%------------------------------------------------------------------------------
% File     : ITP101_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    : patricia.ax [Gau19]
%          : HL4101_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  479 ( 171 unt; 192 typ;   0 def)
%            Number of atoms       :  614 ( 354 equ)
%            Maximal formula atoms :   16 (   1 avg)
%            Number of connectives :  360 (  33   ~;  18   |; 131   &)
%                                         (  49 <=>; 129  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   21 (   2 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :  154 (  92   >;  62   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  188 ( 188 usr;  30 con; 0-6 aty)
%            Number of variables   : 1359 (1180   !;  23   ?;1359   :)
%                                         ( 156  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Eind__type_2Erecspace,type,
    tyop_2Eind__type_2Erecspace: $tType > $tType ).

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_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

tff(tyop_2Eone_2Eone,type,
    tyop_2Eone_2Eone: $tType ).

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

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

tff(tyop_2Epatricia_2Eptree,type,
    tyop_2Epatricia_2Eptree: $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_2Earithmetic_2E_2B_2E0,type,
    c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2B_2E2,type,
    c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

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_2Earithmetic_2E_2D_2E0,type,
    c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2D_2E2,type,
    c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

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_2Enum_2E0_2E0,type,
    c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).

tff(c_2Eprim__rec_2E_3C_2E0,type,
    c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Eprim__rec_2E_3C_2E2,type,
    c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > 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_2Epatricia_2EADD_2E0,type,
    c_2Epatricia_2EADD_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a))) ).

tff(c_2Epatricia_2EADD_2E2,type,
    c_2Epatricia_2EADD_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a) ) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Epatricia_2EADD__LIST_2E0,type,
    c_2Epatricia_2EADD__LIST_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)),tyop_2Epatricia_2Eptree(A_27a))) ).

tff(c_2Epatricia_2EADD__LIST_2E2,type,
    c_2Epatricia_2EADD__LIST_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)) ) > tyop_2Epatricia_2Eptree(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_2EARB_2E0,type,
    c_2Ebool_2EARB_2E0: 
      !>[A_27a: $tType] : A_27a ).

tff(c_2Ebit_2EBIT_2E0,type,
    c_2Ebit_2EBIT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Ebit_2EBIT_2E2,type,
    c_2Ebit_2EBIT_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2EBIT1_2E0,type,
    c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT1_2E1,type,
    c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EBIT2_2E0,type,
    c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT2_2E1,type,
    c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Eind__type_2EBOTTOM_2E0,type,
    c_2Eind__type_2EBOTTOM_2E0: 
      !>[A_27a: $tType] : tyop_2Eind__type_2Erecspace(A_27a) ).

tff(c_2Epatricia_2EBRANCH_2E0,type,
    c_2Epatricia_2EBRANCH_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)) ).

tff(c_2Epatricia_2EBRANCH_2E1,type,
    c_2Epatricia_2EBRANCH_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Epatricia_2EBRANCHING__BIT_2E0,type,
    c_2Epatricia_2EBRANCHING__BIT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Epatricia_2EBRANCHING__BIT_2E2,type,
    c_2Epatricia_2EBRANCHING__BIT_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epatricia_2EBranch_2E0,type,
    c_2Epatricia_2EBranch_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))) ).

tff(c_2Epatricia_2EBranch_2E4,type,
    c_2Epatricia_2EBranch_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * tyop_2Enum_2Enum * tyop_2Epatricia_2Eptree(A_27a) * tyop_2Epatricia_2Eptree(A_27a) ) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Ecombin_2EC_2E0,type,
    c_2Ecombin_2EC_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(A_27b,tyop_2Emin_2Efun(A_27a,A_27c))) ).

tff(c_2Ecombin_2EC_2E1,type,
    c_2Ecombin_2EC_2E1: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27c)) ) ).

tff(c_2Epred__set_2ECARD_2E0,type,
    c_2Epred__set_2ECARD_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).

tff(c_2Epred__set_2ECARD_2E1,type,
    c_2Epred__set_2ECARD_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ) ).

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_2Eind__type_2ECONSTR_2E0,type,
    c_2Eind__type_2ECONSTR_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)),tyop_2Eind__type_2Erecspace(A_27a)))) ).

tff(c_2Eind__type_2ECONSTR_2E3,type,
    c_2Eind__type_2ECONSTR_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)) ) > tyop_2Eind__type_2Erecspace(A_27a) ) ).

tff(c_2Ebool_2EDATATYPE_2E0,type,
    c_2Ebool_2EDATATYPE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2EDATATYPE_2E1,type,
    c_2Ebool_2EDATATYPE_2E1: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EDELETE_2E0,type,
    c_2Epred__set_2EDELETE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EDELETE_2E2,type,
    c_2Epred__set_2EDELETE_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27a ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epatricia_2EDEPTH_2E0,type,
    c_2Epatricia_2EDEPTH_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum) ).

tff(c_2Epatricia_2EDEPTH_2E1,type,
    c_2Epatricia_2EDEPTH_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Enum_2Enum ) ).

tff(c_2Earithmetic_2EDIV2_2E0,type,
    c_2Earithmetic_2EDIV2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EDIV2_2E1,type,
    c_2Earithmetic_2EDIV2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Elist_2EEL_2E0,type,
    c_2Elist_2EEL_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a)) ).

tff(c_2Elist_2EEL_2E2,type,
    c_2Elist_2EEL_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * tyop_2Elist_2Elist(A_27a) ) > A_27a ) ).

tff(c_2Epred__set_2EEMPTY_2E0,type,
    c_2Epred__set_2EEMPTY_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EEVEN_2E0,type,
    c_2Earithmetic_2EEVEN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EEVEN_2E1,type,
    c_2Earithmetic_2EEVEN_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).

tff(c_2Epatricia_2EEVERY__LEAF_2E0,type,
    c_2Epatricia_2EEVERY__LEAF_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Epatricia_2EEVERY__LEAF_2E2,type,
    c_2Epatricia_2EEVERY__LEAF_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Epatricia_2Eptree(A_27a) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epatricia_2EEXISTS__LEAF_2E0,type,
    c_2Epatricia_2EEXISTS__LEAF_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Epatricia_2EEXISTS__LEAF_2E2,type,
    c_2Epatricia_2EEXISTS__LEAF_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Epatricia_2Eptree(A_27a) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2EEXP_2E0,type,
    c_2Earithmetic_2EEXP_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EEXP_2E2,type,
    c_2Earithmetic_2EEXP_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epatricia_2EEmpty_2E0,type,
    c_2Epatricia_2EEmpty_2E0: 
      !>[A_27a: $tType] : tyop_2Epatricia_2Eptree(A_27a) ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Eind__type_2EFCONS_2E0,type,
    c_2Eind__type_2EFCONS_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))) ).

tff(c_2Eind__type_2EFCONS_2E2,type,
    c_2Eind__type_2EFCONS_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) ).

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_2Epatricia_2EFIND_2E0,type,
    c_2Epatricia_2EFIND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)) ).

tff(c_2Epatricia_2EFIND_2E2,type,
    c_2Epatricia_2EFIND_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Enum_2Enum ) > A_27a ) ).

tff(c_2Epred__set_2EFINITE_2E0,type,
    c_2Epred__set_2EFINITE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Epred__set_2EFINITE_2E1,type,
    c_2Epred__set_2EFINITE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Elist_2EFOLDL_2E0,type,
    c_2Elist_2EFOLDL_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b))) ).

tff(c_2Elist_2EFOLDL_2E1,type,
    c_2Elist_2EFOLDL_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27b)) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)) ) ).

tff(c_2Elist_2EFOLDL_2E3,type,
    c_2Elist_2EFOLDL_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27b)) * A_27b * tyop_2Elist_2Elist(A_27a) ) > A_27b ) ).

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_2Ebool_2EIN_2E0,type,
    c_2Ebool_2EIN_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2EIN_2E2,type,
    c_2Ebool_2EIN_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EINSERT_2E0,type,
    c_2Epred__set_2EINSERT_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EINSERT_2E2,type,
    c_2Epred__set_2EINSERT_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epatricia_2EINSERT__PTREE_2E0,type,
    c_2Epatricia_2EINSERT__PTREE_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))) ).

tff(c_2Epatricia_2EINSERT__PTREE_2E2,type,
    c_2Epatricia_2EINSERT__PTREE_2E2: ( tyop_2Enum_2Enum * tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ) > tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ).

tff(c_2Epatricia_2EIN__PTREE_2E0,type,
    c_2Epatricia_2EIN__PTREE_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Ebool)) ).

tff(c_2Epatricia_2EIN__PTREE_2E2,type,
    c_2Epatricia_2EIN__PTREE_2E2: ( tyop_2Enum_2Enum * tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ) > tyop_2Emin_2Ebool ).

tff(c_2Epatricia_2EIS__EMPTY_2E0,type,
    c_2Epatricia_2EIS__EMPTY_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool) ).

tff(c_2Epatricia_2EIS__EMPTY_2E1,type,
    c_2Epatricia_2EIS__EMPTY_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Emin_2Ebool ) ).

tff(c_2Epatricia_2EIS__PTREE_2E0,type,
    c_2Epatricia_2EIS__PTREE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool) ).

tff(c_2Epatricia_2EIS__PTREE_2E1,type,
    c_2Epatricia_2EIS__PTREE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Emin_2Ebool ) ).

tff(c_2Eoption_2EIS__SOME_2E0,type,
    c_2Eoption_2EIS__SOME_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool) ).

tff(c_2Eoption_2EIS__SOME_2E1,type,
    c_2Eoption_2EIS__SOME_2E1: 
      !>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(A_27a) > tyop_2Emin_2Ebool ) ).

tff(c_2Epatricia_2EJOIN_2E0,type,
    c_2Epatricia_2EJOIN_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)) ).

tff(c_2Epatricia_2EJOIN_2E1,type,
    c_2Epatricia_2EJOIN_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Epatricia_2EKEYS_2E0,type,
    c_2Epatricia_2EKEYS_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Elist_2Elist(tyop_2Enum_2Enum)) ).

tff(c_2Epatricia_2EKEYS_2E1,type,
    c_2Epatricia_2EKEYS_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Elist_2Elist(tyop_2Enum_2Enum) ) ).

tff(c_2Elist_2ELENGTH_2E0,type,
    c_2Elist_2ELENGTH_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum) ).

tff(c_2Elist_2ELENGTH_2E1,type,
    c_2Elist_2ELENGTH_2E1: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > tyop_2Enum_2Enum ) ).

tff(c_2Ebool_2ELET_2E0,type,
    c_2Ebool_2ELET_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ).

tff(c_2Ebool_2ELET_2E2,type,
    c_2Ebool_2ELET_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(c_2Elist_2ELIST__TO__SET_2E0,type,
    c_2Elist_2ELIST__TO__SET_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Elist_2ELIST__TO__SET_2E1,type,
    c_2Elist_2ELIST__TO__SET_2E1: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epatricia_2ELeaf_2E0,type,
    c_2Epatricia_2ELeaf_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))) ).

tff(c_2Epatricia_2ELeaf_2E2,type,
    c_2Epatricia_2ELeaf_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a ) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Earithmetic_2EMAX_2E0,type,
    c_2Earithmetic_2EMAX_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EMAX_2E2,type,
    c_2Earithmetic_2EMAX_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Ebit_2EMOD__2EXP_2E0,type,
    c_2Ebit_2EMOD__2EXP_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Ebit_2EMOD__2EXP_2E2,type,
    c_2Ebit_2EMOD__2EXP_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Ebit_2EMOD__2EXP__EQ_2E0,type,
    c_2Ebit_2EMOD__2EXP__EQ_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))) ).

tff(c_2Ebit_2EMOD__2EXP__EQ_2E3,type,
    c_2Ebit_2EMOD__2EXP__EQ_2E3: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

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_2Earithmetic_2ENUMERAL_2E0,type,
    c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2ENUMERAL_2E1,type,
    c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Epatricia_2ENUMSET__OF__PTREE_2E0,type,
    c_2Epatricia_2ENUMSET__OF__PTREE_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Epatricia_2ENUMSET__OF__PTREE_2E1,type,
    c_2Epatricia_2ENUMSET__OF__PTREE_2E1: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EODD_2E0,type,
    c_2Earithmetic_2EODD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EODD_2E1,type,
    c_2Earithmetic_2EODD_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).

tff(c_2Epatricia_2EPEEK_2E0,type,
    c_2Epatricia_2EPEEK_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27a))) ).

tff(c_2Epatricia_2EPEEK_2E2,type,
    c_2Epatricia_2EPEEK_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Enum_2Enum ) > tyop_2Eoption_2Eoption(A_27a) ) ).

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_2Epatricia_2EPTREE__OF__NUMSET_2E0,type,
    c_2Epatricia_2EPTREE__OF__NUMSET_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))) ).

tff(c_2Epatricia_2EPTREE__OF__NUMSET_2E2,type,
    c_2Epatricia_2EPTREE__OF__NUMSET_2E2: ( tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ) > tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ).

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_2Epatricia_2EREMOVE_2E0,type,
    c_2Epatricia_2EREMOVE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))) ).

tff(c_2Epatricia_2EREMOVE_2E2,type,
    c_2Epatricia_2EREMOVE_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Enum_2Enum ) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Elist_2ESET__TO__LIST_2E0,type,
    c_2Elist_2ESET__TO__LIST_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Elist_2Elist(A_27a)) ).

tff(c_2Elist_2ESET__TO__LIST_2E1,type,
    c_2Elist_2ESET__TO__LIST_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Epatricia_2ESIZE_2E0,type,
    c_2Epatricia_2ESIZE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum) ).

tff(c_2Epatricia_2ESIZE_2E1,type,
    c_2Epatricia_2ESIZE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Enum_2Enum ) ).

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_2Enum_2ESUC_2E0,type,
    c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enum_2ESUC_2E1,type,
    c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

tff(c_2Eoption_2ETHE_2E0,type,
    c_2Eoption_2ETHE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),A_27a) ).

tff(c_2Eoption_2ETHE_2E1,type,
    c_2Eoption_2ETHE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Eoption_2Eoption(A_27a) > A_27a ) ).

tff(c_2Epatricia_2ETRANSFORM_2E0,type,
    c_2Epatricia_2ETRANSFORM_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27a))) ).

tff(c_2Epatricia_2ETRANSFORM_2E2,type,
    c_2Epatricia_2ETRANSFORM_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,A_27a) * tyop_2Epatricia_2Eptree(A_27b) ) > tyop_2Epatricia_2Eptree(A_27a) ) ).

tff(c_2Epatricia_2ETRAVERSE_2E0,type,
    c_2Epatricia_2ETRAVERSE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Elist_2Elist(tyop_2Enum_2Enum)) ).

tff(c_2Epatricia_2ETRAVERSE_2E1,type,
    c_2Epatricia_2ETRAVERSE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree(A_27a) > tyop_2Elist_2Elist(tyop_2Enum_2Enum) ) ).

tff(c_2Epatricia_2ETRAVERSE__AUX_2E0,type,
    c_2Epatricia_2ETRAVERSE__AUX_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum))) ).

tff(c_2Epatricia_2ETRAVERSE__AUX_2E2,type,
    c_2Epatricia_2ETRAVERSE__AUX_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * tyop_2Elist_2Elist(tyop_2Enum_2Enum) ) > tyop_2Elist_2Elist(tyop_2Enum_2Enum) ) ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
    c_2Ebool_2ETYPE__DEFINITION_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
    c_2Ebool_2ETYPE__DEFINITION_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EUNION_2E0,type,
    c_2Epred__set_2EUNION_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EUNION_2E2,type,
    c_2Epred__set_2EUNION_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epatricia_2EUNION__PTREE_2E0,type,
    c_2Epatricia_2EUNION__PTREE_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))) ).

tff(c_2Epatricia_2EUNION__PTREE_2E2,type,
    c_2Epatricia_2EUNION__PTREE_2E2: ( tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) * tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ) > tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone) ).

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_2Earithmetic_2EZERO_2E0,type,
    c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).

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_2Eone_2Eone_2E0,type,
    c_2Eone_2Eone_2E0: tyop_2Eone_2Eone ).

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_2Epatricia_2Eptree__CASE_2E0,type,
    c_2Epatricia_2Eptree__CASE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b)))) ).

tff(c_2Epatricia_2Eptree__CASE_2E4,type,
    c_2Epatricia_2Eptree__CASE_2E4: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epatricia_2Eptree(A_27a) * A_27b * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))) ) > A_27b ) ).

tff(c_2Epatricia_2Eptree__size_2E0,type,
    c_2Epatricia_2Eptree__size_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum)) ).

tff(c_2Epatricia_2Eptree__size_2E2,type,
    c_2Epatricia_2Eptree__size_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * tyop_2Epatricia_2Eptree(A_27a) ) > tyop_2Enum_2Enum ) ).

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_2Earithmetic_2E_2B_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),c_2Epair_2E_2C_2E0(A_27a,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27b)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Eone_2Eone] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Eone_2Eone,X0_2E0,X1_2E0) = app_2E2(tyop_2Eone_2Eone,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Eone_2Eone),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Eone_2Eone)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Eone_2Eone),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Enum_2Enum_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27b_29_2Ctyop_2Epatricia_2Eptree_28A_27b_29_29_29,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27b_29_2Ctyop_2Epatricia_2Eptree_28A_27b_29_29,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))] : ( c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),c_2Epair_2E_2C_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),c_2Epair_2E_2C_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27b_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27b_29,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b),X1_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),c_2Epair_2E_2C_2E0(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,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_2Eprim__rec_2E_3C_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_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_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29_29_2Ctyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29_29_2Ctyop_2Emin_2Ebool_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),c_2Emin_2E_40_2E0(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool))),X0_2E0) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Emin_2Efun_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Emin_2Ebool_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),c_2Emin_2E_40_2E0(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EADD_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)] : ( c_2Epatricia_2EADD_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EADD_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EADD_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27b)] : ( c_2Epatricia_2EADD_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27b),tyop_2Epatricia_2Eptree(A_27b),app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27b),tyop_2Epatricia_2Eptree(A_27b)),c_2Epatricia_2EADD_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EADD_2E2_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Eone_2Eone)] : ( c_2Epatricia_2EADD_2E2(tyop_2Eone_2Eone,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),c_2Epatricia_2EADD_2E0(tyop_2Eone_2Eone),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EADD__LIST_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a))] : ( c_2Epatricia_2EADD__LIST_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)),tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EADD__LIST_2E0(A_27a),X0_2E0),X1_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(arityeq1_2Ec_2Elist_2EALL__DISTINCT_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2EALL__DISTINCT_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum),X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2EAPPEND_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum),app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum)),c_2Elist_2EAPPEND_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebit_2EBIT_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebit_2EBIT_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Ebit_2EBIT_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EBRANCH_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))] : ( c_2Epatricia_2EBRANCH_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBRANCH_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EBRANCH_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))))] : ( c_2Epatricia_2EBRANCH_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))),tyop_2Epatricia_2Eptree(A_27b),c_2Epatricia_2EBRANCH_2E0(A_27b),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EBRANCHING__BIT_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EBRANCHING__BIT_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epatricia_2EBRANCHING__BIT_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2EBranch_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Epatricia_2Eptree(A_27a),X3_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EBranch_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),c_2Epatricia_2EBranch_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2EBranch_2E4_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Epatricia_2Eptree(A_27b),X3_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2EBranch_2E4(A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b),app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)))),c_2Epatricia_2EBranch_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2EBranch_2E4_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Epatricia_2Eptree(A_27c),X3_2E0: tyop_2Epatricia_2Eptree(A_27c)] : ( c_2Epatricia_2EBranch_2E4(A_27c,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27c),tyop_2Epatricia_2Eptree(A_27c),app_2E2(tyop_2Epatricia_2Eptree(A_27c),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27c),tyop_2Epatricia_2Eptree(A_27c)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27c),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27c),tyop_2Epatricia_2Eptree(A_27c))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27c),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27c),tyop_2Epatricia_2Eptree(A_27c)))),c_2Epatricia_2EBranch_2E0(A_27c),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2Ecombin_2EC_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epatricia_2Eptree_28tyop_2Eone_2Eone_29_20mono_2Etyop_2Epatricia_2Eptree_28tyop_2Eone_2Eone_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)))] : ( c_2Ecombin_2EC_2E1(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))),c_2Ecombin_2EC_2E0(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ECOND_2E0(tyop_2Enum_2Enum),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(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28tyop_2Eone_2Eone_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),X2_2E0: tyop_2Eoption_2Eoption(tyop_2Eone_2Eone)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),app_2E2(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Eoption_2Eoption(tyop_2Eone_2Eone)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Eoption_2Eoption(tyop_2Eone_2Eone))),c_2Ebool_2ECOND_2E0(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Epatricia_2Eptree(A_27a),X2_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2ECONS_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum)),c_2Elist_2ECONS_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28A_27a_2Ctyop_2Enum_2Enum_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))] : ( c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))),c_2Eind__type_2ECONSTR_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(tyop_2Emin_2Ebool),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: A_27a] : ( c_2Epred__set_2EDELETE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EDELETE_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EDEPTH_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EDEPTH_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,c_2Epatricia_2EDEPTH_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EDIV2_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EDIV2_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EEL_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EEL_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a),c_2Elist_2EEL_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EEVEN_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EEVEN_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EEVERY__LEAF_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),c_2Epatricia_2EEVERY__LEAF_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EEVERY__LEAF_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2EEVERY__LEAF_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Ebool),c_2Epatricia_2EEVERY__LEAF_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EEXISTS__LEAF_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),c_2Epatricia_2EEXISTS__LEAF_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EEXP_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EEXP_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28A_27a_2Ctyop_2Enum_2Enum_29_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))] : ( c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))),c_2Eind__type_2EFCONS_2E0(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EFILTER_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_2Elist_2EFILTER_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2EFILTER_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2EFILTER_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2EFILTER_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum)),c_2Elist_2EFILTER_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EFIND_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EFIND_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),c_2Epatricia_2EFIND_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2EFOLDL_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2CA_27a_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Elist_2EFOLDL_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)),tyop_2Epatricia_2Eptree(A_27a))),c_2Elist_2EFOLDL_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0) ) ).

tff(arityeq3_2Ec_2Elist_2EFOLDL_2E3_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epatricia_2Eptree_28tyop_2Eone_2Eone_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))),X1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X2_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2EFOLDL_2E3(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone))),c_2Elist_2EFOLDL_2E0(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ecombin_2EI_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] : ( c_2Ecombin_2EI_2E1(tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ecombin_2EI_2E0(tyop_2Emin_2Ebool),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecombin_2EI_2E1_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Ecombin_2EI_2E0(tyop_2Epatricia_2Eptree(A_27a)),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EINSERT__PTREE_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EINSERT__PTREE_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),c_2Epatricia_2EINSERT__PTREE_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EIN__PTREE_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EIN__PTREE_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Ebool),c_2Epatricia_2EIN__PTREE_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EIS__EMPTY_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EIS__EMPTY_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,c_2Epatricia_2EIS__EMPTY_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EIS__PTREE_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EIS__PTREE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,c_2Epatricia_2EIS__PTREE_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EIS__PTREE_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2EIS__PTREE_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Ebool,c_2Epatricia_2EIS__PTREE_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EIS__PTREE_2E1_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27c)] : ( c_2Epatricia_2EIS__PTREE_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27c),tyop_2Emin_2Ebool,c_2Epatricia_2EIS__PTREE_2E0(A_27c),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EIS__PTREE_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Ebool,c_2Epatricia_2EIS__PTREE_2E0(tyop_2Eone_2Eone),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2EIS__SOME_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : ( c_2Eoption_2EIS__SOME_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Ebool,c_2Eoption_2EIS__SOME_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2EIS__SOME_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27b)] : ( c_2Eoption_2EIS__SOME_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Ebool,c_2Eoption_2EIS__SOME_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2EIS__SOME_2E1_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27c)] : ( c_2Eoption_2EIS__SOME_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27c),tyop_2Emin_2Ebool,c_2Eoption_2EIS__SOME_2E0(A_27c),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2EIS__SOME_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Eoption_2Eoption(tyop_2Eone_2Eone)] : ( c_2Eoption_2EIS__SOME_2E1(tyop_2Eone_2Eone,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),tyop_2Emin_2Ebool,c_2Eoption_2EIS__SOME_2E0(tyop_2Eone_2Eone),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EJOIN_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))))] : ( c_2Epatricia_2EJOIN_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EJOIN_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EKEYS_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EKEYS_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Epatricia_2EKEYS_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2EKEYS_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2EKEYS_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Epatricia_2EKEYS_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ELENGTH_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2ELENGTH_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),c_2Ebool_2ELET_2E0(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2ELIST__TO__SET_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ELIST__TO__SET_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2ELIST__TO__SET_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2ELeaf_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27a] : ( c_2Epatricia_2ELeaf_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2ELeaf_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2ELeaf_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: A_27b] : ( c_2Epatricia_2ELeaf_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epatricia_2Eptree(A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Epatricia_2Eptree(A_27b)),c_2Epatricia_2ELeaf_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EMAX_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMAX_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMAX_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebit_2EMOD__2EXP_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebit_2EMOD__2EXP_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ebit_2EMOD__2EXP_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ebit_2EMOD__2EXP__EQ_2E3,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebit_2EMOD__2EXP__EQ_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Ebit_2EMOD__2EXP__EQ_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ENUMSET__OF__PTREE_2E1,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2ENUMSET__OF__PTREE_2E1(X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epatricia_2ENUMSET__OF__PTREE_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EODD_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EODD_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EODD_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EPEEK_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27a)),c_2Epatricia_2EPEEK_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EPEEK_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27b),app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27b)),c_2Epatricia_2EPEEK_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EPEEK_2E2_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27c),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27c),app_2E2(tyop_2Epatricia_2Eptree(A_27c),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(A_27c)),c_2Epatricia_2EPEEK_2E0(A_27c),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EPEEK_2E2_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(tyop_2Eone_2Eone,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Eone_2Eone)),c_2Epatricia_2EPEEK_2E0(tyop_2Eone_2Eone),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EPERM_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Esorting_2EPERM_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Esorting_2EPERM_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EPERM_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum),X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Esorting_2EPERM_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),c_2Esorting_2EPERM_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EPTREE__OF__NUMSET_2E2,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),c_2Epatricia_2EPTREE__OF__NUMSET_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EQSORT_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Esorting_2EQSORT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(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)),c_2Esorting_2EQSORT_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Esorting_2EQSORT_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Esorting_2EQSORT_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum)),c_2Esorting_2EQSORT_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EREMOVE_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EREMOVE_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EREMOVE_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27b),app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27b)),c_2Epatricia_2EREMOVE_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EREMOVE_2E2_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(tyop_2Eone_2Eone,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),c_2Epatricia_2EREMOVE_2E0(tyop_2Eone_2Eone),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Elist_2ESET__TO__LIST_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Elist_2ESET__TO__LIST_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Elist_2ESET__TO__LIST_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ESIZE_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ESIZE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,c_2Epatricia_2ESIZE_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ESIZE_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2ESIZE_2E1(tyop_2Eone_2Eone,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Enum_2Enum,c_2Epatricia_2ESIZE_2E0(tyop_2Eone_2Eone),X0_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(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: A_27b] : ( c_2Eoption_2ESOME_2E1(A_27b,X0_2E0) = app_2E2(A_27b,tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Eone_2Eone] : ( c_2Eoption_2ESOME_2E1(tyop_2Eone_2Eone,X0_2E0) = app_2E2(tyop_2Eone_2Eone,tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),c_2Eoption_2ESOME_2E0(tyop_2Eone_2Eone),X0_2E0) ) ).

tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Eoption_2ETHE_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a)] : ( c_2Eoption_2ETHE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),A_27a,c_2Eoption_2ETHE_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2ETRANSFORM_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27b)),c_2Epatricia_2ETRANSFORM_2E0(A_27b,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2ETRANSFORM_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X1_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2ETRANSFORM_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ETRAVERSE_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Epatricia_2ETRAVERSE_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ETRAVERSE_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Epatricia_2ETRAVERSE_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epatricia_2ETRAVERSE_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,X0_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Elist_2Elist(tyop_2Enum_2Enum),c_2Epatricia_2ETRAVERSE_2E0(tyop_2Eone_2Eone),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2ETRAVERSE__AUX_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Enum_2Enum),tyop_2Elist_2Elist(tyop_2Enum_2Enum)),c_2Epatricia_2ETRAVERSE__AUX_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28A_27a_2Ctyop_2Enum_2Enum_29_29_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2EUNION__PTREE_2E2,axiom,
    ! [X0_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),X1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EUNION__PTREE_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),app_2E2(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)),c_2Epatricia_2EUNION__PTREE_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),X0_2E0) ) ).

tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Epatricia_2Eptree(A_27a),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Epatricia_2Eptree(A_27a)),X0_2E0) ) ).

tff(arityeq2_2Ec_2Erelation_2EWFREC_2E2_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Erelation_2EWFREC_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a))),c_2Erelation_2EWFREC_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Erelation_2EWFREC_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWFREC_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),c_2Erelation_2EWFREC_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),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_2Eoption_2Eoption__CASE_2E3_2Emono_2EA_27a_20mono_2Etyop_2Eoption_2Eoption_28A_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(A_27a),X1_2E0: tyop_2Eoption_2Eoption(A_27b),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))] : ( c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27b),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b),app_2E2(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b)),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b))),c_2Eoption_2Eoption__CASE_2E0(A_27a,tyop_2Eoption_2Eoption(A_27b)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a)),c_2Epair_2Epair__CASE_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epatricia_2Eptree_28A_27a_29_2Ctyop_2Epatricia_2Eptree_28A_27a_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a)),c_2Epair_2Epair__CASE_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),X1_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a)),c_2Epair_2Epair__CASE_2E0(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2Eptree__CASE_2E4_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),X3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b))),c_2Epatricia_2Eptree__CASE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2Eptree__CASE_2E4_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Emin_2Ebool,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Emin_2Ebool,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool))),c_2Epatricia_2Eptree__CASE_2E0(A_27a,tyop_2Emin_2Ebool),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq4_2Ec_2Epatricia_2Eptree__CASE_2E4_2Emono_2EA_27a_20mono_2Etyop_2Epatricia_2Eptree_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Epatricia_2Eptree(A_27a),X1_2E0: tyop_2Epatricia_2Eptree(A_27a),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),X3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Epatricia_2Eptree(A_27a),X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Epatricia_2Eptree(A_27a)))),c_2Epatricia_2Eptree__CASE_2E0(A_27a,tyop_2Epatricia_2Eptree(A_27a)),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq2_2Ec_2Epatricia_2Eptree__size_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2Eptree__size_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum),c_2Epatricia_2Eptree__size_2E0(A_27a),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_2Epatricia_2Eptree__TY__DEF,axiom,
    ! [A_27a: $tType,F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool)] :
      ( ! [V12n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),F3_2E0,V12n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))) )
     => ( ! [V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),F2_2E0,V7n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))) )
       => ( ! [V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),F1_2E0,V4n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))) )
         => ( ! [V1a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))] :
                ( p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool,F0_2E0,V1a0_27_2E0))
              <=> ! [V2_27ptree_27_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool)] :
                    ( ! [V3a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))] :
                        ( ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),c_2Enum_2E0_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),c_2Ebool_2EARB_2E0(tyop_2Enum_2Enum),c_2Epair_2E_2C_2E2(A_27a,tyop_2Enum_2Enum,c_2Ebool_2EARB_2E0(A_27a),c_2Ebool_2EARB_2E0(tyop_2Enum_2Enum))),F1_2E0) )
                          | ? [V5a0_2E0: tyop_2Enum_2Enum,V6a1_2E0: A_27a] : ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),V5a0_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Enum_2Enum,V6a1_2E0,c_2Ebool_2EARB_2E0(tyop_2Enum_2Enum))),F2_2E0) )
                          | ? [V8a0_2E0: tyop_2Enum_2Enum,V9a1_2E0: tyop_2Enum_2Enum,V10a2_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),V11a3_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)))] :
                              ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum)),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum),V8a0_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Enum_2Enum,c_2Ebool_2EARB_2E0(A_27a),V9a1_2E0)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),V10a2_2E0,c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),V11a3_2E0,F3_2E0))) )
                              & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool,V2_27ptree_27_2E0,V10a2_2E0))
                              & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool,V2_27ptree_27_2E0,V11a3_2E0)) ) )
                       => p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool,V2_27ptree_27_2E0,V3a0_27_2E0)) )
                   => p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Emin_2Ebool,V2_27ptree_27_2E0,V1a0_27_2E0)) ) )
           => ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(A_27a,tyop_2Enum_2Enum))),tyop_2Epatricia_2Eptree(A_27a),F0_2E0,V0rep_2E0)) ) ) ) ) ).

tff(thm_2Epatricia_2Eptree__case__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0v_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V2f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,c_2Epatricia_2EEmpty_2E0(A_27a),V0v_2E0,V1f_2E0,V2f1_2E0) = V0v_2E0 )
      & ! [V3a0_2E0: tyop_2Enum_2Enum,V4a1_2E0: A_27a,V5v_2E0: A_27b,V6f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V7f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,c_2Epatricia_2ELeaf_2E2(A_27a,V3a0_2E0,V4a1_2E0),V5v_2E0,V6f_2E0,V7f1_2E0) = app_2E2(A_27a,A_27b,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b),V6f_2E0,V3a0_2E0),V4a1_2E0) )
      & ! [V8a0_2E0: tyop_2Enum_2Enum,V9a1_2E0: tyop_2Enum_2Enum,V10a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V11a3_2E0: tyop_2Epatricia_2Eptree(A_27a),V12v_2E0: A_27b,V13f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V14f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))))] : ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,c_2Epatricia_2EBranch_2E4(A_27a,V8a0_2E0,V9a1_2E0,V10a2_2E0,V11a3_2E0),V12v_2E0,V13f_2E0,V14f1_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))),V14f1_2E0,V8a0_2E0),V9a1_2E0),V10a2_2E0),V11a3_2E0) ) ) ).

tff(thm_2Epatricia_2Eptree__size__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Epatricia_2Eptree__size_2E2(A_27a,V0f_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Enum_2E0_2E0 )
      & ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2a0_2E0: tyop_2Enum_2Enum,V3a1_2E0: A_27a] : ( c_2Epatricia_2Eptree__size_2E2(A_27a,V1f_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V2a0_2E0,V3a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(V2a0_2E0,app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3a1_2E0))) )
      & ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5a0_2E0: tyop_2Enum_2Enum,V6a1_2E0: tyop_2Enum_2Enum,V7a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V8a3_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2Eptree__size_2E2(A_27a,V4f_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V5a0_2E0,V6a1_2E0,V7a2_2E0,V8a3_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(V5a0_2E0,c_2Earithmetic_2E_2B_2E2(V6a1_2E0,c_2Earithmetic_2E_2B_2E2(c_2Epatricia_2Eptree__size_2E2(A_27a,V4f_2E0,V7a2_2E0),c_2Epatricia_2Eptree__size_2E2(A_27a,V4f_2E0,V8a3_2E0))))) ) ) ).

tff(thm_2Epatricia_2EJOIN__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))))] :
      ( ! [V0p0_2E0: tyop_2Enum_2Enum,V1t0_2E0: tyop_2Epatricia_2Eptree(A_27a),V3t1_2E0: tyop_2Epatricia_2Eptree(A_27a),V4m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),F0_2E0,V0p0_2E0),V1t0_2E0),V3t1_2E0),V4m_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EBIT_2E2(V4m_2E0,V0p0_2E0),c_2Epatricia_2EBranch_2E4(A_27a,c_2Ebit_2EMOD__2EXP_2E2(V4m_2E0,V0p0_2E0),V4m_2E0,V1t0_2E0,V3t1_2E0),c_2Epatricia_2EBranch_2E4(A_27a,c_2Ebit_2EMOD__2EXP_2E2(V4m_2E0,V0p0_2E0),V4m_2E0,V3t1_2E0,V1t0_2E0)) )
     => ! [V0p0_2E0: tyop_2Enum_2Enum,V1t0_2E0: tyop_2Epatricia_2Eptree(A_27a),V2p1_2E0: tyop_2Enum_2Enum,V3t1_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EJOIN_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),V0p0_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),V1t0_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),V2p1_2E0,V3t1_2E0)))) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)))),F0_2E0,V0p0_2E0),V1t0_2E0),V3t1_2E0),c_2Epatricia_2EBRANCHING__BIT_2E2(V0p0_2E0,V2p1_2E0)) ) ) ).

tff(thm_2Epatricia_2EBRANCH__primitive__def,axiom,
    ! [A_27a: $tType,F10_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))))))),F9_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))))))),F6_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))),F8_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))))),F7_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))))),F5_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] :
      ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V17v20_2E0: tyop_2Enum_2Enum,V18v21_2E0: tyop_2Enum_2Enum,V19v22_2E0: tyop_2Epatricia_2Eptree(A_27a),V20v23_2E0: tyop_2Epatricia_2Eptree(A_27a),V23v44_2E0: tyop_2Enum_2Enum,V24v45_2E0: tyop_2Enum_2Enum,V25v46_2E0: tyop_2Epatricia_2Eptree(A_27a),V26v47_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))))),F10_2E0,V3p_2E0),V5m_2E0),V17v20_2E0),V18v21_2E0),V19v22_2E0),V20v23_2E0),V23v44_2E0),V24v45_2E0),V25v46_2E0),V26v47_2E0) = c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V3p_2E0,V5m_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V17v20_2E0,V18v21_2E0,V19v22_2E0,V20v23_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V23v44_2E0,V24v45_2E0,V25v46_2E0,V26v47_2E0))) )
     => ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V17v20_2E0: tyop_2Enum_2Enum,V18v21_2E0: tyop_2Enum_2Enum,V19v22_2E0: tyop_2Epatricia_2Eptree(A_27a),V20v23_2E0: tyop_2Epatricia_2Eptree(A_27a),V21v42_2E0: tyop_2Enum_2Enum,V22v43_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))))),F9_2E0,V3p_2E0),V5m_2E0),V17v20_2E0),V18v21_2E0),V19v22_2E0),V20v23_2E0),V21v42_2E0),V22v43_2E0) = c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V3p_2E0,V5m_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V17v20_2E0,V18v21_2E0,V19v22_2E0,V20v23_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V21v42_2E0,V22v43_2E0))) )
       => ( ! [V8t_2E0: tyop_2Epatricia_2Eptree(A_27a),V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V17v20_2E0: tyop_2Enum_2Enum,V18v21_2E0: tyop_2Enum_2Enum,V19v22_2E0: tyop_2Epatricia_2Eptree(A_27a),V20v23_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),F6_2E0,V8t_2E0),V3p_2E0),V5m_2E0),V17v20_2E0),V18v21_2E0),V19v22_2E0),V20v23_2E0) = c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Epatricia_2Eptree(A_27a),V8t_2E0,c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V17v20_2E0,V18v21_2E0,V19v22_2E0,V20v23_2E0)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))))),F9_2E0,V3p_2E0),V5m_2E0),V17v20_2E0),V18v21_2E0),V19v22_2E0),V20v23_2E0),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))))),F10_2E0,V3p_2E0),V5m_2E0),V17v20_2E0),V18v21_2E0),V19v22_2E0),V20v23_2E0)) )
         => ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V9v18_2E0: tyop_2Enum_2Enum,V10v19_2E0: A_27a,V13v32_2E0: tyop_2Enum_2Enum,V14v33_2E0: tyop_2Enum_2Enum,V15v34_2E0: tyop_2Epatricia_2Eptree(A_27a),V16v35_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))),F8_2E0,V3p_2E0),V5m_2E0),V9v18_2E0),V10v19_2E0),V13v32_2E0),V14v33_2E0),V15v34_2E0),V16v35_2E0) = c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V3p_2E0,V5m_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V9v18_2E0,V10v19_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V13v32_2E0,V14v33_2E0,V15v34_2E0,V16v35_2E0))) )
           => ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V9v18_2E0: tyop_2Enum_2Enum,V10v19_2E0: A_27a,V11v30_2E0: tyop_2Enum_2Enum,V12v31_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))),F7_2E0,V3p_2E0),V5m_2E0),V9v18_2E0),V10v19_2E0),V11v30_2E0),V12v31_2E0) = c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V3p_2E0,V5m_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V9v18_2E0,V10v19_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V11v30_2E0,V12v31_2E0))) )
             => ( ! [V8t_2E0: tyop_2Epatricia_2Eptree(A_27a),V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V9v18_2E0: tyop_2Enum_2Enum,V10v19_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),F5_2E0,V8t_2E0),V3p_2E0),V5m_2E0),V9v18_2E0),V10v19_2E0) = c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Epatricia_2Eptree(A_27a),V8t_2E0,c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V9v18_2E0,V10v19_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))))),F7_2E0,V3p_2E0),V5m_2E0),V9v18_2E0),V10v19_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))))),F8_2E0,V3p_2E0),V5m_2E0),V9v18_2E0),V10v19_2E0)) )
               => ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V7v4_2E0: tyop_2Epatricia_2Eptree(A_27a),V8t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),F4_2E0,V3p_2E0),V5m_2E0),V7v4_2E0),V8t_2E0) = c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Epatricia_2Eptree(A_27a),V7v4_2E0,c_2Ecombin_2EI_2E1(tyop_2Epatricia_2Eptree(A_27a),V8t_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a)))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))))),F5_2E0,V8t_2E0),V3p_2E0),V5m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))))),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))))),F6_2E0,V8t_2E0),V3p_2E0),V5m_2E0)) )
                 => ( ! [V3p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V6v3_2E0: tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))] : ( app_2E2(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a))),F3_2E0,V3p_2E0),V5m_2E0),V6v3_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V6v3_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),F4_2E0,V3p_2E0),V5m_2E0)) )
                   => ( ! [V3p_2E0: tyop_2Enum_2Enum,V4v1_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))] : ( app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),tyop_2Epatricia_2Eptree(A_27a)),F2_2E0,V3p_2E0),V4v1_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V4v1_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Epatricia_2Eptree(A_27a))),F3_2E0,V3p_2E0)) )
                     => ( ! [V1BRANCH_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),V2a_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))] : ( app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a)),F1_2E0,V1BRANCH_2E0),V2a_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V2a_2E0,F2_2E0) )
                       => ( ! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool))] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,F0_2E0,V0R_2E0) = c_2Erelation_2EWF_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),V0R_2E0) )
                         => ( c_2Epatricia_2EBRANCH_2E0(A_27a) = c_2Erelation_2EWFREC_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Epatricia_2Eptree(A_27a),c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)),F0_2E0),F1_2E0) ) ) ) ) ) ) ) ) ) ) ) ) ).

tff(thm_2Epatricia_2EREMOVE__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1j_2E0: tyop_2Enum_2Enum,V3k_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1j_2E0),V3k_2E0))
        <=> ( V1j_2E0 = V3k_2E0 ) )
     => ( ! [V0k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),V0k_2E0) = c_2Epatricia_2EEmpty_2E0(A_27a) )
        & ! [V1j_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a,V3k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V1j_2E0,V2d_2E0),V3k_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1j_2E0),V3k_2E0),c_2Epatricia_2EEmpty_2E0(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V1j_2E0,V2d_2E0)) )
        & ! [V4p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V6l_2E0: tyop_2Epatricia_2Eptree(A_27a),V7r_2E0: tyop_2Epatricia_2Eptree(A_27a),V8k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V4p_2E0,V5m_2E0,V6l_2E0,V7r_2E0),V8k_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EMOD__2EXP__EQ_2E3(V5m_2E0,V8k_2E0,V4p_2E0),c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EBIT_2E2(V5m_2E0,V8k_2E0),c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V4p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V5m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EREMOVE_2E2(A_27a,V6l_2E0,V8k_2E0),V7r_2E0)))),c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V4p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V5m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V6l_2E0,c_2Epatricia_2EREMOVE_2E2(A_27a,V7r_2E0,V8k_2E0)))))),c_2Epatricia_2EBranch_2E4(A_27a,V4p_2E0,V5m_2E0,V6l_2E0,V7r_2E0)) ) ) ) ).

tff(thm_2Epatricia_2ETRAVERSE__AUX__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),V0a_2E0) = V0a_2E0 )
      & ! [V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a,V3a_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V1k_2E0,V2d_2E0),V3a_2E0) = c_2Elist_2ECONS_2E2(tyop_2Enum_2Enum,V1k_2E0,V3a_2E0) )
      & ! [V4p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V6l_2E0: tyop_2Epatricia_2Eptree(A_27a),V7r_2E0: tyop_2Epatricia_2Eptree(A_27a),V8a_2E0: tyop_2Elist_2Elist(tyop_2Enum_2Enum)] : ( c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V4p_2E0,V5m_2E0,V6l_2E0,V7r_2E0),V8a_2E0) = c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,V6l_2E0,c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,V7r_2E0,V8a_2E0)) ) ) ).

tff(thm_2Epatricia_2ETRAVERSE__def,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Elist_2ENIL_2E0(tyop_2Enum_2Enum) )
      & ! [V0j_2E0: tyop_2Enum_2Enum,V1d_2E0: A_27a] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V0j_2E0,V1d_2E0)) = c_2Elist_2ECONS_2E2(tyop_2Enum_2Enum,V0j_2E0,c_2Elist_2ENIL_2E0(tyop_2Enum_2Enum)) )
      & ! [V2p_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4l_2E0: tyop_2Epatricia_2Eptree(A_27a),V5r_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V2p_2E0,V3m_2E0,V4l_2E0,V5r_2E0)) = c_2Elist_2EAPPEND_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V4l_2E0),c_2Epatricia_2ETRAVERSE_2E1(A_27a,V5r_2E0)) ) ) ).

tff(thm_2Epatricia_2EKEYS__def,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EKEYS_2E1(A_27a,V0t_2E0) = c_2Esorting_2EQSORT_2E2(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E0,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)) ) ).

tff(thm_2Epatricia_2ETRANSFORM__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,V0f_2E0,c_2Epatricia_2EEmpty_2E0(A_27b)) = c_2Epatricia_2EEmpty_2E0(A_27a) )
      & ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27b] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,V1f_2E0,c_2Epatricia_2ELeaf_2E2(A_27b,V2j_2E0,V3d_2E0)) = c_2Epatricia_2ELeaf_2E2(A_27a,V2j_2E0,app_2E2(A_27b,A_27a,V1f_2E0,V3d_2E0)) )
      & ! [V4f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V5p_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7l_2E0: tyop_2Epatricia_2Eptree(A_27b),V8r_2E0: tyop_2Epatricia_2Eptree(A_27b)] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,V4f_2E0,c_2Epatricia_2EBranch_2E4(A_27b,V5p_2E0,V6m_2E0,V7l_2E0,V8r_2E0)) = c_2Epatricia_2EBranch_2E4(A_27a,V5p_2E0,V6m_2E0,c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,V4f_2E0,V7l_2E0),c_2Epatricia_2ETRANSFORM_2E2(A_27a,A_27b,V4f_2E0,V8r_2E0)) ) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Ebool_2ET_2E0 )
      & ! [V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a] : ( c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V1P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V2j_2E0,V3d_2E0)) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0,V2j_2E0),V3d_2E0) )
      & ! [V4P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V5p_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7l_2E0: tyop_2Epatricia_2Eptree(A_27a),V8r_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
          ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V4P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V5p_2E0,V6m_2E0,V7l_2E0,V8r_2E0)))
        <=> ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V4P_2E0,V7l_2E0))
            & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V4P_2E0,V8r_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EEXISTS__LEAF__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Ebool_2EF_2E0 )
      & ! [V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a] : ( c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,V1P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V2j_2E0,V3d_2E0)) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0,V2j_2E0),V3d_2E0) )
      & ! [V4P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V5p_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7l_2E0: tyop_2Epatricia_2Eptree(A_27a),V8r_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
          ( p(c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,V4P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V5p_2E0,V6m_2E0,V7l_2E0,V8r_2E0)))
        <=> ( p(c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,V4P_2E0,V7l_2E0))
            | p(c_2Epatricia_2EEXISTS__LEAF_2E2(A_27a,V4P_2E0,V8r_2E0)) ) ) ) ).

tff(thm_2Epatricia_2ESIZE__def,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ESIZE_2E1(A_27a,V0t_2E0) = c_2Elist_2ELENGTH_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)) ) ).

tff(thm_2Epatricia_2EDEPTH__def,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Epatricia_2EDEPTH_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Enum_2E0_2E0 )
      & ! [V0j_2E0: tyop_2Enum_2Enum,V1d_2E0: A_27a] : ( c_2Epatricia_2EDEPTH_2E1(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V0j_2E0,V1d_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
      & ! [V2p_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4l_2E0: tyop_2Epatricia_2Eptree(A_27a),V5r_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EDEPTH_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V2p_2E0,V3m_2E0,V4l_2E0,V5r_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2EMAX_2E2(c_2Epatricia_2EDEPTH_2E1(A_27a,V4l_2E0),c_2Epatricia_2EDEPTH_2E1(A_27a,V5r_2E0))) ) ) ).

tff(thm_2Epatricia_2EIS__PTREE__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V3m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum,V6k_2E0: tyop_2Enum_2Enum,V7d_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V3m_2E0),V2p_2E0),V6k_2E0),V7d_2E0))
        <=> ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V3m_2E0,V6k_2E0,V2p_2E0))
            & p(c_2Ebit_2EBIT_2E2(V3m_2E0,V6k_2E0)) ) )
     => ( ! [V3m_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum,V8k_2E0: tyop_2Enum_2Enum,V9d_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V3m_2E0),V2p_2E0),V8k_2E0),V9d_2E0))
          <=> ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V3m_2E0,V8k_2E0,V2p_2E0))
              & ~ p(c_2Ebit_2EBIT_2E2(V3m_2E0,V8k_2E0)) ) )
       => ( ( c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Ebool_2ET_2E0 )
          & ! [V0k_2E0: tyop_2Enum_2Enum,V1d_2E0: A_27a] : ( c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V0k_2E0,V1d_2E0)) = c_2Ebool_2ET_2E0 )
          & ! [V2p_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4l_2E0: tyop_2Epatricia_2Eptree(A_27a),V5r_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
              ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V2p_2E0,V3m_2E0,V4l_2E0,V5r_2E0)))
            <=> ( p(c_2Eprim__rec_2E_3C_2E2(V2p_2E0,c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3m_2E0)))
                & p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V4l_2E0))
                & p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V5r_2E0))
                & ( V4l_2E0 != c_2Epatricia_2EEmpty_2E0(A_27a) )
                & ( V5r_2E0 != c_2Epatricia_2EEmpty_2E0(A_27a) )
                & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V3m_2E0),V2p_2E0),V4l_2E0))
                & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V3m_2E0),V2p_2E0),V5r_2E0)) ) ) ) ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__def,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EIN__PTREE_2E2(V0n_2E0,V1t_2E0) = c_2Eoption_2EIS__SOME_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EPEEK_2E2(tyop_2Eone_2Eone,V1t_2E0,V0n_2E0)) ) ).

tff(thm_2Epatricia_2EINSERT__PTREE__def,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EINSERT__PTREE_2E2(V0n_2E0,V1t_2E0) = c_2Epatricia_2EADD_2E2(tyop_2Eone_2Eone,V1t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Eone_2Eone,V0n_2E0,c_2Eone_2Eone_2E0)) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__def,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0) = c_2Elist_2EFOLDL_2E3(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),c_2Ecombin_2EC_2E1(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),c_2Epatricia_2EINSERT__PTREE_2E0),V0t_2E0,c_2Elist_2ESET__TO__LIST_2E1(tyop_2Enum_2Enum,V1s_2E0)) ) ).

tff(thm_2Epatricia_2ENUMSET__OF__PTREE__def,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0) = c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0)) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__def,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,V1t2_2E0) = c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t1_2E0,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V1t2_2E0)) ) ).

tff(thm_2Epatricia_2EIS__EMPTY__primitive__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))] :
      ( ! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,F0_2E0,V0R_2E0) = c_2Erelation_2EWF_2E1(tyop_2Epatricia_2Eptree(A_27a),V0R_2E0) )
     => ( ! [V5v8_2E0: tyop_2Enum_2Enum,V6v9_2E0: tyop_2Enum_2Enum,V7v10_2E0: tyop_2Epatricia_2Eptree(A_27a),V8v11_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool))),F3_2E0,V5v8_2E0),V6v9_2E0),V7v10_2E0),V8v11_2E0) = c_2Ecombin_2EI_2E1(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) )
       => ( ! [V3v6_2E0: tyop_2Enum_2Enum,V4v7_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F2_2E0,V3v6_2E0),V4v7_2E0) = c_2Ecombin_2EI_2E1(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) )
         => ( ! [V1IS__EMPTY_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),V2a_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool),F1_2E0,V1IS__EMPTY_2E0),V2a_2E0) = c_2Epatricia_2Eptree__CASE_2E4(A_27a,tyop_2Emin_2Ebool,V2a_2E0,c_2Ecombin_2EI_2E1(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0),F2_2E0,F3_2E0) )
           => ( c_2Epatricia_2EIS__EMPTY_2E0(A_27a) = c_2Erelation_2EWFREC_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)),F0_2E0),F1_2E0) ) ) ) ) ) ).

tff(thm_2Epatricia_2EFIND__def,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EFIND_2E2(A_27a,V0t_2E0,V1k_2E0) = c_2Eoption_2ETHE_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V0t_2E0,V1k_2E0)) ) ).

tff(thm_2Epatricia_2EADD__LIST__def,axiom,
    ! [A_27a: $tType] : ( c_2Epatricia_2EADD__LIST_2E0(A_27a) = c_2Elist_2EFOLDL_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EADD_2E0(A_27a)) ) ).

tff(thm_2Epatricia_2Edatatype__ptree,axiom,
    ! [A_27a: $tType,V0ptree_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Emin_2Ebool)))] : p(c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Emin_2Ebool),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Epatricia_2Eptree(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))))),tyop_2Emin_2Ebool)),V0ptree_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)),c_2Epatricia_2ELeaf_2E0(A_27a)),c_2Epatricia_2EBranch_2E0(A_27a)))) ).

tff(thm_2Epatricia_2Eptree__11,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a0_2E0: tyop_2Enum_2Enum,V1a1_2E0: A_27a,V2a0_27_2E0: tyop_2Enum_2Enum,V3a1_27_2E0: A_27a] :
          ( ( c_2Epatricia_2ELeaf_2E2(A_27a,V0a0_2E0,V1a1_2E0) = c_2Epatricia_2ELeaf_2E2(A_27a,V2a0_27_2E0,V3a1_27_2E0) )
        <=> ( ( V0a0_2E0 = V2a0_27_2E0 )
            & ( V1a1_2E0 = V3a1_27_2E0 ) ) )
      & ! [V4a0_2E0: tyop_2Enum_2Enum,V5a1_2E0: tyop_2Enum_2Enum,V6a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V7a3_2E0: tyop_2Epatricia_2Eptree(A_27a),V8a0_27_2E0: tyop_2Enum_2Enum,V9a1_27_2E0: tyop_2Enum_2Enum,V10a2_27_2E0: tyop_2Epatricia_2Eptree(A_27a),V11a3_27_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
          ( ( c_2Epatricia_2EBranch_2E4(A_27a,V4a0_2E0,V5a1_2E0,V6a2_2E0,V7a3_2E0) = c_2Epatricia_2EBranch_2E4(A_27a,V8a0_27_2E0,V9a1_27_2E0,V10a2_27_2E0,V11a3_27_2E0) )
        <=> ( ( V4a0_2E0 = V8a0_27_2E0 )
            & ( V5a1_2E0 = V9a1_27_2E0 )
            & ( V6a2_2E0 = V10a2_27_2E0 )
            & ( V7a3_2E0 = V11a3_27_2E0 ) ) ) ) ).

tff(thm_2Epatricia_2Eptree__distinct,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a1_2E0: A_27a,V1a0_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EEmpty_2E0(A_27a) != c_2Epatricia_2ELeaf_2E2(A_27a,V1a0_2E0,V0a1_2E0) )
      & ! [V2a3_2E0: tyop_2Epatricia_2Eptree(A_27a),V3a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V4a1_2E0: tyop_2Enum_2Enum,V5a0_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EEmpty_2E0(A_27a) != c_2Epatricia_2EBranch_2E4(A_27a,V5a0_2E0,V4a1_2E0,V3a2_2E0,V2a3_2E0) )
      & ! [V6a3_2E0: tyop_2Epatricia_2Eptree(A_27a),V7a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V8a1_27_2E0: tyop_2Enum_2Enum,V9a1_2E0: A_27a,V10a0_27_2E0: tyop_2Enum_2Enum,V11a0_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2ELeaf_2E2(A_27a,V11a0_2E0,V9a1_2E0) != c_2Epatricia_2EBranch_2E4(A_27a,V10a0_27_2E0,V8a1_27_2E0,V7a2_2E0,V6a3_2E0) ) ) ).

tff(thm_2Epatricia_2Eptree__nchotomy,axiom,
    ! [A_27a: $tType,V0pp_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( ( V0pp_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) )
      | ? [V1n_2E0: tyop_2Enum_2Enum,V2a_2E0: A_27a] : ( V0pp_2E0 = c_2Epatricia_2ELeaf_2E2(A_27a,V1n_2E0,V2a_2E0) )
      | ? [V3n0_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum,V5p_2E0: tyop_2Epatricia_2Eptree(A_27a),V6p0_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( V0pp_2E0 = c_2Epatricia_2EBranch_2E4(A_27a,V3n0_2E0,V4n_2E0,V5p_2E0,V6p0_2E0) ) ) ).

tff(thm_2Epatricia_2Eptree__Axiom,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f0_2E0: A_27b,V1f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V2f2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))))))] :
    ? [V3fn_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)] :
      ( ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,V3fn_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)) = V0f0_2E0 )
      & ! [V4a0_2E0: tyop_2Enum_2Enum,V5a1_2E0: A_27a] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,V3fn_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V4a0_2E0,V5a1_2E0)) = app_2E2(A_27a,A_27b,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b),V1f1_2E0,V4a0_2E0),V5a1_2E0) )
      & ! [V6a0_2E0: tyop_2Enum_2Enum,V7a1_2E0: tyop_2Enum_2Enum,V8a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V9a3_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,V3fn_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V6a0_2E0,V7a1_2E0,V8a2_2E0,V9a3_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))))),V2f2_2E0,V6a0_2E0),V7a1_2E0),V8a2_2E0),V9a3_2E0),app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,V3fn_2E0,V8a2_2E0)),app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,V3fn_2E0,V9a3_2E0)) ) ) ).

tff(thm_2Epatricia_2Eptree__induction,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)))
        & ! [V1n_2E0: tyop_2Enum_2Enum,V2a_2E0: A_27a] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V1n_2E0,V2a_2E0)))
        & ! [V3p_2E0: tyop_2Epatricia_2Eptree(A_27a),V4p0_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
            ( ( p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V3p_2E0))
              & p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V4p0_2E0)) )
           => ! [V5n_2E0: tyop_2Enum_2Enum,V6n0_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V6n0_2E0,V5n_2E0,V3p_2E0,V4p0_2E0))) ) )
     => ! [V7p_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V7p_2E0)) ) ).

tff(thm_2Epatricia_2Eptree__case__cong,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0v_27_2E0: A_27b,V1f1_27_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V3M_2E0: tyop_2Epatricia_2Eptree(A_27a),V4M_27_2E0: tyop_2Epatricia_2Eptree(A_27a),V5v_2E0: A_27b,V6f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b)),V7f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))))] :
      ( ( ( V3M_2E0 = V4M_27_2E0 )
        & ( ( V4M_27_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) )
         => ( V5v_2E0 = V0v_27_2E0 ) )
        & ! [V8a0_2E0: tyop_2Enum_2Enum,V9a1_2E0: A_27a] :
            ( ( V4M_27_2E0 = c_2Epatricia_2ELeaf_2E2(A_27a,V8a0_2E0,V9a1_2E0) )
           => ( app_2E2(A_27a,A_27b,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b),V6f_2E0,V8a0_2E0),V9a1_2E0) = app_2E2(A_27a,A_27b,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b),V2f_27_2E0,V8a0_2E0),V9a1_2E0) ) )
        & ! [V10a0_2E0: tyop_2Enum_2Enum,V11a1_2E0: tyop_2Enum_2Enum,V12a2_2E0: tyop_2Epatricia_2Eptree(A_27a),V13a3_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
            ( ( V4M_27_2E0 = c_2Epatricia_2EBranch_2E4(A_27a,V10a0_2E0,V11a1_2E0,V12a2_2E0,V13a3_2E0) )
           => ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))),V7f1_2E0,V10a0_2E0),V11a1_2E0),V12a2_2E0),V13a3_2E0) = app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))),V1f1_27_2E0,V10a0_2E0),V11a1_2E0),V12a2_2E0),V13a3_2E0) ) ) )
     => ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,V3M_2E0,V5v_2E0,V6f_2E0,V7f1_2E0) = c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,V4M_27_2E0,V0v_27_2E0,V2f_27_2E0,V1f1_27_2E0) ) ) ).

tff(thm_2Epatricia_2Eptree__case__eq,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Epatricia_2Eptree(A_27a),V1v_27_2E0: A_27b,V2v_2E0: A_27b,V3f1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)))),V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b))] :
      ( ( c_2Epatricia_2Eptree__CASE_2E4(A_27a,A_27b,V0x_2E0,V2v_2E0,V4f_2E0,V3f1_2E0) = V1v_27_2E0 )
    <=> ( ( ( V0x_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) )
          & ( V2v_2E0 = V1v_27_2E0 ) )
        | ? [V5n_2E0: tyop_2Enum_2Enum,V6a_2E0: A_27a] :
            ( ( V0x_2E0 = c_2Epatricia_2ELeaf_2E2(A_27a,V5n_2E0,V6a_2E0) )
            & ( app_2E2(A_27a,A_27b,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27b),V4f_2E0,V5n_2E0),V6a_2E0) = V1v_27_2E0 ) )
        | ? [V7n0_2E0: tyop_2Enum_2Enum,V8n_2E0: tyop_2Enum_2Enum,V9p_2E0: tyop_2Epatricia_2Eptree(A_27a),V10p0_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
            ( ( V0x_2E0 = c_2Epatricia_2EBranch_2E4(A_27a,V7n0_2E0,V8n_2E0,V9p_2E0,V10p0_2E0) )
            & ( app_2E2(tyop_2Epatricia_2Eptree(A_27a),A_27b,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),A_27b))),V3f1_2E0,V7n0_2E0),V8n_2E0),V9p_2E0),V10p0_2E0) = V1v_27_2E0 ) ) ) ) ).

tff(thm_2Epatricia_2EBRANCHING__BIT__ind,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1p0_2E0: tyop_2Enum_2Enum,V2p1_2E0: tyop_2Enum_2Enum] :
          ( ( ~ ( ( c_2Earithmetic_2EODD_2E1(V1p0_2E0) = c_2Earithmetic_2EEVEN_2E1(V2p1_2E0) )
                | ( V1p0_2E0 = V2p1_2E0 ) )
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Earithmetic_2EDIV2_2E1(V1p0_2E0)),c_2Earithmetic_2EDIV2_2E1(V2p1_2E0))) )
         => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V1p0_2E0),V2p1_2E0)) )
     => ! [V3v_2E0: tyop_2Enum_2Enum,V4v1_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3v_2E0),V4v1_2E0)) ) ).

tff(thm_2Epatricia_2EBRANCHING__BIT__def,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1p0_2E0: tyop_2Enum_2Enum,V0p1_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1p0_2E0),V0p1_2E0))
        <=> ( ( c_2Earithmetic_2EODD_2E1(V1p0_2E0) = c_2Earithmetic_2EEVEN_2E1(V0p1_2E0) )
            | ( V1p0_2E0 = V0p1_2E0 ) ) )
     => ! [V0p1_2E0: tyop_2Enum_2Enum,V1p0_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EBRANCHING__BIT_2E2(V1p0_2E0,V0p1_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1p0_2E0),V0p1_2E0),c_2Enum_2E0_2E0,c_2Enum_2ESUC_2E1(c_2Epatricia_2EBRANCHING__BIT_2E2(c_2Earithmetic_2EDIV2_2E1(V1p0_2E0),c_2Earithmetic_2EDIV2_2E1(V0p1_2E0)))) ) ) ).

tff(thm_2Epatricia_2EPEEK__ind,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ( ! [V1k_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)),V1k_2E0))
        & ! [V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a,V4k_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V2j_2E0,V3d_2E0)),V4k_2E0))
        & ! [V5p_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7l_2E0: tyop_2Epatricia_2Eptree(A_27a),V8r_2E0: tyop_2Epatricia_2Eptree(A_27a),V9k_2E0: tyop_2Enum_2Enum] :
            ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EBIT_2E2(V6m_2E0,V9k_2E0),V7l_2E0,V8r_2E0)),V9k_2E0))
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V5p_2E0,V6m_2E0,V7l_2E0,V8r_2E0)),V9k_2E0)) ) )
     => ! [V10v_2E0: tyop_2Epatricia_2Eptree(A_27a),V11v1_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V10v_2E0),V11v1_2E0)) ) ).

tff(thm_2Epatricia_2EPEEK__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0))
        <=> ( V1k_2E0 = V2j_2E0 ) )
     => ( ! [V0k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),V0k_2E0) = c_2Eoption_2ENONE_2E0(A_27a) )
        & ! [V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a] : ( c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V2j_2E0,V3d_2E0),V1k_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0),c_2Eoption_2ESOME_2E1(A_27a,V3d_2E0),c_2Eoption_2ENONE_2E0(A_27a)) )
        & ! [V4r_2E0: tyop_2Epatricia_2Eptree(A_27a),V5p_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7l_2E0: tyop_2Epatricia_2Eptree(A_27a),V8k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V5p_2E0,V6m_2E0,V7l_2E0,V4r_2E0),V8k_2E0) = c_2Epatricia_2EPEEK_2E2(A_27a,c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EBIT_2E2(V6m_2E0,V8k_2E0),V7l_2E0,V4r_2E0),V8k_2E0) ) ) ) ).

tff(thm_2Epatricia_2EADD__ind,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool))] :
      ( ( ! [V1k_2E0: tyop_2Enum_2Enum,V2e_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2e_2E0)))
        & ! [V3j_2E0: tyop_2Enum_2Enum,V4d_2E0: A_27a,V5k_2E0: tyop_2Enum_2Enum,V6e_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V3j_2E0,V4d_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V5k_2E0,V6e_2E0)))
        & ! [V7p_2E0: tyop_2Enum_2Enum,V8m_2E0: tyop_2Enum_2Enum,V9l_2E0: tyop_2Epatricia_2Eptree(A_27a),V10r_2E0: tyop_2Epatricia_2Eptree(A_27a),V11k_2E0: tyop_2Enum_2Enum,V12e_2E0: A_27a] :
            ( ( ( ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V8m_2E0,V11k_2E0,V7p_2E0))
                  & ~ p(c_2Ebit_2EBIT_2E2(V8m_2E0,V11k_2E0)) )
               => p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,V10r_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V11k_2E0,V12e_2E0))) )
              & ( ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V8m_2E0,V11k_2E0,V7p_2E0))
                  & p(c_2Ebit_2EBIT_2E2(V8m_2E0,V11k_2E0)) )
               => p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,V9l_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V11k_2E0,V12e_2E0))) ) )
           => p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V7p_2E0,V8m_2E0,V9l_2E0,V10r_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V11k_2E0,V12e_2E0))) ) )
     => ! [V13v_2E0: tyop_2Epatricia_2Eptree(A_27a),V14v1_2E0: tyop_2Enum_2Enum,V15v2_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),V0P_2E0,V13v_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V14v1_2E0,V15v2_2E0))) ) ).

tff(thm_2Epatricia_2EADD__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V3j_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V3j_2E0),V2k_2E0))
        <=> ( V3j_2E0 = V2k_2E0 ) )
     => ( ! [V0k_2E0: tyop_2Enum_2Enum,V1e_2E0: A_27a] : ( c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V0k_2E0,V1e_2E0)) = c_2Epatricia_2ELeaf_2E2(A_27a,V0k_2E0,V1e_2E0) )
        & ! [V2k_2E0: tyop_2Enum_2Enum,V3j_2E0: tyop_2Enum_2Enum,V4e_2E0: A_27a,V5d_2E0: A_27a] : ( c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V3j_2E0,V5d_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V2k_2E0,V4e_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V3j_2E0),V2k_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V2k_2E0,V4e_2E0),c_2Epatricia_2EJOIN_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),V2k_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2ELeaf_2E2(A_27a,V2k_2E0,V4e_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),V3j_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V3j_2E0,V5d_2E0)))))) )
        & ! [V6r_2E0: tyop_2Epatricia_2Eptree(A_27a),V7p_2E0: tyop_2Enum_2Enum,V8m_2E0: tyop_2Enum_2Enum,V9l_2E0: tyop_2Epatricia_2Eptree(A_27a),V10k_2E0: tyop_2Enum_2Enum,V11e_2E0: A_27a] : ( c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V7p_2E0,V8m_2E0,V9l_2E0,V6r_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V10k_2E0,V11e_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EMOD__2EXP__EQ_2E3(V8m_2E0,V10k_2E0,V7p_2E0),c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),c_2Ebit_2EBIT_2E2(V8m_2E0,V10k_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V7p_2E0,V8m_2E0,c_2Epatricia_2EADD_2E2(A_27a,V9l_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V10k_2E0,V11e_2E0)),V6r_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V7p_2E0,V8m_2E0,V9l_2E0,c_2Epatricia_2EADD_2E2(A_27a,V6r_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V10k_2E0,V11e_2E0)))),c_2Epatricia_2EJOIN_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a))),V10k_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2ELeaf_2E2(A_27a,V10k_2E0,V11e_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(A_27a),V7p_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V7p_2E0,V8m_2E0,V9l_2E0,V6r_2E0)))))) ) ) ) ).

tff(thm_2Epatricia_2EBRANCH__ind,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool)] :
      ( ( ! [V1p_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum,V3t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V1p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V2m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EEmpty_2E0(A_27a),V3t_2E0)))))
        & ! [V4p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V6v6_2E0: tyop_2Enum_2Enum,V7v7_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V4p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V5m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V6v6_2E0,V7v7_2E0),c_2Epatricia_2EEmpty_2E0(A_27a))))))
        & ! [V8p_2E0: tyop_2Enum_2Enum,V9m_2E0: tyop_2Enum_2Enum,V10v8_2E0: tyop_2Enum_2Enum,V11v9_2E0: tyop_2Enum_2Enum,V12v10_2E0: tyop_2Epatricia_2Eptree(A_27a),V13v11_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V8p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V9m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V10v8_2E0,V11v9_2E0,V12v10_2E0,V13v11_2E0),c_2Epatricia_2EEmpty_2E0(A_27a))))))
        & ! [V14p_2E0: tyop_2Enum_2Enum,V15m_2E0: tyop_2Enum_2Enum,V16v12_2E0: tyop_2Enum_2Enum,V17v13_2E0: A_27a,V18v24_2E0: tyop_2Enum_2Enum,V19v25_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V14p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V15m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V16v12_2E0,V17v13_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V18v24_2E0,V19v25_2E0))))))
        & ! [V20p_2E0: tyop_2Enum_2Enum,V21m_2E0: tyop_2Enum_2Enum,V22v12_2E0: tyop_2Enum_2Enum,V23v13_2E0: A_27a,V24v26_2E0: tyop_2Enum_2Enum,V25v27_2E0: tyop_2Enum_2Enum,V26v28_2E0: tyop_2Epatricia_2Eptree(A_27a),V27v29_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V20p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V21m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V22v12_2E0,V23v13_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V24v26_2E0,V25v27_2E0,V26v28_2E0,V27v29_2E0))))))
        & ! [V28p_2E0: tyop_2Enum_2Enum,V29m_2E0: tyop_2Enum_2Enum,V30v14_2E0: tyop_2Enum_2Enum,V31v15_2E0: tyop_2Enum_2Enum,V32v16_2E0: tyop_2Epatricia_2Eptree(A_27a),V33v17_2E0: tyop_2Epatricia_2Eptree(A_27a),V34v36_2E0: tyop_2Enum_2Enum,V35v37_2E0: A_27a] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V28p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V29m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V30v14_2E0,V31v15_2E0,V32v16_2E0,V33v17_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V34v36_2E0,V35v37_2E0))))))
        & ! [V36p_2E0: tyop_2Enum_2Enum,V37m_2E0: tyop_2Enum_2Enum,V38v14_2E0: tyop_2Enum_2Enum,V39v15_2E0: tyop_2Enum_2Enum,V40v16_2E0: tyop_2Epatricia_2Eptree(A_27a),V41v17_2E0: tyop_2Epatricia_2Eptree(A_27a),V42v38_2E0: tyop_2Enum_2Enum,V43v39_2E0: tyop_2Enum_2Enum,V44v40_2E0: tyop_2Epatricia_2Eptree(A_27a),V45v41_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V36p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V37m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V38v14_2E0,V39v15_2E0,V40v16_2E0,V41v17_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V42v38_2E0,V43v39_2E0,V44v40_2E0,V45v41_2E0)))))) )
     => ! [V46v_2E0: tyop_2Enum_2Enum,V47v1_2E0: tyop_2Enum_2Enum,V48v2_2E0: tyop_2Epatricia_2Eptree(A_27a),V49v3_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)))),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V46v_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V47v1_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V48v2_2E0,V49v3_2E0))))) ) ).

tff(thm_2Epatricia_2EBRANCH__def,axiom,
    ! [A_27a: $tType,V0v9_2E0: tyop_2Enum_2Enum,V1v8_2E0: tyop_2Enum_2Enum,V2v7_2E0: A_27a,V3v6_2E0: tyop_2Enum_2Enum,V4v41_2E0: tyop_2Epatricia_2Eptree(A_27a),V5v40_2E0: tyop_2Epatricia_2Eptree(A_27a),V6v39_2E0: tyop_2Enum_2Enum,V7v38_2E0: tyop_2Enum_2Enum,V8v37_2E0: A_27a,V9v36_2E0: tyop_2Enum_2Enum,V10v29_2E0: tyop_2Epatricia_2Eptree(A_27a),V11v28_2E0: tyop_2Epatricia_2Eptree(A_27a),V12v27_2E0: tyop_2Enum_2Enum,V13v26_2E0: tyop_2Enum_2Enum,V14v25_2E0: A_27a,V15v24_2E0: tyop_2Enum_2Enum,V16v17_2E0: tyop_2Epatricia_2Eptree(A_27a),V17v16_2E0: tyop_2Epatricia_2Eptree(A_27a),V18v15_2E0: tyop_2Enum_2Enum,V19v14_2E0: tyop_2Enum_2Enum,V20v13_2E0: A_27a,V21v12_2E0: tyop_2Enum_2Enum,V22v11_2E0: tyop_2Epatricia_2Eptree(A_27a),V23v10_2E0: tyop_2Epatricia_2Eptree(A_27a),V24t_2E0: tyop_2Epatricia_2Eptree(A_27a),V25p_2E0: tyop_2Enum_2Enum,V26m_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EEmpty_2E0(A_27a),V24t_2E0)))) = V24t_2E0 )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V3v6_2E0,V2v7_2E0),c_2Epatricia_2EEmpty_2E0(A_27a))))) = c_2Epatricia_2ELeaf_2E2(A_27a,V3v6_2E0,V2v7_2E0) )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V1v8_2E0,V0v9_2E0,V23v10_2E0,V22v11_2E0),c_2Epatricia_2EEmpty_2E0(A_27a))))) = c_2Epatricia_2EBranch_2E4(A_27a,V1v8_2E0,V0v9_2E0,V23v10_2E0,V22v11_2E0) )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V21v12_2E0,V20v13_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V15v24_2E0,V14v25_2E0))))) = c_2Epatricia_2EBranch_2E4(A_27a,V25p_2E0,V26m_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V21v12_2E0,V20v13_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V15v24_2E0,V14v25_2E0)) )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2ELeaf_2E2(A_27a,V21v12_2E0,V20v13_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V13v26_2E0,V12v27_2E0,V11v28_2E0,V10v29_2E0))))) = c_2Epatricia_2EBranch_2E4(A_27a,V25p_2E0,V26m_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V21v12_2E0,V20v13_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V13v26_2E0,V12v27_2E0,V11v28_2E0,V10v29_2E0)) )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V19v14_2E0,V18v15_2E0,V17v16_2E0,V16v17_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V9v36_2E0,V8v37_2E0))))) = c_2Epatricia_2EBranch_2E4(A_27a,V25p_2E0,V26m_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V19v14_2E0,V18v15_2E0,V17v16_2E0,V16v17_2E0),c_2Epatricia_2ELeaf_2E2(A_27a,V9v36_2E0,V8v37_2E0)) )
      & ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V25p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V26m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),c_2Epatricia_2EBranch_2E4(A_27a,V19v14_2E0,V18v15_2E0,V17v16_2E0,V16v17_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V7v38_2E0,V6v39_2E0,V5v40_2E0,V4v41_2E0))))) = c_2Epatricia_2EBranch_2E4(A_27a,V25p_2E0,V26m_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V19v14_2E0,V18v15_2E0,V17v16_2E0,V16v17_2E0),c_2Epatricia_2EBranch_2E4(A_27a,V7v38_2E0,V6v39_2E0,V5v40_2E0,V4v41_2E0)) ) ) ).

tff(thm_2Epatricia_2EIS__EMPTY__ind,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2EEmpty_2E0(A_27a)))
        & ! [V1v_2E0: tyop_2Enum_2Enum,V2v1_2E0: A_27a] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2ELeaf_2E2(A_27a,V1v_2E0,V2v1_2E0)))
        & ! [V3v2_2E0: tyop_2Enum_2Enum,V4v3_2E0: tyop_2Enum_2Enum,V5v4_2E0: tyop_2Epatricia_2Eptree(A_27a),V6v5_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V3v2_2E0,V4v3_2E0,V5v4_2E0,V6v5_2E0))) )
     => ! [V7v_2E0: tyop_2Epatricia_2Eptree(A_27a)] : p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V7v_2E0)) ) ).

tff(thm_2Epatricia_2EIS__EMPTY__def,axiom,
    ! [A_27a: $tType,V0v5_2E0: tyop_2Epatricia_2Eptree(A_27a),V1v4_2E0: tyop_2Epatricia_2Eptree(A_27a),V2v3_2E0: tyop_2Enum_2Enum,V3v2_2E0: tyop_2Enum_2Enum,V4v1_2E0: A_27a,V5v_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Epatricia_2EIS__EMPTY_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Ebool_2ET_2E0 )
      & ( c_2Epatricia_2EIS__EMPTY_2E1(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V5v_2E0,V4v1_2E0)) = c_2Ebool_2EF_2E0 )
      & ( c_2Epatricia_2EIS__EMPTY_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V3v2_2E0,V2v3_2E0,V1v4_2E0,V0v5_2E0)) = c_2Ebool_2EF_2E0 ) ) ).

tff(thm_2Epatricia_2EBRANCHING__BIT,axiom,
    ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
      ( ( V0a_2E0 != V1b_2E0 )
     => ( c_2Ebit_2EBIT_2E2(c_2Epatricia_2EBRANCHING__BIT_2E2(V0a_2E0,V1b_2E0),V0a_2E0) != c_2Ebit_2EBIT_2E2(c_2Epatricia_2EBRANCHING__BIT_2E2(V0a_2E0,V1b_2E0),V1b_2E0) ) ) ).

tff(thm_2Epatricia_2EBRANCHING__BIT__ZERO,axiom,
    ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Epatricia_2EBRANCHING__BIT_2E2(V0a_2E0,V1b_2E0) = c_2Enum_2E0_2E0 )
    <=> ( ( c_2Earithmetic_2EODD_2E1(V0a_2E0) = c_2Earithmetic_2EEVEN_2E1(V1b_2E0) )
        | ( V0a_2E0 = V1b_2E0 ) ) ) ).

tff(thm_2Epatricia_2EBRANCHING__BIT__SYM,axiom,
    ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EBRANCHING__BIT_2E2(V0a_2E0,V1b_2E0) = c_2Epatricia_2EBRANCHING__BIT_2E2(V1b_2E0,V0a_2E0) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__ADD,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a] :
      ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V2k_2E0),V3d_2E0))
        & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V1t_2E0)) )
     => p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,c_2Epatricia_2EADD_2E2(A_27a,V1t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V2k_2E0,V3d_2E0)))) ) ).

tff(thm_2Epatricia_2EMONO__EVERY__LEAF,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2t_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( ( ! [V3k_2E0: tyop_2Enum_2Enum,V4d_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V3k_2E0),V4d_2E0))
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0,V3k_2E0),V4d_2E0)) )
        & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V2t_2E0)) )
     => p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V1Q_2E0,V2t_2E0)) ) ).

tff(thm_2Epatricia_2ENOT__ADD__EMPTY,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a] : ( c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0)) != c_2Epatricia_2EEmpty_2E0(A_27a) ) ).

tff(thm_2Epatricia_2EEMPTY__IS__PTREE,axiom,
    ! [A_27a: $tType] : p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a))) ).

tff(thm_2Epatricia_2EADD__IS__PTREE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1x_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,V1x_2E0))) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__BRANCH,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1p_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum,V3l_2E0: tyop_2Epatricia_2Eptree(A_27a),V4r_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V1p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V2m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V3l_2E0,V4r_2E0))))))
    <=> ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V3l_2E0))
        & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V4r_2E0)) ) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__REMOVE,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V1t_2E0))
     => p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,c_2Epatricia_2EREMOVE_2E2(A_27a,V1t_2E0,V2k_2E0))) ) ).

tff(thm_2Epatricia_2EIS__PTREE__BRANCH,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V1m_2E0: tyop_2Enum_2Enum,V0p_2E0: tyop_2Enum_2Enum,V4k_2E0: tyop_2Enum_2Enum,V5d_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1m_2E0),V0p_2E0),V4k_2E0),V5d_2E0))
        <=> ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V1m_2E0,V4k_2E0,V0p_2E0))
            & p(c_2Ebit_2EBIT_2E2(V1m_2E0,V4k_2E0)) ) )
     => ( ! [V1m_2E0: tyop_2Enum_2Enum,V0p_2E0: tyop_2Enum_2Enum,V6k_2E0: tyop_2Enum_2Enum,V7d_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1m_2E0),V0p_2E0),V6k_2E0),V7d_2E0))
          <=> ( p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V1m_2E0,V6k_2E0,V0p_2E0))
              & ~ p(c_2Ebit_2EBIT_2E2(V1m_2E0,V6k_2E0)) ) )
       => ! [V0p_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2l_2E0: tyop_2Epatricia_2Eptree(A_27a),V3r_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
            ( ( p(c_2Eprim__rec_2E_3C_2E2(V0p_2E0,c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0)))
              & ~ ( ( V2l_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) )
                  & ( V3r_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) ) )
              & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1m_2E0),V0p_2E0),V2l_2E0))
              & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1m_2E0),V0p_2E0),V3r_2E0))
              & p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V2l_2E0))
              & p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V3r_2E0)) )
           => p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V0p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V1m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V2l_2E0,V3r_2E0)))))) ) ) ) ).

tff(thm_2Epatricia_2EREMOVE__IS__PTREE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0))) ) ).

tff(thm_2Epatricia_2EPEEK__NONE,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum] :
      ( ( ! [V3d_2E0: A_27a] : ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V2k_2E0),V3d_2E0))
        & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V1t_2E0)) )
     => ( c_2Epatricia_2EPEEK_2E2(A_27a,V1t_2E0,V2k_2E0) = c_2Eoption_2ENONE_2E0(A_27a) ) ) ).

tff(thm_2Epatricia_2EPEEK__ADD,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V3j_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V3j_2E0))
        <=> ( V1k_2E0 = V3j_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a,V3j_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
         => ( c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0)),V3j_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V3j_2E0),c_2Eoption_2ESOME_2E1(A_27a,V2d_2E0),c_2Epatricia_2EPEEK_2E2(A_27a,V0t_2E0,V3j_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EBRANCH,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool)] :
      ( ! [V2l_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
          ( p(app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,F0_2E0,V2l_2E0))
        <=> ( V2l_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) ) )
     => ! [V0p_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2l_2E0: tyop_2Epatricia_2Eptree(A_27a),V3r_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V0p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V1m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V2l_2E0,V3r_2E0)))) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,F0_2E0,V2l_2E0),V3r_2E0,c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Emin_2Ebool,F0_2E0,V3r_2E0),V2l_2E0,c_2Epatricia_2EBranch_2E4(A_27a,V0p_2E0,V1m_2E0,V2l_2E0,V3r_2E0))) ) ) ).

tff(thm_2Epatricia_2EPEEK__REMOVE,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0))
        <=> ( V1k_2E0 = V2j_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
         => ( c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0),V2j_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0),c_2Eoption_2ENONE_2E0(A_27a),c_2Epatricia_2EPEEK_2E2(A_27a,V0t_2E0,V2j_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__TRANSFORM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3t_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( ( ! [V4k_2E0: tyop_2Enum_2Enum,V5d_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V4k_2E0),V5d_2E0))
           => p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1Q_2E0,V4k_2E0),app_2E2(A_27a,A_27b,V2f_2E0,V5d_2E0))) )
        & p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V3t_2E0)) )
     => p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27b,V1Q_2E0,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V2f_2E0,V3t_2E0))) ) ).

tff(thm_2Epatricia_2ETRANSFORM__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( ( c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0) = c_2Epatricia_2EEmpty_2E0(A_27b) )
    <=> ( V1t_2E0 = c_2Epatricia_2EEmpty_2E0(A_27a) ) ) ).

tff(thm_2Epatricia_2ETRANSFORM__IS__PTREE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V1t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(A_27b,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0))) ) ).

tff(thm_2Epatricia_2EPEEK__TRANSFORM,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),F0_2E0,V0f_2E0),V3x_2E0) = c_2Eoption_2ESOME_2E1(A_27b,app_2E2(A_27a,A_27b,V0f_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27b,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0),V2k_2E0) = c_2Eoption_2Eoption__CASE_2E3(A_27a,tyop_2Eoption_2Eoption(A_27b),c_2Epatricia_2EPEEK_2E2(A_27a,V1t_2E0,V2k_2E0),c_2Eoption_2ENONE_2E0(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),F0_2E0,V0f_2E0)) ) ) ).

tff(thm_2Epatricia_2EADD__TRANSFORM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,c_2Epatricia_2EADD_2E2(A_27a,V1t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V2k_2E0,V3d_2E0))) = c_2Epatricia_2EADD_2E2(A_27b,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27b,V2k_2E0,app_2E2(A_27a,A_27b,V0f_2E0,V3d_2E0))) ) ).

tff(thm_2Epatricia_2ETRANSFORM__BRANCH,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1p_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum,V3l_2E0: tyop_2Epatricia_2Eptree(A_27a),V4r_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,c_2Epatricia_2EBRANCH_2E1(A_27a,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a))),V1p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a)),V2m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27a),tyop_2Epatricia_2Eptree(A_27a),V3l_2E0,V4r_2E0))))) = c_2Epatricia_2EBRANCH_2E1(A_27b,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b))),V1p_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b)),V2m_2E0,c_2Epair_2E_2C_2E2(tyop_2Epatricia_2Eptree(A_27b),tyop_2Epatricia_2Eptree(A_27b),c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V3l_2E0),c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V4r_2E0))))) ) ).

tff(thm_2Epatricia_2EREMOVE__TRANSFORM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,c_2Epatricia_2EREMOVE_2E2(A_27a,V1t_2E0,V2k_2E0)) = c_2Epatricia_2EREMOVE_2E2(A_27b,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0),V2k_2E0) ) ).

tff(thm_2Epatricia_2EREMOVE__ADD__EQ,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a] : ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0)),V1k_2E0) = c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0) ) ).

tff(thm_2Epatricia_2EADD__ADD,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a,V3e_2E0: A_27a] : ( c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V3e_2E0)) = c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V3e_2E0)) ) ).

tff(thm_2Epatricia_2EEVERY__LEAF__PEEK,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a),V2k_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EEVERY__LEAF_2E2(A_27a,V0P_2E0,V1t_2E0))
        & p(c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V1t_2E0,V2k_2E0))) )
     => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V2k_2E0),c_2Eoption_2ETHE_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V1t_2E0,V2k_2E0)))) ) ).

tff(thm_2Epatricia_2EIS__PTREE__PEEK,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
      ( ! [V0k_2E0: tyop_2Enum_2Enum] : ~ p(c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),V0k_2E0)))
      & ! [V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum,V3b_2E0: A_27b] :
          ( p(c_2Eoption_2EIS__SOME_2E1(A_27b,c_2Epatricia_2EPEEK_2E2(A_27b,c_2Epatricia_2ELeaf_2E2(A_27b,V2j_2E0,V3b_2E0),V1k_2E0)))
        <=> ( V2j_2E0 = V1k_2E0 ) )
      & ! [V4p_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum,V6l_2E0: tyop_2Epatricia_2Eptree(A_27c),V7r_2E0: tyop_2Epatricia_2Eptree(A_27c)] :
          ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27c,c_2Epatricia_2EBranch_2E4(A_27c,V4p_2E0,V5m_2E0,V6l_2E0,V7r_2E0)))
         => ( ? [V8k_2E0: tyop_2Enum_2Enum] :
                ( p(c_2Ebit_2EBIT_2E2(V5m_2E0,V8k_2E0))
                & p(c_2Eoption_2EIS__SOME_2E1(A_27c,c_2Epatricia_2EPEEK_2E2(A_27c,V6l_2E0,V8k_2E0))) )
            & ? [V9k_2E0: tyop_2Enum_2Enum] :
                ( ~ p(c_2Ebit_2EBIT_2E2(V5m_2E0,V9k_2E0))
                & p(c_2Eoption_2EIS__SOME_2E1(A_27c,c_2Epatricia_2EPEEK_2E2(A_27c,V7r_2E0,V9k_2E0))) )
            & ! [V10k_2E0: tyop_2Enum_2Enum,V11n_2E0: tyop_2Enum_2Enum] :
                ( ( ~ p(c_2Ebit_2EMOD__2EXP__EQ_2E3(V5m_2E0,V10k_2E0,V4p_2E0))
                  | ( p(c_2Eprim__rec_2E_3C_2E2(V11n_2E0,V5m_2E0))
                    & ( c_2Ebit_2EBIT_2E2(V11n_2E0,V4p_2E0) != c_2Ebit_2EBIT_2E2(V11n_2E0,V10k_2E0) ) ) )
               => ( ~ p(c_2Eoption_2EIS__SOME_2E1(A_27c,c_2Epatricia_2EPEEK_2E2(A_27c,V6l_2E0,V10k_2E0)))
                  & ~ p(c_2Eoption_2EIS__SOME_2E1(A_27c,c_2Epatricia_2EPEEK_2E2(A_27c,V7r_2E0,V10k_2E0))) ) ) ) ) ) ).

tff(thm_2Epatricia_2EPTREE__EQ,axiom,
    ! [A_27a: $tType,V0t1_2E0: tyop_2Epatricia_2Eptree(A_27a),V1t2_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V1t2_2E0)) )
     => ( ! [V2k_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EPEEK_2E2(A_27a,V0t1_2E0,V2k_2E0) = c_2Epatricia_2EPEEK_2E2(A_27a,V1t2_2E0,V2k_2E0) )
      <=> ( V0t1_2E0 = V1t2_2E0 ) ) ) ).

tff(thm_2Epatricia_2EREMOVE__REMOVE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0),V1k_2E0) = c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0) ) ) ).

tff(thm_2Epatricia_2EREMOVE__ADD,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V3j_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V3j_2E0))
        <=> ( V1k_2E0 = V3j_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a,V3j_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
         => ( c_2Epatricia_2EREMOVE_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0)),V3j_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Epatricia_2Eptree(A_27a),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V3j_2E0),c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V3j_2E0),c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V3j_2E0),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0))) ) ) ) ).

tff(thm_2Epatricia_2EADD__ADD__SYM,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum,V3d_2E0: A_27a,V4e_2E0: A_27a] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
        & ( V1k_2E0 != V2j_2E0 ) )
     => ( c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V3d_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V2j_2E0,V4e_2E0)) = c_2Epatricia_2EADD_2E2(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V2j_2E0,V4e_2E0)),c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V3d_2E0)) ) ) ).

tff(thm_2Epatricia_2EFILTER__ALL,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Elist_2Elist(A_27a)] :
      ( ! [V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,c_2Elist_2ELENGTH_2E1(A_27a,V1l_2E0)))
         => ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,c_2Elist_2EEL_2E2(A_27a,V2n_2E0,V1l_2E0))) )
    <=> ( c_2Elist_2EFILTER_2E2(A_27a,V0P_2E0,V1l_2E0) = c_2Elist_2ENIL_2E0(A_27a) ) ) ).

tff(thm_2Epatricia_2ETRAVERSE__TRANSFORM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27b,c_2Epatricia_2ETRANSFORM_2E2(A_27b,A_27a,V0f_2E0,V1t_2E0)) = c_2Epatricia_2ETRAVERSE_2E1(A_27a,V1t_2E0) ) ).

tff(thm_2Epatricia_2EMEM__TRAVERSE__PEEK,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0))) = c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V0t_2E0,V1k_2E0)) ) ) ).

tff(thm_2Epatricia_2EIN__NUMSET__OF__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1n_2E0,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0)) = c_2Epatricia_2EIN__PTREE_2E2(V1n_2E0,V0t_2E0) ) ) ).

tff(thm_2Epatricia_2EADD__LIST__IS__PTREE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a))] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EADD__LIST_2E2(A_27a,V0t_2E0,V1l_2E0))) ) ).

tff(thm_2Epatricia_2EADD__LIST__TO__EMPTY__IS__PTREE,axiom,
    ! [A_27a: $tType,V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,A_27a))] : p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EADD__LIST_2E2(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a),V0l_2E0))) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0))) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE__EMPTY,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0))) ).

tff(thm_2Epatricia_2ENOT__KEY__LEFT__AND__RIGHT,axiom,
    ! [A_27a: $tType,V0p_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2l_2E0: tyop_2Epatricia_2Eptree(A_27a),V3r_2E0: tyop_2Epatricia_2Eptree(A_27a),V4k_2E0: tyop_2Enum_2Enum,V5j_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V0p_2E0,V1m_2E0,V2l_2E0,V3r_2E0)))
        & p(c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V2l_2E0,V4k_2E0)))
        & p(c_2Eoption_2EIS__SOME_2E1(A_27a,c_2Epatricia_2EPEEK_2E2(A_27a,V3r_2E0,V5j_2E0))) )
     => ( V4k_2E0 != V5j_2E0 ) ) ).

tff(thm_2Epatricia_2EALL__DISTINCT__TRAVERSE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => p(c_2Elist_2EALL__DISTINCT_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0))) ) ).

tff(thm_2Epatricia_2EMEM__ALL__DISTINCT__IMP__PERM,axiom,
    ! [A_27a: $tType,V0l1_2E0: tyop_2Elist_2Elist(A_27a),V1l2_2E0: tyop_2Elist_2Elist(A_27a)] :
      ( ( p(c_2Elist_2EALL__DISTINCT_2E1(A_27a,V0l1_2E0))
        & p(c_2Elist_2EALL__DISTINCT_2E1(A_27a,V1l2_2E0))
        & ! [V2x_2E0: A_27a] : ( c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,c_2Elist_2ELIST__TO__SET_2E1(A_27a,V0l1_2E0)) = c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,c_2Elist_2ELIST__TO__SET_2E1(A_27a,V1l2_2E0)) ) )
     => p(c_2Esorting_2EPERM_2E2(A_27a,V0l1_2E0,V1l2_2E0)) ) ).

tff(thm_2Epatricia_2EMEM__TRAVERSE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0))) = c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0)) ) ) ).

tff(thm_2Epatricia_2EINSERT__PTREE__IS__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1x_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EINSERT__PTREE_2E2(V1x_2E0,V0t_2E0))) ) ).

tff(thm_2Epatricia_2EFINITE__NUMSET__OF__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0))) ).

tff(thm_2Epatricia_2EADD__INSERT,axiom,
    ! [V0v_2E0: tyop_2Eone_2Eone,V1t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V2n_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EADD_2E2(tyop_2Eone_2Eone,V1t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Eone_2Eone,V2n_2E0,V0v_2E0)) = c_2Epatricia_2EINSERT__PTREE_2E2(V2n_2E0,V1t_2E0) ) ).

tff(thm_2Epatricia_2EPEEK__INSERT__PTREE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0))
        <=> ( V1k_2E0 = V2j_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1k_2E0: tyop_2Enum_2Enum,V2j_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
         => ( c_2Epatricia_2EPEEK_2E2(tyop_2Eone_2Eone,c_2Epatricia_2EINSERT__PTREE_2E2(V1k_2E0,V0t_2E0),V2j_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Eone_2Eone),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2j_2E0),c_2Eoption_2ESOME_2E1(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0),c_2Epatricia_2EPEEK_2E2(tyop_2Eone_2Eone,V0t_2E0,V2j_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EMEM__TRAVERSE__INSERT__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1x_2E0: tyop_2Enum_2Enum,V2h_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1x_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EINSERT__PTREE_2E2(V2h_2E0,V0t_2E0)))))
      <=> ( ( V1x_2E0 = V2h_2E0 )
          | ( ( V1x_2E0 != V2h_2E0 )
            & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1x_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0)))) ) ) ) ) ).

tff(thm_2Epatricia_2EPERM__INSERT__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0))
     => ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
       => p(c_2Esorting_2EPERM_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,c_2Elist_2EFOLDL_2E3(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),c_2Ecombin_2EC_2E1(tyop_2Enum_2Enum,tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),c_2Epatricia_2EINSERT__PTREE_2E0),V0t_2E0,c_2Elist_2ESET__TO__LIST_2E1(tyop_2Enum_2Enum,V1s_2E0))),c_2Elist_2ESET__TO__LIST_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0),V1s_2E0)))) ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__OF__NUMSET,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0)) )
     => ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0)))
      <=> ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,V0t_2E0))
          | p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,V1s_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__EMPTY,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Epatricia_2EIN__PTREE_2E2(V0n_2E0,c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone))) ).

tff(thm_2Epatricia_2EIN__PTREE__OF__NUMSET__EMPTY,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V0s_2E0))
     => ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1n_2E0,V0s_2E0) = c_2Epatricia_2EIN__PTREE_2E2(V1n_2E0,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0)) ) ) ).

tff(thm_2Epatricia_2EPTREE__EXTENSION,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t2_2E0)) )
     => ( ( V0t1_2E0 = V1t2_2E0 )
      <=> ! [V2x_2E0: tyop_2Enum_2Enum] : ( c_2Epatricia_2EIN__PTREE_2E2(V2x_2E0,V0t1_2E0) = c_2Epatricia_2EIN__PTREE_2E2(V2x_2E0,V1t2_2E0) ) ) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__NUMSET__OF__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0)) )
     => ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0),V1s_2E0)) = c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0) ) ) ).

tff(thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0)) )
     => ( c_2Epatricia_2ENUMSET__OF__PTREE_2E1(c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0),V1s_2E0) ) ) ).

tff(thm_2Epatricia_2ENUMSET__OF__PTREE__EMPTY,axiom,
    c_2Epatricia_2ENUMSET__OF__PTREE_2E1(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__EMPTY,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum)) = V0t_2E0 ) ).

tff(thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET__EMPTY,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V0s_2E0))
     => ( c_2Epatricia_2ENUMSET__OF__PTREE_2E1(c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0)) = V0s_2E0 ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__INSERT__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,c_2Epatricia_2EINSERT__PTREE_2E2(V1m_2E0,V0t_2E0)))
      <=> ( ( V1m_2E0 = V2n_2E0 )
          | p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,V0t_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__REMOVE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,c_2Epatricia_2EREMOVE_2E2(tyop_2Eone_2Eone,V0t_2E0,V1m_2E0)))
      <=> ( ( V2n_2E0 != V1m_2E0 )
          & p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,V0t_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EIN__PTREE__UNION__PTREE,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t2_2E0)) )
     => ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,V1t2_2E0)))
      <=> ( p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,V0t1_2E0))
          | p(c_2Epatricia_2EIN__PTREE_2E2(V2n_2E0,V1t2_2E0)) ) ) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__IS__PTREE,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t2_2E0)) )
     => p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,V1t2_2E0))) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__COMM,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t2_2E0)) )
     => ( c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,V1t2_2E0) = c_2Epatricia_2EUNION__PTREE_2E2(V1t2_2E0,V0t1_2E0) ) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__COMM__EMPTY,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( c_2Epatricia_2EUNION__PTREE_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0t_2E0) = c_2Epatricia_2EUNION__PTREE_2E2(V0t_2E0,c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone)) ) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__EMPTY,axiom,
    ( ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] : ( c_2Epatricia_2EUNION__PTREE_2E2(V0t_2E0,c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone)) = V0t_2E0 )
    & ! [V1t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
        ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t_2E0))
       => ( c_2Epatricia_2EUNION__PTREE_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V1t_2E0) = V1t_2E0 ) ) ) ).

tff(thm_2Epatricia_2EUNION__PTREE__ASSOC,axiom,
    ! [V0t1_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1t2_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V2t3_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V1t2_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V2t3_2E0)) )
     => ( c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,c_2Epatricia_2EUNION__PTREE_2E2(V1t2_2E0,V2t3_2E0)) = c_2Epatricia_2EUNION__PTREE_2E2(c_2Epatricia_2EUNION__PTREE_2E2(V0t1_2E0,V1t2_2E0),V2t3_2E0) ) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__UNION,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2s2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s1_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V2s2_2E0)) )
     => ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,V1s1_2E0,V2s2_2E0)) = c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s1_2E0),V2s2_2E0) ) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0)) )
     => ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,V2x_2E0,V1s_2E0)) = c_2Epatricia_2EINSERT__PTREE_2E2(V2x_2E0,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0)) ) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT__EMPTY,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V0s_2E0))
     => ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,V1x_2E0,V0s_2E0)) = c_2Epatricia_2EINSERT__PTREE_2E2(V1x_2E0,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0)) ) ) ).

tff(thm_2Epatricia_2EPTREE__OF__NUMSET__DELETE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V0s_2E0))
     => ( c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),c_2Epred__set_2EDELETE_2E2(tyop_2Enum_2Enum,V0s_2E0,V1x_2E0)) = c_2Epatricia_2EREMOVE_2E2(tyop_2Eone_2Eone,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0),V1x_2E0) ) ) ).

tff(thm_2Epatricia_2ETRAVERSE__AUX,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0) = c_2Epatricia_2ETRAVERSE__AUX_2E2(A_27a,V0t_2E0,c_2Elist_2ENIL_2E0(tyop_2Enum_2Enum)) ) ).

tff(thm_2Epatricia_2EPTREE__TRAVERSE__EQ,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0t1_2E0: tyop_2Epatricia_2Eptree(A_27a),V1t2_2E0: tyop_2Epatricia_2Eptree(A_27b)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(A_27b,V1t2_2E0)) )
     => ( ! [V2k_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t1_2E0))) = c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27b,V1t2_2E0))) )
      <=> ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t1_2E0) = c_2Epatricia_2ETRAVERSE_2E1(A_27b,V1t2_2E0) ) ) ) ).

tff(thm_2Epatricia_2EQSORT__MEM__EQ,axiom,
    ! [A_27a: $tType,V0l2_2E0: tyop_2Elist_2Elist(A_27a),V1l1_2E0: tyop_2Elist_2Elist(A_27a),V2R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ( c_2Esorting_2EQSORT_2E2(A_27a,V2R_2E0,V1l1_2E0) = c_2Esorting_2EQSORT_2E2(A_27a,V2R_2E0,V0l2_2E0) )
     => ! [V3x_2E0: A_27a] : ( c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Elist_2ELIST__TO__SET_2E1(A_27a,V1l1_2E0)) = c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Elist_2ELIST__TO__SET_2E1(A_27a,V0l2_2E0)) ) ) ).

tff(thm_2Epatricia_2EKEYS__PEEK,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0t1_2E0: tyop_2Epatricia_2Eptree(A_27a),V1t2_2E0: tyop_2Epatricia_2Eptree(A_27b)] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t1_2E0))
        & p(c_2Epatricia_2EIS__PTREE_2E1(A_27b,V1t2_2E0)) )
     => ( ( c_2Epatricia_2EKEYS_2E1(A_27a,V0t1_2E0) = c_2Epatricia_2EKEYS_2E1(A_27b,V1t2_2E0) )
      <=> ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t1_2E0) = c_2Epatricia_2ETRAVERSE_2E1(A_27b,V1t2_2E0) ) ) ) ).

tff(thm_2Epatricia_2EPERM__ADD,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
        & ~ p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) )
     => p(c_2Esorting_2EPERM_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0))),c_2Elist_2ECONS_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) ) ).

tff(thm_2Epatricia_2EPERM__NOT__ADD,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
        & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) )
     => ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0))) = c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0) ) ) ).

tff(thm_2Epatricia_2EPERM__NOT__REMOVE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
        & ~ p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) )
     => ( c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0)) = c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0) ) ) ).

tff(thm_2Epatricia_2EPERM__DELETE__PTREE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2x_2E0))
        <=> ( V2x_2E0 != V1k_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1k_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
            & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0)))) )
         => p(c_2Esorting_2EPERM_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EREMOVE_2E2(tyop_2Eone_2Eone,V0t_2E0,V1k_2E0)),c_2Elist_2EFILTER_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0)))) ) ) ).

tff(thm_2Epatricia_2EFILTER__NONE,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Elist_2Elist(A_27a)] :
      ( ! [V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,c_2Elist_2ELENGTH_2E1(A_27a,V1l_2E0)))
         => p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,c_2Elist_2EEL_2E2(A_27a,V2n_2E0,V1l_2E0))) )
     => ( c_2Elist_2EFILTER_2E2(A_27a,V0P_2E0,V1l_2E0) = V1l_2E0 ) ) ).

tff(thm_2Epatricia_2EPERM__REMOVE,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V1k_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),V2x_2E0))
        <=> ( V2x_2E0 != V1k_2E0 ) )
     => ! [V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
            & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) )
         => p(c_2Esorting_2EPERM_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0)),c_2Elist_2EFILTER_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V1k_2E0),c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0)))) ) ) ).

tff(thm_2Epatricia_2ESIZE__ADD,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum,V2d_2E0: A_27a] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => ( c_2Epatricia_2ESIZE_2E1(A_27a,c_2Epatricia_2EADD_2E2(A_27a,V0t_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,A_27a,V1k_2E0,V2d_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0))),c_2Epatricia_2ESIZE_2E1(A_27a,V0t_2E0),c_2Earithmetic_2E_2B_2E2(c_2Epatricia_2ESIZE_2E1(A_27a,V0t_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).

tff(thm_2Epatricia_2ESIZE__REMOVE,axiom,
    ! [A_27a: $tType,V0t_2E0: tyop_2Epatricia_2Eptree(A_27a),V1k_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(A_27a,V0t_2E0))
     => ( c_2Epatricia_2ESIZE_2E1(A_27a,c_2Epatricia_2EREMOVE_2E2(A_27a,V0t_2E0,V1k_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Elist_2ELIST__TO__SET_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(A_27a,V0t_2E0))),c_2Earithmetic_2E_2D_2E2(c_2Epatricia_2ESIZE_2E1(A_27a,V0t_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Epatricia_2ESIZE_2E1(A_27a,V0t_2E0)) ) ) ).

tff(thm_2Epatricia_2ESIZE,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Epatricia_2ESIZE_2E1(A_27a,c_2Epatricia_2EEmpty_2E0(A_27a)) = c_2Enum_2E0_2E0 )
      & ! [V0k_2E0: tyop_2Enum_2Enum,V1d_2E0: A_27a] : ( c_2Epatricia_2ESIZE_2E1(A_27a,c_2Epatricia_2ELeaf_2E2(A_27a,V0k_2E0,V1d_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
      & ! [V2p_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4l_2E0: tyop_2Epatricia_2Eptree(A_27a),V5r_2E0: tyop_2Epatricia_2Eptree(A_27a)] : ( c_2Epatricia_2ESIZE_2E1(A_27a,c_2Epatricia_2EBranch_2E4(A_27a,V2p_2E0,V3m_2E0,V4l_2E0,V5r_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Epatricia_2ESIZE_2E1(A_27a,V4l_2E0),c_2Epatricia_2ESIZE_2E1(A_27a,V5r_2E0)) ) ) ).

tff(thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0))
     => ( ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
          & p(c_2Elist_2EALL__DISTINCT_2E1(tyop_2Enum_2Enum,c_2Elist_2EAPPEND_2E2(tyop_2Enum_2Enum,c_2Epatricia_2ETRAVERSE_2E1(tyop_2Eone_2Eone,V0t_2E0),c_2Elist_2ESET__TO__LIST_2E1(tyop_2Enum_2Enum,V1s_2E0)))) )
       => ( c_2Epatricia_2ESIZE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(V0t_2E0,V1s_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Epatricia_2ESIZE_2E1(tyop_2Eone_2Eone,V0t_2E0),c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,V1s_2E0)) ) ) ) ).

tff(thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET__EMPTY,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V0s_2E0))
     => ( c_2Epatricia_2ESIZE_2E1(tyop_2Eone_2Eone,c_2Epatricia_2EPTREE__OF__NUMSET_2E2(c_2Epatricia_2EEmpty_2E0(tyop_2Eone_2Eone),V0s_2E0)) = c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,V0s_2E0) ) ) ).

tff(thm_2Epatricia_2ECARD__LIST__TO__SET,axiom,
    ! [A_27a: $tType,V0ls_2E0: tyop_2Elist_2Elist(A_27a)] :
      ( p(c_2Elist_2EALL__DISTINCT_2E1(A_27a,V0ls_2E0))
     => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Elist_2ELIST__TO__SET_2E1(A_27a,V0ls_2E0)) = c_2Elist_2ELENGTH_2E1(A_27a,V0ls_2E0) ) ) ).

tff(thm_2Epatricia_2ECARD__NUMSET__OF__PTREE,axiom,
    ! [V0t_2E0: tyop_2Epatricia_2Eptree(tyop_2Eone_2Eone)] :
      ( p(c_2Epatricia_2EIS__PTREE_2E1(tyop_2Eone_2Eone,V0t_2E0))
     => ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,c_2Epatricia_2ENUMSET__OF__PTREE_2E1(V0t_2E0)) = c_2Epatricia_2ESIZE_2E1(tyop_2Eone_2Eone,V0t_2E0) ) ) ).

tff(thm_2Epatricia_2EDELETE__UNION,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a,V1s1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2s2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDELETE_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1s1_2E0,V2s2_2E0),V0x_2E0) = c_2Epred__set_2EUNION_2E2(A_27a,c_2Epred__set_2EDELETE_2E2(A_27a,V1s1_2E0,V0x_2E0),c_2Epred__set_2EDELETE_2E2(A_27a,V2s2_2E0,V0x_2E0)) ) ).

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