ITP001 Axioms: ITP019_7.ax


%------------------------------------------------------------------------------
% File     : ITP019_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    : while.ax [Gau19]
%          : HL4019_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  119 (  29 unt;  53 typ;   0 def)
%            Number of atoms       :  167 (  59 equ)
%            Maximal formula atoms :   11 (   1 avg)
%            Number of connectives :  117 (  16   ~;   3   |;  22   &)
%                                         (  27 <=>;  49  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   6 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   44 (  24   >;  20   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   50 (  50 usr;  12 con; 0-5 aty)
%            Number of variables   :  280 ( 235   !;   7   ?; 280   :)
%                                         (  38  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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_2Eoption_2Eoption,type,
    tyop_2Eoption_2Eoption: $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_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_2Earithmetic_2E_3C_3D_2E0,type,
    c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Earithmetic_2E_3C_3D_2E2,type,
    c_2Earithmetic_2E_3C_3D_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_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_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2EFUNPOW_2E0,type,
    c_2Earithmetic_2EFUNPOW_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Earithmetic_2EFUNPOW_2E3,type,
    c_2Earithmetic_2EFUNPOW_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Enum_2Enum * A_27a ) > A_27a ) ).

tff(c_2Ewhile_2EHOARE__SPEC_2E0,type,
    c_2Ewhile_2EHOARE__SPEC_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_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))) ).

tff(c_2Ewhile_2EHOARE__SPEC_2E3,type,
    c_2Ewhile_2EHOARE__SPEC_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Ewhile_2ELEAST_2E0,type,
    c_2Ewhile_2ELEAST_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).

tff(c_2Ewhile_2ELEAST_2E1,type,
    c_2Ewhile_2ELEAST_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ).

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

tff(c_2Ewhile_2EOLEAST_2E0,type,
    c_2Ewhile_2EOLEAST_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)) ).

tff(c_2Ewhile_2EOLEAST_2E1,type,
    c_2Ewhile_2EOLEAST_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) > tyop_2Eoption_2Eoption(tyop_2Enum_2Enum) ).

tff(c_2Ewhile_2EOWHILE_2E0,type,
    c_2Ewhile_2EOWHILE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)))) ).

tff(c_2Ewhile_2EOWHILE_2E3,type,
    c_2Ewhile_2EOWHILE_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,A_27a) * A_27a ) > tyop_2Eoption_2Eoption(A_27a) ) ).

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

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

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

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

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_2Ewhile_2EWHILE_2E0,type,
    c_2Ewhile_2EWHILE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ewhile_2EWHILE_2E2,type,
    c_2Ewhile_2EWHILE_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,A_27a) ) > tyop_2Emin_2Efun(A_27a,A_27a) ) ).

tff(c_2Ewhile_2EWHILE_2E3,type,
    c_2Ewhile_2EWHILE_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,A_27a) * A_27a ) > A_27a ) ).

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_2Ecombin_2Eo_2E0,type,
    c_2Ecombin_2Eo_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).

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

tff(c_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_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_2Earithmetic_2E_3C_3D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_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_2Earithmetic_2E_3C_3D_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(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ebool_2ECOND_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ebool_2ECOND_2E0(A_27a),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_2Enum_2Enum_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X2_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),c_2Ebool_2ECOND_2E0(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Earithmetic_2EFUNPOW_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Enum_2Enum,X2_2E0: A_27a] : ( c_2Earithmetic_2EFUNPOW_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Earithmetic_2EFUNPOW_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ewhile_2EHOARE__SPEC_2E3_2Emono_2EA_27a_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2EHOARE__SPEC_2E3(A_27a,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ewhile_2EHOARE__SPEC_2E0(A_27a,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ewhile_2EHOARE__SPEC_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2EHOARE__SPEC_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ewhile_2EHOARE__SPEC_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ewhile_2ELEAST_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2ELEAST_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ewhile_2EOLEAST_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2EOLEAST_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Ewhile_2EOLEAST_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Ewhile_2EOWHILE_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X2_2E0: A_27a] : ( c_2Ewhile_2EOWHILE_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a))),c_2Ewhile_2EOWHILE_2E0(A_27a),X0_2E0),X1_2E0),X2_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_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ewhile_2EWHILE_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,A_27a)] : ( c_2Ewhile_2EWHILE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ewhile_2EWHILE_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ewhile_2EWHILE_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X2_2E0: A_27a] : ( c_2Ewhile_2EWHILE_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ewhile_2EWHILE_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ewhile_2EWHILE_2E3_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_2Enum_2Enum),X2_2E0: tyop_2Enum_2Enum] : ( c_2Ewhile_2EWHILE_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ewhile_2EWHILE_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_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(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,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_2Emin_2Ebool,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_2Ecombin_2Eo_2E0(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),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_2Ewhile_2EWHILE,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2x_2E0: A_27a] : ( c_2Ewhile_2EWHILE_2E3(A_27a,V0P_2E0,V1g_2E0,V2x_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0),c_2Ewhile_2EWHILE_2E3(A_27a,V0P_2E0,V1g_2E0,app_2E2(A_27a,A_27a,V1g_2E0,V2x_2E0)),V2x_2E0) ) ).

tff(thm_2Ewhile_2EHOARE__SPEC__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1C_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
      ( p(c_2Ewhile_2EHOARE__SPEC_2E3(A_27a,A_27b,V0P_2E0,V1C_2E0,V2Q_2E0))
    <=> ! [V3s_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3s_2E0))
         => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V2Q_2E0,app_2E2(A_27a,A_27b,V1C_2E0,V3s_2E0))) ) ) ).

tff(thm_2Ewhile_2ELEAST__DEF,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2ELEAST_2E1(V0P_2E0) = c_2Ewhile_2EWHILE_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,V0P_2E0),c_2Enum_2ESUC_2E0,c_2Enum_2E0_2E0) ) ).

tff(thm_2Ewhile_2EOLEAST__def,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0P_2E0))
        <=> ? [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0)) )
     => ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(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_2Ebool),F1_2E0,V0P_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V2n_2E0) )
       => ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2EOLEAST_2E1(V0P_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0P_2E0),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0P_2E0))),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)) ) ) ) ).

tff(thm_2Ewhile_2EOWHILE__def,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))))] :
      ( ! [V0G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),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,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0G_2E0),V1f_2E0),V2s_2E0))
        <=> ? [V3n_2E0: tyop_2Enum_2Enum] : ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0G_2E0,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,V3n_2E0,V2s_2E0))) )
     => ( ! [V0G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: A_27a,V4n_2E0: tyop_2Enum_2Enum] :
            ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F1_2E0,V0G_2E0),V1f_2E0),V2s_2E0),V4n_2E0))
          <=> ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0G_2E0,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,V4n_2E0,V2s_2E0))) )
       => ! [V0G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: A_27a] : ( c_2Ewhile_2EOWHILE_2E3(A_27a,V0G_2E0,V1f_2E0,V2s_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),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,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0G_2E0),V1f_2E0),V2s_2E0),c_2Eoption_2ESOME_2E1(A_27a,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,c_2Ewhile_2ELEAST_2E1(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F1_2E0,V0G_2E0),V1f_2E0),V2s_2E0)),V2s_2E0)),c_2Eoption_2ENONE_2E0(A_27a)) ) ) ) ).

tff(thm_2Ewhile_2EITERATION,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
    ? [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
    ! [V3x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,V2f_2E0,V3x_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0),V3x_2E0,app_2E2(A_27a,A_27a,V2f_2E0,app_2E2(A_27a,A_27a,V1g_2E0,V3x_2E0))) ) ).

tff(thm_2Ewhile_2EWHILE__INDUCTION,axiom,
    ! [A_27a: $tType,V0B_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1C_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ( p(c_2Erelation_2EWF_2E1(A_27a,V2R_2E0))
        & ! [V3s_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0B_2E0,V3s_2E0))
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2R_2E0,app_2E2(A_27a,A_27a,V1C_2E0,V3s_2E0)),V3s_2E0)) ) )
     => ! [V4P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ! [V5s_2E0: A_27a] :
              ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0B_2E0,V5s_2E0))
               => p(app_2E2(A_27a,tyop_2Emin_2Ebool,V4P_2E0,app_2E2(A_27a,A_27a,V1C_2E0,V5s_2E0))) )
             => p(app_2E2(A_27a,tyop_2Emin_2Ebool,V4P_2E0,V5s_2E0)) )
         => ! [V6v_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V4P_2E0,V6v_2E0)) ) ) ).

tff(thm_2Ewhile_2EWHILE__RULE,axiom,
    ! [A_27a: $tType,F1_2E0: 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))),F0_2E0: 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)))] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2B_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6s_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,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)),F1_2E0,V0P_2E0),V2B_2E0),V6s_2E0))
        <=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V6s_2E0))
            & ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2B_2E0,V6s_2E0)) ) )
     => ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2B_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5s_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,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)),F0_2E0,V0P_2E0),V2B_2E0),V5s_2E0))
          <=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V5s_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2B_2E0,V5s_2E0)) ) )
       => ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2B_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3C_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
            ( ( p(c_2Erelation_2EWF_2E1(A_27a,V1R_2E0))
              & ! [V4s_2E0: A_27a] :
                  ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2B_2E0,V4s_2E0))
                 => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0,app_2E2(A_27a,A_27a,V3C_2E0,V4s_2E0)),V4s_2E0)) ) )
           => ( p(c_2Ewhile_2EHOARE__SPEC_2E3(A_27a,A_27a,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)),F0_2E0,V0P_2E0),V2B_2E0),V3C_2E0,V0P_2E0))
             => p(c_2Ewhile_2EHOARE__SPEC_2E3(A_27a,A_27a,V0P_2E0,c_2Ewhile_2EWHILE_2E2(A_27a,V2B_2E0,V3C_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)),F1_2E0,V0P_2E0),V2B_2E0))) ) ) ) ) ).

tff(thm_2Ewhile_2ELEAST__INTRO,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Enum_2Enum] :
      ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
     => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Ewhile_2ELEAST_2E1(V0P_2E0))) ) ).

tff(thm_2Ewhile_2ELESS__LEAST,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,c_2Ewhile_2ELEAST_2E1(V0P_2E0)))
     => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1m_2E0)) ) ).

tff(thm_2Ewhile_2EFULL__LEAST__INTRO,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Enum_2Enum] :
      ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
     => ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Ewhile_2ELEAST_2E1(V0P_2E0)))
        & p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ewhile_2ELEAST_2E1(V0P_2E0),V1x_2E0)) ) ) ).

tff(thm_2Ewhile_2ELEAST__ELIM,axiom,
    ! [V0Q_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( ? [V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V2n_2E0))
        & ! [V3n_2E0: tyop_2Enum_2Enum] :
            ( ( ! [V4m_2E0: tyop_2Enum_2Enum] :
                  ( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V3n_2E0))
                 => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V4m_2E0)) )
              & p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V3n_2E0)) )
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0Q_2E0,V3n_2E0)) ) )
     => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0Q_2E0,c_2Ewhile_2ELEAST_2E1(V1P_2E0))) ) ).

tff(thm_2Ewhile_2ELEAST__EXISTS,axiom,
    ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ? [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,V1n_2E0))
    <=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,c_2Ewhile_2ELEAST_2E1(V0p_2E0)))
        & ! [V2n_2E0: tyop_2Enum_2Enum] :
            ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,c_2Ewhile_2ELEAST_2E1(V0p_2E0)))
           => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,V2n_2E0)) ) ) ) ).

tff(thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
    ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ? [V1n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,V1n_2E0))
     => ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,c_2Ewhile_2ELEAST_2E1(V0p_2E0)))
        & ! [V2n_2E0: tyop_2Enum_2Enum] :
            ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,c_2Ewhile_2ELEAST_2E1(V0p_2E0)))
           => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,V2n_2E0)) ) ) ) ).

tff(thm_2Ewhile_2ELEAST__EQ,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V0x_2E0: tyop_2Enum_2Enum,V2n_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),F1_2E0,V0x_2E0),V2n_2E0))
        <=> ( V0x_2E0 = V2n_2E0 ) )
     => ( ! [V0x_2E0: tyop_2Enum_2Enum,V1n_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,V0x_2E0),V1n_2E0))
          <=> ( V1n_2E0 = V0x_2E0 ) )
       => ! [V0x_2E0: tyop_2Enum_2Enum] :
            ( ( c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) = V0x_2E0 )
            & ( c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0x_2E0)) = V0x_2E0 ) ) ) ) ).

tff(thm_2Ewhile_2ELEAST__T,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ! [V0x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0) = c_2Ebool_2ET_2E0 )
     => ( c_2Ewhile_2ELEAST_2E1(F0_2E0) = c_2Enum_2E0_2E0 ) ) ).

tff(thm_2Ewhile_2EOLEAST__INTRO,axiom,
    ! [V0Q_2E0: tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( ( ! [V2n_2E0: tyop_2Enum_2Enum] : ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V2n_2E0))
         => p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,V0Q_2E0,c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum))) )
        & ! [V3n_2E0: tyop_2Enum_2Enum] :
            ( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V3n_2E0))
              & ! [V4m_2E0: tyop_2Enum_2Enum] :
                  ( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V3n_2E0))
                 => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V4m_2E0)) ) )
           => p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,V0Q_2E0,c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V3n_2E0))) ) )
     => p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,V0Q_2E0,c_2Ewhile_2EOLEAST_2E1(V1P_2E0))) ) ).

tff(thm_2Ewhile_2EOLEAST__EQNS,axiom,
    ! [F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
      ( ! [V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F3_2E0,V4n_2E0) = c_2Ebool_2ET_2E0 )
     => ( ! [V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F2_2E0,V3n_2E0) = c_2Ebool_2EF_2E0 )
       => ( ! [V0x_2E0: tyop_2Enum_2Enum,V2n_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),F1_2E0,V0x_2E0),V2n_2E0))
            <=> ( V0x_2E0 = V2n_2E0 ) )
         => ( ! [V0x_2E0: tyop_2Enum_2Enum,V1n_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,V0x_2E0),V1n_2E0))
              <=> ( V1n_2E0 = V0x_2E0 ) )
           => ! [V0x_2E0: tyop_2Enum_2Enum] :
                ( ( c_2Ewhile_2EOLEAST_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0x_2E0) )
                & ( c_2Ewhile_2EOLEAST_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0x_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0x_2E0) )
                & ( c_2Ewhile_2EOLEAST_2E1(F2_2E0) = c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum) )
                & ( c_2Ewhile_2EOLEAST_2E1(F3_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,c_2Enum_2E0_2E0) ) ) ) ) ) ) ).

tff(thm_2Ewhile_2EOLEAST__EQ__NONE,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( c_2Ewhile_2EOLEAST_2E1(V0P_2E0) = c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum) )
    <=> ! [V1n_2E0: tyop_2Enum_2Enum] : ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0)) ) ).

tff(thm_2Ewhile_2EOLEAST__EQ__SOME,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum,V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ( c_2Ewhile_2EOLEAST_2E1(V1P_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0n_2E0) )
    <=> ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V0n_2E0))
        & ! [V2m_2E0: tyop_2Enum_2Enum] :
            ( p(c_2Eprim__rec_2E_3C_2E2(V2m_2E0,V0n_2E0))
           => ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V2m_2E0)) ) ) ) ).

tff(thm_2Ewhile_2EOWHILE__THM,axiom,
    ! [A_27a: $tType,V0s_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2EOWHILE_2E3(A_27a,V2G_2E0,V1f_2E0,V0s_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27a,tyop_2Emin_2Ebool,V2G_2E0,V0s_2E0),c_2Ewhile_2EOWHILE_2E3(A_27a,V2G_2E0,V1f_2E0,app_2E2(A_27a,A_27a,V1f_2E0,V0s_2E0)),c_2Eoption_2ESOME_2E1(A_27a,V0s_2E0)) ) ).

tff(thm_2Ewhile_2EOWHILE__EQ__NONE,axiom,
    ! [A_27a: $tType,V0s_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( c_2Ewhile_2EOWHILE_2E3(A_27a,V2G_2E0,V1f_2E0,V0s_2E0) = c_2Eoption_2ENONE_2E0(A_27a) )
    <=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2G_2E0,c_2Earithmetic_2EFUNPOW_2E3(A_27a,V1f_2E0,V3n_2E0,V0s_2E0))) ) ).

tff(thm_2Ewhile_2EOWHILE__ENDCOND,axiom,
    ! [A_27a: $tType,V0s_27_2E0: A_27a,V1s_2E0: A_27a,V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( c_2Ewhile_2EOWHILE_2E3(A_27a,V3G_2E0,V2f_2E0,V1s_2E0) = c_2Eoption_2ESOME_2E1(A_27a,V0s_27_2E0) )
     => ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V3G_2E0,V0s_27_2E0)) ) ).

tff(thm_2Ewhile_2EOWHILE__WHILE,axiom,
    ! [A_27a: $tType,V0s_27_2E0: A_27a,V1s_2E0: A_27a,V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( c_2Ewhile_2EOWHILE_2E3(A_27a,V3G_2E0,V2f_2E0,V1s_2E0) = c_2Eoption_2ESOME_2E1(A_27a,V0s_27_2E0) )
     => ( c_2Ewhile_2EWHILE_2E3(A_27a,V3G_2E0,V2f_2E0,V1s_2E0) = V0s_27_2E0 ) ) ).

tff(thm_2Ewhile_2EOWHILE__INV__IND,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3s_2E0: A_27a] :
      ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3s_2E0))
        & ! [V4x_2E0: A_27a] :
            ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1G_2E0,V4x_2E0)) )
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,app_2E2(A_27a,A_27a,V2f_2E0,V4x_2E0))) ) )
     => ! [V5s_27_2E0: A_27a] :
          ( ( c_2Ewhile_2EOWHILE_2E3(A_27a,V1G_2E0,V2f_2E0,V3s_2E0) = c_2Eoption_2ESOME_2E1(A_27a,V5s_27_2E0) )
         => p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V5s_27_2E0)) ) ) ).

tff(thm_2Ewhile_2EOWHILE__IND,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1G_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
      ( ( ! [V3s_2E0: A_27a] :
            ( ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1G_2E0,V3s_2E0))
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V3s_2E0),V3s_2E0)) )
        & ! [V4s1_2E0: A_27a,V5s2_2E0: A_27a] :
            ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1G_2E0,V4s1_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,app_2E2(A_27a,A_27a,V2f_2E0,V4s1_2E0)),V5s2_2E0)) )
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V4s1_2E0),V5s2_2E0)) ) )
     => ! [V6s1_2E0: A_27a,V7s2_2E0: A_27a] :
          ( ( c_2Ewhile_2EOWHILE_2E3(A_27a,V1G_2E0,V2f_2E0,V6s1_2E0) = c_2Eoption_2ESOME_2E1(A_27a,V7s2_2E0) )
         => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V6s1_2E0),V7s2_2E0)) ) ) ).

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