ITP001 Axioms: ITP083^5.ax


%------------------------------------------------------------------------------
% File     : ITP083^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : fmaptree^2.ax [Gau20]
%          : HL4083^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   53 (   2 unt;  13 typ;   0 def)
%            Number of atoms       : 1024 (  30 equ;   0 cnn)
%            Maximal formula atoms :   59 (  19 avg)
%            Number of connectives : 2340 (   0   ~;   0   |;  23   &;2198   @)
%                                         (   4 <=>; 115  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (  15 avg;2198 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   28 (  28   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   48 (  47 usr;  34 con; 0-3 aty)
%            Number of variables   :  213 (  24   ^ 180   !;   9   ?; 213   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Efmaptree_2Efmaptree,type,
    ty_2Efmaptree_2Efmaptree: del > del > del ).

thf(tp_c_2Efmaptree_2EFTNode,type,
    c_2Efmaptree_2EFTNode: del > del > $i ).

thf(mem_c_2Efmaptree_2EFTNode,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ).

thf(tp_c_2Efmaptree_2Eapply__path,type,
    c_2Efmaptree_2Eapply__path: del > del > $i ).

thf(mem_c_2Efmaptree_2Eapply__path,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ).

thf(tp_c_2Efmaptree_2Econstruct,type,
    c_2Efmaptree_2Econstruct: del > del > $i ).

thf(mem_c_2Efmaptree_2Econstruct,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).

thf(tp_c_2Efmaptree_2Efmtreerec,type,
    c_2Efmaptree_2Efmtreerec: del > del > del > $i ).

thf(mem_c_2Efmaptree_2Efmtreerec,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) @ A_27a ) ) ) ).

thf(tp_c_2Efmaptree_2EfromF,type,
    c_2Efmaptree_2EfromF: del > del > $i ).

thf(mem_c_2Efmaptree_2EfromF,axiom,
    ! [A_27key: del,A_27value: del] : ( mem @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) ) ) ).

thf(tp_c_2Efmaptree_2Efupd__at__path,type,
    c_2Efmaptree_2Efupd__at__path: del > del > $i ).

thf(mem_c_2Efmaptree_2Efupd__at__path,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ).

thf(tp_c_2Efmaptree_2Eitem,type,
    c_2Efmaptree_2Eitem: del > del > $i ).

thf(mem_c_2Efmaptree_2Eitem,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ A_27b ) ) ).

thf(tp_c_2Efmaptree_2Emap,type,
    c_2Efmaptree_2Emap: del > del > $i ).

thf(mem_c_2Efmaptree_2Emap,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ).

thf(tp_c_2Efmaptree_2Erelrec,type,
    c_2Efmaptree_2Erelrec: del > del > del > $i ).

thf(mem_c_2Efmaptree_2Erelrec,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) ) ) ).

thf(tp_c_2Efmaptree_2EtoF,type,
    c_2Efmaptree_2EtoF: del > del > $i ).

thf(mem_c_2Efmaptree_2EtoF,axiom,
    ! [A_27key: del,A_27value: del] : ( mem @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) ) ) ).

thf(tp_c_2Efmaptree_2Eupdate__at__path,type,
    c_2Efmaptree_2Eupdate__at__path: del > del > $i ).

thf(mem_c_2Efmaptree_2Eupdate__at__path,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ).

thf(tp_c_2Efmaptree_2Ewf,type,
    c_2Efmaptree_2Ewf: del > del > $i ).

thf(mem_c_2Efmaptree_2Ewf,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) ) ).

thf(ax_thm_2Efmaptree_2Econstruct__def,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ A_27a )
     => ! [V1kfm: $i] :
          ( ( mem @ V1kfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
         => ! [V2kl: $i] :
              ( ( mem @ V2kl @ ( ty_2Elist_2Elist @ A_27b ) )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V0a ) @ V1kfm ) @ V2kl )
                = ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V2kl ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0a ) )
                  @ ( lam @ A_27b
                    @ ^ [V3h: $i] :
                        ( lam @ ( ty_2Elist_2Elist @ A_27b )
                        @ ^ [V4t: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1kfm ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1kfm ) @ V3h ) @ V4t ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Ewf__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Efmaptree_2Ewf @ A_27a @ A_27b )
      = ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
        @ ^ [V0a0: $i] :
            ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
            @ ( lam @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool )
              @ ^ [V1wf_27: $i] :
                  ( ap
                  @ ( ap @ c_2Emin_2E_3D_3D_3E
                    @ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
                      @ ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
                        @ ^ [V2a0: $i] :
                            ( ap
                            @ ( ap @ c_2Emin_2E_3D_3D_3E
                              @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                @ ( lam @ A_27a
                                  @ ^ [V3a: $i] :
                                      ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
                                      @ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
                                        @ ^ [V4fm: $i] :
                                            ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2a0 ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V3a ) @ V4fm ) ) )
                                            @ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
                                              @ ( lam @ A_27b
                                                @ ^ [V5k: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V4fm ) ) ) @ ( ap @ V1wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V4fm ) @ V5k ) ) ) ) ) ) ) ) ) ) )
                            @ ( ap @ V1wf_27 @ V2a0 ) ) ) ) )
                  @ ( ap @ V1wf_27 @ V0a0 ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Ewf__rules,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ A_27a )
     => ! [V1fm: $i] :
          ( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
         => ( ! [V2k: $i] :
                ( ( mem @ V2k @ A_27b )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1fm ) ) )
                 => ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1fm ) @ V2k ) ) ) ) )
           => ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V0a ) @ V1fm ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Ewf__ind,axiom,
    ! [A_27a: del,A_27b: del,V0wf_27: $i] :
      ( ( mem @ V0wf_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
     => ( ! [V1a: $i] :
            ( ( mem @ V1a @ A_27a )
           => ! [V2fm: $i] :
                ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
               => ( ! [V3k: $i] :
                      ( ( mem @ V3k @ A_27b )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
                       => ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) )
                 => ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
       => ! [V4a0: $i] :
            ( ( mem @ V4a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
           => ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V4a0 ) )
             => ( p @ ( ap @ V0wf_27 @ V4a0 ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Ewf__strongind,axiom,
    ! [A_27a: del,A_27b: del,V0wf_27: $i] :
      ( ( mem @ V0wf_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
     => ( ! [V1a: $i] :
            ( ( mem @ V1a @ A_27a )
           => ! [V2fm: $i] :
                ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
               => ( ! [V3k: $i] :
                      ( ( mem @ V3k @ A_27b )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
                       => ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) )
                          & ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) ) )
                 => ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
       => ! [V4a0: $i] :
            ( ( mem @ V4a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
           => ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V4a0 ) )
             => ( p @ ( ap @ V0wf_27 @ V4a0 ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Ewf__cases,axiom,
    ! [A_27a: del,A_27b: del,V0a0: $i] :
      ( ( mem @ V0a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
     => ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V0a0 ) )
      <=> ? [V1a: $i] :
            ( ( mem @ V1a @ A_27a )
            & ? [V2fm: $i] :
                ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
                & ( V0a0
                  = ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) )
                & ! [V3k: $i] :
                    ( ( mem @ V3k @ A_27b )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
                     => ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Efmaptree__TY__DEF,axiom,
    ! [A_27key: del,A_27value: del] :
    ? [V0rep: $i] :
      ( ( mem @ V0rep @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) ) )
      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) ) @ ( c_2Efmaptree_2Ewf @ A_27value @ A_27key ) ) @ V0rep ) ) ) ).

thf(ax_thm_2Efmaptree_2Efmap__bij__thm,axiom,
    ! [A_27key: del,A_27value: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) )
         => ( ( ap @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ ( ap @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ V0a ) )
            = V0a ) )
      & ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) )
         => ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27value @ A_27key ) @ V1r ) )
          <=> ( ( ap @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ ( ap @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ V1r ) )
              = V1r ) ) ) ) ).

thf(ax_thm_2Efmaptree_2EFTNode__def,axiom,
    ! [A_27a: del,A_27b: del,V0i: $i] :
      ( ( mem @ V0i @ A_27b )
     => ! [V1fm: $i] :
          ( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
         => ( ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0i ) @ V1fm )
            = ( ap @ ( c_2Efmaptree_2EfromF @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27b @ A_27a ) @ V0i ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) @ ( c_2Efmaptree_2EtoF @ A_27a @ A_27b ) ) @ V1fm ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2EFTNode__11,axiom,
    ! [A_27a: del,A_27b: del,V0i1: $i] :
      ( ( mem @ V0i1 @ A_27b )
     => ! [V1f1: $i] :
          ( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
         => ! [V2i2: $i] :
              ( ( mem @ V2i2 @ A_27b )
             => ! [V3f2: $i] :
                  ( ( mem @ V3f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
                 => ( ( ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0i1 ) @ V1f1 )
                      = ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V2i2 ) @ V3f2 ) )
                  <=> ( ( V0i1 = V2i2 )
                      & ( V1f1 = V3f2 ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Efmaptree__nchotomy,axiom,
    ! [A_27a: del,A_27b: del,V0ft: $i] :
      ( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
     => ? [V1i: $i] :
          ( ( mem @ V1i @ A_27b )
          & ? [V2fm: $i] :
              ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
              & ( V0ft
                = ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V1i ) @ V2fm ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Eitem__map__def,axiom,
    ! [A_27a: del,A_27b: del,V0ft: $i] :
      ( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
     => ( V0ft
        = ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V0ft ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V0ft ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Eitem__thm,axiom,
    ! [A_27a: del,A_27b: del,V0i: $i] :
      ( ( mem @ V0i @ A_27a )
     => ! [V1fm: $i] :
          ( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
         => ( ( ap @ ( c_2Efmaptree_2Eitem @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V0i ) @ V1fm ) )
            = V0i ) ) ) ).

thf(conj_thm_2Efmaptree_2Emap__thm,axiom,
    ! [A_27a: del,A_27b: del,V0i: $i] :
      ( ( mem @ V0i @ A_27a )
     => ! [V1fm: $i] :
          ( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
         => ( ( ap @ ( c_2Efmaptree_2Emap @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V0i ) @ V1fm ) )
            = V1fm ) ) ) ).

thf(ax_thm_2Efmaptree_2Eapply__path__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ! [V0ft: $i] :
          ( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
         => ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0ft )
            = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V0ft ) ) )
      & ! [V1h: $i] :
          ( ( mem @ V1h @ A_27a )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
             => ! [V3ft: $i] :
                  ( ( mem @ V3ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
                 => ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3ft )
                    = ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft ) ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V2t ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft ) ) @ V1h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Eupdate__at__path__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ A_27b )
         => ! [V1ft: $i] :
              ( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0a ) @ V1ft )
                = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V1ft ) ) ) ) ) )
      & ! [V2h: $i] :
          ( ( mem @ V2h @ A_27a )
         => ! [V3t: $i] :
              ( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
             => ! [V4a: $i] :
                  ( ( mem @ V4a @ A_27b )
                 => ! [V5ft: $i] :
                      ( ( mem @ V5ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
                     => ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4a ) @ V5ft )
                        = ( ap
                          @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) ) )
                            @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ V3t ) @ V4a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ V2h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
                              @ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
                                @ ^ [V6ft_27: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2h ) @ V6ft_27 ) ) ) ) ) ) )
                          @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Efupd__at__path__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ! [V0f: $i] :
          ( ( mem @ V0f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) )
         => ! [V1ft: $i] :
              ( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0f ) @ V1ft )
                = ( ap @ V0f @ V1ft ) ) ) )
      & ! [V2h: $i] :
          ( ( mem @ V2h @ A_27a )
         => ! [V3t: $i] :
              ( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
             => ! [V4f: $i] :
                  ( ( mem @ V4f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) )
                 => ! [V5ft: $i] :
                      ( ( mem @ V5ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
                     => ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4f ) @ V5ft )
                        = ( ap
                          @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) ) )
                            @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ V3t ) @ V4f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ V2h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
                              @ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
                                @ ^ [V6ft_27: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2h ) @ V6ft_27 ) ) ) ) ) ) )
                          @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Eft__ind,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ bool ) )
     => ( ! [V1a: $i] :
            ( ( mem @ V1a @ A_27b )
           => ! [V2fm: $i] :
                ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
               => ( ! [V3k: $i] :
                      ( ( mem @ V3k @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2fm ) ) )
                       => ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2fm ) @ V3k ) ) ) ) )
                 => ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
       => ! [V4ft: $i] :
            ( ( mem @ V4ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
           => ( p @ ( ap @ V0P @ V4ft ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Eapplicable__paths__FINITE,axiom,
    ! [A_27a: del,A_27b: del,V0ft: $i] :
      ( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
     => ( p
        @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Elist_2Elist @ A_27a ) )
          @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
            @ ( lam @ ( ty_2Elist_2Elist @ A_27a )
              @ ^ [V1p: $i] :
                  ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) @ V1p )
                  @ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
                    @ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
                      @ ^ [V2ft_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V1p ) @ V0ft ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2ft_27 ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Eapply__path__SNOC,axiom,
    ! [A_27a: del,A_27b: del,V0ft: $i] :
      ( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2p: $i] :
              ( ( mem @ V2p @ ( ty_2Elist_2Elist @ A_27a ) )
             => ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2p ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V0ft )
                = ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V2p ) @ V0ft ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
                  @ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
                    @ ^ [V3ft_27: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft_27 ) ) @ V1x ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Erelrec__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] :
      ( ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c )
      = ( lam @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) )
        @ ^ [V0h: $i] :
            ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a )
            @ ^ [V1a0: $i] :
                ( lam @ A_27c
                @ ^ [V2a1: $i] :
                    ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
                    @ ( lam @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) )
                      @ ^ [V3relrec_27: $i] :
                          ( ap
                          @ ( ap @ c_2Emin_2E_3D_3D_3E
                            @ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
                              @ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a )
                                @ ^ [V4a0: $i] :
                                    ( ap @ ( c_2Ebool_2E_21 @ A_27c )
                                    @ ( lam @ A_27c
                                      @ ^ [V5a1: $i] :
                                          ( ap
                                          @ ( ap @ c_2Emin_2E_3D_3D_3E
                                            @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                              @ ( lam @ A_27a
                                                @ ^ [V6i: $i] :
                                                    ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
                                                    @ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
                                                      @ ^ [V7fm: $i] :
                                                          ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
                                                          @ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c )
                                                            @ ^ [V8rfm: $i] :
                                                                ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4a0 ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V6i ) @ V7fm ) ) )
                                                                @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27c ) @ V5a1 ) @ ( ap @ ( ap @ ( ap @ V0h @ V6i ) @ V8rfm ) @ V7fm ) ) )
                                                                  @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V8rfm ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) ) )
                                                                    @ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
                                                                      @ ( lam @ A_27b
                                                                        @ ^ [V9d: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V9d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) ) ) @ ( ap @ ( ap @ V3relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) @ V9d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V8rfm ) @ V9d ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ ( ap @ ( ap @ V3relrec_27 @ V4a0 ) @ V5a1 ) ) ) ) ) ) )
                          @ ( ap @ ( ap @ V3relrec_27 @ V1a0 ) @ V2a1 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Erelrec__rules,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
     => ! [V1i: $i] :
          ( ( mem @ V1i @ A_27a )
         => ! [V2fm: $i] :
              ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
             => ! [V3rfm: $i] :
                  ( ( mem @ V3rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
                 => ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V3rfm )
                        = ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) )
                      & ! [V4d: $i] :
                          ( ( mem @ V4d @ A_27b )
                         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) ) )
                           => ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) @ V4d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V3rfm ) @ V4d ) ) ) ) ) )
                   => ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V1i ) @ V2fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V1i ) @ V3rfm ) @ V2fm ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Erelrec__ind,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
     => ! [V1relrec_27: $i] :
          ( ( mem @ V1relrec_27 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
         => ( ! [V2i: $i] :
                ( ( mem @ V2i @ A_27a )
               => ! [V3fm: $i] :
                    ( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
                   => ! [V4rfm: $i] :
                        ( ( mem @ V4rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
                       => ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V4rfm )
                              = ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) )
                            & ! [V5d: $i] :
                                ( ( mem @ V5d @ A_27b )
                               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) ) )
                                 => ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) ) ) ) )
                         => ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V4rfm ) @ V3fm ) ) ) ) ) ) )
           => ! [V6a0: $i] :
                ( ( mem @ V6a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
               => ! [V7a1: $i] :
                    ( ( mem @ V7a1 @ A_27c )
                   => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V6a0 ) @ V7a1 ) )
                     => ( p @ ( ap @ ( ap @ V1relrec_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Erelrec__strongind,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
     => ! [V1relrec_27: $i] :
          ( ( mem @ V1relrec_27 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
         => ( ! [V2i: $i] :
                ( ( mem @ V2i @ A_27a )
               => ! [V3fm: $i] :
                    ( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
                   => ! [V4rfm: $i] :
                        ( ( mem @ V4rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
                       => ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V4rfm )
                              = ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) )
                            & ! [V5d: $i] :
                                ( ( mem @ V5d @ A_27b )
                               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) ) )
                                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) )
                                    & ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) ) ) ) ) )
                         => ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V4rfm ) @ V3fm ) ) ) ) ) ) )
           => ! [V6a0: $i] :
                ( ( mem @ V6a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
               => ! [V7a1: $i] :
                    ( ( mem @ V7a1 @ A_27c )
                   => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V6a0 ) @ V7a1 ) )
                     => ( p @ ( ap @ ( ap @ V1relrec_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Erelrec__cases,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
     => ! [V1a0: $i] :
          ( ( mem @ V1a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
         => ! [V2a1: $i] :
              ( ( mem @ V2a1 @ A_27c )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V1a0 ) @ V2a1 ) )
              <=> ? [V3i: $i] :
                    ( ( mem @ V3i @ A_27a )
                    & ? [V4fm: $i] :
                        ( ( mem @ V4fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
                        & ? [V5rfm: $i] :
                            ( ( mem @ V5rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
                            & ( V1a0
                              = ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V3i ) @ V4fm ) )
                            & ( V2a1
                              = ( ap @ ( ap @ ( ap @ V0h @ V3i ) @ V5rfm ) @ V4fm ) )
                            & ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V5rfm )
                              = ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) )
                            & ! [V6d: $i] :
                                ( ( mem @ V6d @ A_27b )
                               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) ) )
                                 => ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) @ V6d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V5rfm ) @ V6d ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Efmaptree_2Efmtreerec__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) )
     => ! [V1ft: $i] :
          ( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) )
         => ( ( ap @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V1ft )
            = ( ap @ ( c_2Emin_2E_40 @ A_27a )
              @ ( lam @ A_27a
                @ ^ [V2r: $i] : ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27b @ A_27c @ A_27a ) @ V0h ) @ V1ft ) @ V2r ) ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Efmtreerec__thm,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) )
     => ! [V1i: $i] :
          ( ( mem @ V1i @ A_27b )
         => ! [V2fm: $i] :
              ( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) )
             => ( ( ap @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27c @ A_27b ) @ V1i ) @ V2fm ) )
                = ( ap @ ( ap @ ( ap @ V0h @ V1i ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) @ A_27a ) @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) ) @ V2fm ) ) @ V2fm ) ) ) ) ) ).

thf(conj_thm_2Efmaptree_2Efmtree__Axiom,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
      ( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ A_27c ) ) ) )
     => ? [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ A_27c ) )
          & ! [V2i: $i] :
              ( ( mem @ V2i @ A_27a )
             => ! [V3fm: $i] :
                  ( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
                 => ( ( ap @ V1f @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) )
                    = ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V3fm ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ A_27c ) @ V1f ) @ V3fm ) ) ) ) ) ) ) ).

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