(A_27e,A_27f)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27g_2CA_27h_29,axiom,
    ! [A_27g: $tType,A_27h: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27g,A_27h),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h))] : ( c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),app_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h))),c_2Elist_2ECONS_2E0(tyop_2Epair_2Eprod(A_27g,A_27h)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EFILTER_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Elist_2EFILTER_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2EFILTER_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eupdate_2EFIND_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Eupdate_2EFIND_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27a)),c_2Eupdate_2EFIND_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eupdate_2EFIND_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2EFIND_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Eupdate_2EFIND_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EFST_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,c_2Epair_2EFST_2E0(A_27a,A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecombin_2EI_2E1_2Emono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Ecombin_2EI_2E1(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ecombin_2EI_2E0(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eupdate_2ELIST__UPDATE_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),c_2Eupdate_2ELIST__UPDATE_2E0(A_27a,A_27b),X0_2E0) ) ).

tff(arityeq3_2Ec_2Eupdate_2ELIST__UPDATE_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X2_2E0: A_27a] : ( c_2Eupdate_2ELIST__UPDATE_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),c_2Eupdate_2ELIST__UPDATE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Eupdate_2ELIST__UPDATE_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))] : ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27c,A_27d,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),c_2Eupdate_2ELIST__UPDATE_2E0(A_27c,A_27d),X0_2E0) ) ).

tff(arityeq2_2Ec_2Eupdate_2ELIST__UPDATE_2E2_2Emono_2EA_27e_20mono_2EA_27f,axiom,
    ! [A_27e: $tType,A_27f: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),X1_2E0: tyop_2Emin_2Efun(A_27e,A_27f)] : ( c_2Eupdate_2ELIST__UPDATE_2E2(A_27e,A_27f,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,A_27f),tyop_2Emin_2Efun(A_27e,A_27f),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,A_27f),tyop_2Emin_2Efun(A_27e,A_27f)),c_2Eupdate_2ELIST__UPDATE_2E0(A_27e,A_27f),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eupdate_2ELIST__UPDATE_2E2_2Emono_2EA_27g_20mono_2EA_27h,axiom,
    ! [A_27g: $tType,A_27h: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),X1_2E0: tyop_2Emin_2Efun(A_27g,A_27h)] : ( c_2Eupdate_2ELIST__UPDATE_2E2(A_27g,A_27h,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27g,A_27h),tyop_2Emin_2Efun(A_27g,A_27h),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,A_27h),tyop_2Emin_2Efun(A_27g,A_27h)),c_2Eupdate_2ELIST__UPDATE_2E0(A_27g,A_27h),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Elist_2EMAP_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27a)),c_2Elist_2EMAP_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Eupdate_2EOVERRIDE_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Eupdate_2EOVERRIDE_2E0(A_27a,A_27b),X0_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EPERM_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Esorting_2EPERM_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool),c_2Esorting_2EPERM_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EQSORT_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Esorting_2EQSORT_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Esorting_2EQSORT_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2ESND_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,c_2Epair_2ESND_2E0(A_27a,A_27b),X0_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ESNOC_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27c,A_27d),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))] : ( c_2Elist_2ESNOC_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))),c_2Elist_2ESNOC_2E0(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ESNOC_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27g_2CA_27h_29,axiom,
    ! [A_27g: $tType,A_27h: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27g,A_27h),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h))] : ( c_2Elist_2ESNOC_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),app_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h))),c_2Elist_2ESNOC_2E0(tyop_2Epair_2Eprod(A_27g,A_27h)),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Eoption_2ESOME_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2EUPDATE_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Ecombin_2EUPDATE_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),c_2Ecombin_2EUPDATE_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ecombin_2EUPDATE_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),c_2Ecombin_2EUPDATE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq4_2Ec_2Ecombin_2EUPDATE_2E4_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X3_2E0: A_27a] : ( c_2Ecombin_2EUPDATE_2E4(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(A_27a,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),c_2Ecombin_2EUPDATE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2EUPDATE_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c,X1_2E0: A_27d] : ( c_2Ecombin_2EUPDATE_2E2(A_27c,A_27d,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d))),c_2Ecombin_2EUPDATE_2E0(A_27c,A_27d),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ecombin_2EUPDATE_2E3_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c,X1_2E0: A_27d,X2_2E0: tyop_2Emin_2Efun(A_27c,A_27d)] : ( c_2Ecombin_2EUPDATE_2E3(A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d))),c_2Ecombin_2EUPDATE_2E0(A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ecombin_2EUPDATE_2E3_2Emono_2EA_27e_20mono_2EA_27f,axiom,
    ! [A_27e: $tType,A_27f: $tType,X0_2E0: A_27e,X1_2E0: A_27f,X2_2E0: tyop_2Emin_2Efun(A_27e,A_27f)] : ( c_2Ecombin_2EUPDATE_2E3(A_27e,A_27f,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,A_27f),tyop_2Emin_2Efun(A_27e,A_27f),app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,A_27f),tyop_2Emin_2Efun(A_27e,A_27f)),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,A_27f),tyop_2Emin_2Efun(A_27e,A_27f))),c_2Ecombin_2EUPDATE_2E0(A_27e,A_27f),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ecombin_2EUPDATE_2E3_2Emono_2EA_27g_20mono_2EA_27h,axiom,
    ! [A_27g: $tType,A_27h: $tType,X0_2E0: A_27g,X1_2E0: A_27h,X2_2E0: tyop_2Emin_2Efun(A_27g,A_27h)] : ( c_2Ecombin_2EUPDATE_2E3(A_27g,A_27h,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27g,A_27h),tyop_2Emin_2Efun(A_27g,A_27h),app_2E2(A_27h,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,A_27h),tyop_2Emin_2Efun(A_27g,A_27h)),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,A_27h),tyop_2Emin_2Efun(A_27g,A_27h))),c_2Ecombin_2EUPDATE_2E0(A_27g,A_27h),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),X0_2E0) ) ).

tff(arityeq2_2Ec_2Erelation_2EWFREC_2E2_2Emono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29_20mono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))))] : ( c_2Erelation_2EWFREC_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),c_2Erelation_2EWFREC_2E0(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        | p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq3_2Ec_2Elist_2Elist__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))))] : ( c_2Elist_2Elist__CASE_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),c_2Elist_2Elist__CASE_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),c_2Ecombin_2Eo_2E0(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27c_2CA_27d_29_20mono_2Etyop_2Emin_2Efun_28A_27c_2CA_27d_29_20mono_2Etyop_2Emin_2Efun_28A_27c_2CA_27d_29,axiom,
    ! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d))] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d))),c_2Ecombin_2Eo_2E0(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d)),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b)] : ( c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),A_27b),app_2E2(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),A_27b)),c_2Eoption_2Eoption__CASE_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2EA_27b_20mono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b))] : ( c_2Epair_2Epair__CASE_2E2(A_27b,A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),A_27b,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),A_27b),c_2Epair_2Epair__CASE_2E0(A_27b,A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] :
      ( ~ p(X0_2E0)
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).

tff(thm_2Eupdate_2EFIND__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eupdate_2EFIND_2E2(A_27a,V0P_2E0,c_2Elist_2ENIL_2E0(A_27a)) = c_2Eoption_2ENONE_2E0(A_27a) )
      & ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2h_2E0: A_27a,V3t_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Eupdate_2EFIND_2E2(A_27a,V1P_2E0,c_2Elist_2ECONS_2E2(A_27a,V2h_2E0,V3t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V2h_2E0),c_2Eoption_2ESOME_2E1(A_27a,V2h_2E0),c_2Eupdate_2EFIND_2E2(A_27a,V1P_2E0,V3t_2E0)) ) ) ).

tff(thm_2Eupdate_2EOVERRIDE__primitive__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,F4_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] :
      ( ! [V6x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V8y_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
          ( p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F4_2E0,V6x_2E0),V8y_2E0))
        <=> ( c_2Epair_2EFST_2E1(A_27a,A_27b,V6x_2E0) != c_2Epair_2EFST_2E1(A_27a,A_27b,V8y_2E0) ) )
     => ( ! [V4OVERRIDE_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),V6x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V7t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),F3_2E0,V4OVERRIDE_2E0),V6x_2E0),V7t_2E0) = c_2Ecombin_2EI_2E1(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V6x_2E0,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V4OVERRIDE_2E0,c_2Elist_2EFILTER_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F4_2E0,V6x_2E0),V7t_2E0)))) )
       => ( ! [V4OVERRIDE_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),V5a_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),F1_2E0,V4OVERRIDE_2E0),V5a_2E0) = c_2Elist_2Elist__CASE_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V5a_2E0,c_2Ecombin_2EI_2E1(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),F3_2E0,V4OVERRIDE_2E0)) )
         => ( ! [V2x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V3y_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
                ( p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F2_2E0,V2x_2E0),V3y_2E0))
              <=> ( c_2Epair_2EFST_2E1(A_27a,A_27b,V2x_2E0) != c_2Epair_2EFST_2E1(A_27a,A_27b,V3y_2E0) ) )
           => ( ! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool))] :
                  ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,F0_2E0,V0R_2E0))
                <=> ( p(c_2Erelation_2EWF_2E1(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0R_2E0))
                    & ! [V1t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool),V0R_2E0,c_2Elist_2EFILTER_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F2_2E0,V2x_2E0),V1t_2E0)),c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V2x_2E0,V1t_2E0))) ) )
             => ( c_2Eupdate_2EOVERRIDE_2E0(A_27a,A_27b) = c_2Erelation_2EWFREC_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),F0_2E0),F1_2E0) ) ) ) ) ) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b))) = c_2Ecombin_2EI_2E0(tyop_2Emin_2Efun(A_27a,A_27b)) )
      & ! [V0h_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V0h_2E0,V1t_2E0)) = c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),c_2Ecombin_2EUPDATE_2E2(A_27a,A_27b,c_2Epair_2EFST_2E1(A_27a,A_27b,V0h_2E0),c_2Epair_2ESND_2E1(A_27a,A_27b,V0h_2E0)),c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V1t_2E0)) ) ) ).

tff(thm_2Eupdate_2EOVERRIDE__ind,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] :
      ( ! [V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V3y_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
          ( p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V1x_2E0),V3y_2E0))
        <=> ( c_2Epair_2EFST_2E1(A_27a,A_27b,V1x_2E0) != c_2Epair_2EFST_2E1(A_27a,A_27b,V3y_2E0) ) )
     => ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)] :
          ( ( p(app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,V0P_2E0,c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b))))
            & ! [V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V2t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] :
                ( p(app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,V0P_2E0,c_2Elist_2EFILTER_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V1x_2E0),V2t_2E0)))
               => p(app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,V0P_2E0,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V1x_2E0,V2t_2E0))) ) )
         => ! [V4v_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : p(app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool,V0P_2E0,V4v_2E0)) ) ) ).

tff(thm_2Eupdate_2EOVERRIDE__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] :
      ( ! [V0x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V2y_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
          ( p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V2y_2E0))
        <=> ( c_2Epair_2EFST_2E1(A_27a,A_27b,V0x_2E0) != c_2Epair_2EFST_2E1(A_27a,A_27b,V2y_2E0) ) )
     => ( ( c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b))) = c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)) )
        & ! [V0x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V0x_2E0,V1t_2E0)) = c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V0x_2E0,c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,c_2Elist_2EFILTER_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1t_2E0))) ) ) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__LOOKUP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),F1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] :
      ( ! [V5v1_2E0: A_27a,V6e_2E0: A_27b] : ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),F2_2E0,V5v1_2E0),V6e_2E0) = V6e_2E0 )
     => ( ! [V4v_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,F1_2E0,V4v_2E0) = c_2Epair_2Epair__CASE_2E2(A_27b,A_27a,A_27b,V4v_2E0,F2_2E0) )
       => ( ! [V2i_2E0: A_27a,V3x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
              ( p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V2i_2E0),V3x_2E0))
            <=> ( c_2Epair_2EFST_2E1(A_27a,A_27b,V3x_2E0) = V2i_2E0 ) )
         => ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2i_2E0: A_27a] : ( c_2Eupdate_2ELIST__UPDATE_2E3(A_27a,A_27b,V0l_2E0,V1f_2E0,V2i_2E0) = c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,c_2Eupdate_2EFIND_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),F0_2E0,V2i_2E0),V0l_2E0),app_2E2(A_27a,A_27b,V1f_2E0,V2i_2E0),F1_2E0) ) ) ) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__OVERRIDE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V0l_2E0) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,V0l_2E0)) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__ALL__DISTINCT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0l1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] :
      ( ( p(c_2Elist_2EALL__DISTINCT_2E1(A_27a,c_2Elist_2EMAP_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,c_2Epair_2EFST_2E0(A_27a,A_27b),V1l2_2E0)))
        & p(c_2Esorting_2EPERM_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V0l1_2E0,V1l2_2E0)) )
     => ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V0l1_2E0) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V1l2_2E0) ) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__SORT__OVERRIDE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),V1l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V1l_2E0) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Esorting_2EQSORT_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V0R_2E0,c_2Eupdate_2EOVERRIDE_2E1(A_27a,A_27b,V1l_2E0))) ) ).

tff(thm_2Eupdate_2ELIST__UPDATE__THMS,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType] :
      ( ! [V0l1_2E0: A_27a,V1l2_2E0: A_27a,V2r1_2E0: A_27b,V3r2_2E0: A_27b] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),c_2Ecombin_2EUPDATE_2E2(A_27a,A_27b,V0l1_2E0,V2r1_2E0),c_2Ecombin_2EUPDATE_2E2(A_27a,A_27b,V1l2_2E0,V3r2_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(A_27a,A_27b,V0l1_2E0,V2r1_2E0),c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(A_27a,A_27b,V1l2_2E0,V3r2_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27a,A_27b))))) )
      & ! [V4l_2E0: A_27c,V5r_2E0: A_27d,V6t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),c_2Ecombin_2EUPDATE_2E2(A_27c,A_27d,V4l_2E0,V5r_2E0),c_2Eupdate_2ELIST__UPDATE_2E1(A_27c,A_27d,V6t_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27c,A_27d,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2E_2C_2E2(A_27c,A_27d,V4l_2E0,V5r_2E0),V6t_2E0)) )
      & ! [V7l1_2E0: A_27e,V8l2_2E0: A_27e,V9r1_2E0: A_27f,V10r2_2E0: A_27f,V11f_2E0: tyop_2Emin_2Efun(A_27e,A_27f)] : ( c_2Ecombin_2EUPDATE_2E3(A_27e,A_27f,V7l1_2E0,V9r1_2E0,c_2Ecombin_2EUPDATE_2E3(A_27e,A_27f,V8l2_2E0,V10r2_2E0,V11f_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E2(A_27e,A_27f,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),c_2Epair_2E_2C_2E2(A_27e,A_27f,V7l1_2E0,V9r1_2E0),c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),c_2Epair_2E_2C_2E2(A_27e,A_27f,V8l2_2E0,V10r2_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(A_27e,A_27f)))),V11f_2E0) )
      & ! [V12l_2E0: A_27g,V13r_2E0: A_27h,V14t_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),V15f_2E0: tyop_2Emin_2Efun(A_27g,A_27h)] : ( c_2Ecombin_2EUPDATE_2E3(A_27g,A_27h,V12l_2E0,V13r_2E0,c_2Eupdate_2ELIST__UPDATE_2E2(A_27g,A_27h,V14t_2E0,V15f_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E2(A_27g,A_27h,c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),c_2Epair_2E_2C_2E2(A_27g,A_27h,V12l_2E0,V13r_2E0),V14t_2E0),V15f_2E0) )
      & ! [V16l1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V17l2_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V16l1_2E0),c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,V17l2_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27a,A_27b,c_2Elist_2EAPPEND_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),V16l1_2E0,V17l2_2E0)) )
      & ! [V18l1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),V19l2_2E0: A_27c,V20r_2E0: A_27d] : ( c_2Ecombin_2Eo_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,A_27d),c_2Eupdate_2ELIST__UPDATE_2E1(A_27c,A_27d,V18l1_2E0),c_2Ecombin_2EUPDATE_2E2(A_27c,A_27d,V19l2_2E0,V20r_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E1(A_27c,A_27d,c_2Elist_2ESNOC_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2E_2C_2E2(A_27c,A_27d,V19l2_2E0,V20r_2E0),V18l1_2E0)) )
      & ! [V21l1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),V22l2_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27e,A_27f)),V23f_2E0: tyop_2Emin_2Efun(A_27e,A_27f)] : ( c_2Eupdate_2ELIST__UPDATE_2E2(A_27e,A_27f,V21l1_2E0,c_2Eupdate_2ELIST__UPDATE_2E2(A_27e,A_27f,V22l2_2E0,V23f_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E2(A_27e,A_27f,c_2Elist_2EAPPEND_2E2(tyop_2Epair_2Eprod(A_27e,A_27f),V21l1_2E0,V22l2_2E0),V23f_2E0) )
      & ! [V24l1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27g,A_27h)),V25l2_2E0: A_27g,V26r_2E0: A_27h,V27f_2E0: tyop_2Emin_2Efun(A_27g,A_27h)] : ( c_2Eupdate_2ELIST__UPDATE_2E2(A_27g,A_27h,V24l1_2E0,c_2Ecombin_2EUPDATE_2E3(A_27g,A_27h,V25l2_2E0,V26r_2E0,V27f_2E0)) = c_2Eupdate_2ELIST__UPDATE_2E2(A_27g,A_27h,c_2Elist_2ESNOC_2E2(tyop_2Epair_2Eprod(A_27g,A_27h),c_2Epair_2E_2C_2E2(A_27g,A_27h,V25l2_2E0,V26r_2E0),V24l1_2E0),V27f_2E0) ) ) ).

tff(thm_2Eupdate_2EAPPLY__UPDATE__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a_2E0: A_27a] : ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,app_2E2(A_27a,A_27b,V0f_2E0,V1a_2E0),V0f_2E0) = V0f_2E0 ) ).

tff(thm_2Eupdate_2EAPPLY__UPDATE__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ! [V1a_2E0: A_27a,V3c_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1a_2E0),V3c_2E0))
        <=> ( V1a_2E0 = V3c_2E0 ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a_2E0: A_27a,V2b_2E0: A_27b,V3c_2E0: A_27a] : ( c_2Ecombin_2EUPDATE_2E4(A_27a,A_27b,V1a_2E0,V2b_2E0,V0f_2E0,V3c_2E0) = c_2Ebool_2ECOND_2E3(A_27b,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1a_2E0),V3c_2E0),V2b_2E0,app_2E2(A_27a,A_27b,V0f_2E0,V3c_2E0)) ) ) ).

tff(thm_2Eupdate_2ESAME__KEY__UPDATE__DIFFER,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27c,A_27d),V1f1_2E0: A_27a,V2f2_2E0: A_27b,V3a_2E0: A_27c,V4b_2E0: A_27d,V5c_2E0: A_27d] :
      ( ( V4b_2E0 != V5c_2E0 )
     => ( c_2Ecombin_2EUPDATE_2E3(A_27c,A_27d,V3a_2E0,V4b_2E0,V0f_2E0) != c_2Ecombin_2EUPDATE_2E3(A_27c,A_27d,V3a_2E0,V5c_2E0,V0f_2E0) ) ) ).

tff(thm_2Eupdate_2EUPDATE__APPLY__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a_2E0: A_27a,V2b_2E0: A_27b] :
      ( ( app_2E2(A_27a,A_27b,V0f_2E0,V1a_2E0) = V2b_2E0 )
    <=> ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V2b_2E0,V0f_2E0) = V0f_2E0 ) ) ).

tff(thm_2Eupdate_2EUPDATE__APPLY__IMP__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1b_2E0: A_27b,V2a_2E0: A_27a] :
      ( ( app_2E2(A_27a,A_27b,V0f_2E0,V2a_2E0) = V1b_2E0 )
     => ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V2a_2E0,V1b_2E0,V0f_2E0) = V0f_2E0 ) ) ).

tff(thm_2Eupdate_2EUPDATE__COMMUTES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a_2E0: A_27a,V2b_2E0: A_27a,V3c_2E0: A_27b,V4d_2E0: A_27b] :
      ( ( V1a_2E0 != V2b_2E0 )
     => ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V3c_2E0,c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V2b_2E0,V4d_2E0,V0f_2E0)) = c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V2b_2E0,V4d_2E0,c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V3c_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eupdate_2EUPDATE__EQ,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a_2E0: A_27a,V2b_2E0: A_27b,V3c_2E0: A_27b] : ( c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V3c_2E0,c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V2b_2E0,V0f_2E0)) = c_2Ecombin_2EUPDATE_2E3(A_27a,A_27b,V1a_2E0,V3c_2E0,V0f_2E0) ) ).

tff(thm_2Eupdate_2EUPDATE__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ! [V0a_2E0: A_27a,V3x_27_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V3x_27_2E0))
        <=> ( V0a_2E0 = V3x_27_2E0 ) )
     => ! [V0a_2E0: A_27a,V1b_2E0: A_27b,V2x_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3x_27_2E0: A_27a] : ( c_2Ecombin_2EUPDATE_2E4(A_27a,A_27b,V0a_2E0,V1b_2E0,V2x_2E0,V3x_27_2E0) = c_2Ebool_2ECOND_2E3(A_27b,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V3x_27_2E0),V1b_2E0,app_2E2(A_27a,A_27b,V2x_2E0,V3x_27_2E0)) ) ) ).

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