ITP001 Axioms: ITP048^5.ax


%------------------------------------------------------------------------------
% File     : ITP048^5 : 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    : inftree^2.ax [Gau20]
%          : HL4048^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   2 unt;   9 typ;   0 def)
%            Number of atoms       :  904 (  25 equ;   0 cnn)
%            Maximal formula atoms :   77 (  23 avg)
%            Number of connectives : 2090 (   1   ~;   3   |;  30   &;1937   @)
%                                         (   5 <=>; 114  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   38 (  18 avg;1937 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   30 (  30   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   34 (  33 usr;  24 con; 0-4 aty)
%            Number of variables   :  246 (  35   ^ 199   !;  12   ?; 246   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Einftree_2Einftree,type,
    ty_2Einftree_2Einftree: del > del > del > del ).

thf(tp_c_2Einftree_2Efrom__inftree,type,
    c_2Einftree_2Efrom__inftree: del > del > del > $i ).

thf(mem_c_2Einftree_2Efrom__inftree,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] : ( mem @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27d ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ) ).

thf(tp_c_2Einftree_2EiLf,type,
    c_2Einftree_2EiLf: del > del > del > $i ).

thf(mem_c_2Einftree_2EiLf,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ ( arr @ A_27a @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) ) ).

thf(tp_c_2Einftree_2EiNd,type,
    c_2Einftree_2EiNd: del > del > del > $i ).

thf(mem_c_2Einftree_2EiNd,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) ) ) ).

thf(tp_c_2Einftree_2Einftree__CASE,type,
    c_2Einftree_2Einftree__CASE: del > del > del > del > $i ).

thf(mem_c_2Einftree_2Einftree__CASE,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Einftree_2Einftree__CASE @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ A_27d ) @ ( arr @ ( arr @ A_27b @ ( arr @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ A_27d ) ) @ A_27d ) ) ) ) ).

thf(tp_c_2Einftree_2Einftree__rec,type,
    c_2Einftree_2Einftree__rec: del > del > del > del > $i ).

thf(mem_c_2Einftree_2Einftree__rec,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Einftree_2Einftree__rec @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27a ) @ A_27a ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27b @ A_27c @ A_27d ) @ A_27a ) ) ) ) ).

thf(tp_c_2Einftree_2Eis__tree,type,
    c_2Einftree_2Eis__tree: del > del > del > $i ).

thf(mem_c_2Einftree_2Eis__tree,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] : ( mem @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ bool ) ) ).

thf(tp_c_2Einftree_2Erelrec,type,
    c_2Einftree_2Erelrec: del > del > del > del > $i ).

thf(mem_c_2Einftree_2Erelrec,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ ( arr @ A_27b @ bool ) ) ) ) ) ).

thf(tp_c_2Einftree_2Eto__inftree,type,
    c_2Einftree_2Eto__inftree: del > del > del > $i ).

thf(mem_c_2Einftree_2Eto__inftree,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] : ( mem @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27d ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) ) ) ).

thf(ax_thm_2Einftree_2Eis__tree__def,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] :
      ( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d )
      = ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
        @ ^ [V0a0: $i] :
            ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ bool ) )
            @ ( lam @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ bool )
              @ ^ [V1is__tree_27: $i] :
                  ( ap
                  @ ( ap @ c_2Emin_2E_3D_3D_3E
                    @ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
                      @ ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
                        @ ^ [V2a0: $i] :
                            ( ap
                            @ ( ap @ c_2Emin_2E_3D_3D_3E
                              @ ( ap
                                @ ( ap @ c_2Ebool_2E_5C_2F
                                  @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                    @ ( lam @ A_27a
                                      @ ^ [V3a: $i] :
                                          ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) @ V2a0 )
                                          @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                                            @ ^ [V4p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V3a ) ) ) ) ) )
                                @ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
                                  @ ( lam @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
                                    @ ^ [V5f: $i] :
                                        ( ap @ ( c_2Ebool_2E_3F @ A_27b )
                                        @ ( lam @ A_27b
                                          @ ^ [V6b: $i] :
                                              ( ap
                                              @ ( ap @ c_2Ebool_2E_2F_5C
                                                @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) @ V2a0 )
                                                  @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                                                    @ ^ [V7p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27d ) ) @ V7p ) @ ( c_2Elist_2ENIL @ A_27d ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V6b ) ) @ ( ap @ ( ap @ V5f @ ( ap @ ( c_2Elist_2EHD @ A_27d ) @ V7p ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27d ) @ V7p ) ) ) ) ) )
                                              @ ( ap @ ( c_2Ebool_2E_21 @ A_27d )
                                                @ ( lam @ A_27d
                                                  @ ^ [V8d: $i] : ( ap @ V1is__tree_27 @ ( ap @ V5f @ V8d ) ) ) ) ) ) ) ) ) ) )
                            @ ( ap @ V1is__tree_27 @ V2a0 ) ) ) ) )
                  @ ( ap @ V1is__tree_27 @ V0a0 ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Eis__tree__rules,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ A_27a )
         => ( p
            @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d )
              @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                @ ^ [V1p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0a ) ) ) ) )
      & ! [V2f: $i] :
          ( ( mem @ V2f @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
         => ! [V3b: $i] :
              ( ( mem @ V3b @ A_27b )
             => ( ! [V4d: $i] :
                    ( ( mem @ V4d @ A_27d )
                   => ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ ( ap @ V2f @ V4d ) ) ) )
               => ( p
                  @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d )
                    @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                      @ ^ [V5p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27d ) ) @ V5p ) @ ( c_2Elist_2ENIL @ A_27d ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3b ) ) @ ( ap @ ( ap @ V2f @ ( ap @ ( c_2Elist_2EHD @ A_27d ) @ V5p ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27d ) @ V5p ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Eis__tree__ind,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del,V0is__tree_27: $i] :
      ( ( mem @ V0is__tree_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ bool ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ A_27a )
             => ( p
                @ ( ap @ V0is__tree_27
                  @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                    @ ^ [V2p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1a ) ) ) ) )
          & ! [V3f: $i] :
              ( ( mem @ V3f @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
             => ! [V4b: $i] :
                  ( ( mem @ V4b @ A_27b )
                 => ( ! [V5d: $i] :
                        ( ( mem @ V5d @ A_27d )
                       => ( p @ ( ap @ V0is__tree_27 @ ( ap @ V3f @ V5d ) ) ) )
                   => ( p
                      @ ( ap @ V0is__tree_27
                        @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                          @ ^ [V6p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27d ) ) @ V6p ) @ ( c_2Elist_2ENIL @ A_27d ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4b ) ) @ ( ap @ ( ap @ V3f @ ( ap @ ( c_2Elist_2EHD @ A_27d ) @ V6p ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27d ) @ V6p ) ) ) ) ) ) ) ) ) )
       => ! [V7a0: $i] :
            ( ( mem @ V7a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
           => ( ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ V7a0 ) )
             => ( p @ ( ap @ V0is__tree_27 @ V7a0 ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Eis__tree__strongind,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del,V0is__tree_27: $i] :
      ( ( mem @ V0is__tree_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ bool ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ A_27a )
             => ( p
                @ ( ap @ V0is__tree_27
                  @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                    @ ^ [V2p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1a ) ) ) ) )
          & ! [V3f: $i] :
              ( ( mem @ V3f @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
             => ! [V4b: $i] :
                  ( ( mem @ V4b @ A_27b )
                 => ( ! [V5d: $i] :
                        ( ( mem @ V5d @ A_27d )
                       => ( ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ ( ap @ V3f @ V5d ) ) )
                          & ( p @ ( ap @ V0is__tree_27 @ ( ap @ V3f @ V5d ) ) ) ) )
                   => ( p
                      @ ( ap @ V0is__tree_27
                        @ ( lam @ ( ty_2Elist_2Elist @ A_27d )
                          @ ^ [V6p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27d ) ) @ V6p ) @ ( c_2Elist_2ENIL @ A_27d ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4b ) ) @ ( ap @ ( ap @ V3f @ ( ap @ ( c_2Elist_2EHD @ A_27d ) @ V6p ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27d ) @ V6p ) ) ) ) ) ) ) ) ) )
       => ! [V7a0: $i] :
            ( ( mem @ V7a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
           => ( ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ V7a0 ) )
             => ( p @ ( ap @ V0is__tree_27 @ V7a0 ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Eis__tree__cases,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del,V0a0: $i] :
      ( ( mem @ V0a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
     => ( ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ V0a0 ) )
      <=> ( ? [V1a: $i] :
              ( ( mem @ V1a @ A_27a )
              & ( V0a0
                = ( lam @ ( ty_2Elist_2Elist @ A_27d )
                  @ ^ [V2p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1a ) ) ) )
          | ? [V3f: $i] :
              ( ( mem @ V3f @ ( arr @ A_27d @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
              & ? [V4b: $i] :
                  ( ( mem @ V4b @ A_27b )
                  & ( V0a0
                    = ( lam @ ( ty_2Elist_2Elist @ A_27d )
                      @ ^ [V5p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27d ) ) @ V5p ) @ ( c_2Elist_2ENIL @ A_27d ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4b ) ) @ ( ap @ ( ap @ V3f @ ( ap @ ( c_2Elist_2EHD @ A_27d ) @ V5p ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27d ) @ V5p ) ) ) ) )
                  & ! [V6d: $i] :
                      ( ( mem @ V6d @ A_27d )
                     => ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ ( ap @ V3f @ V6d ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Einftree_2Einftree__TY__DEF,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] :
    ? [V0rep: $i] :
      ( ( mem @ V0rep @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) ) @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) ) @ V0rep ) ) ) ).

thf(ax_thm_2Einftree_2Einftree__bijections,axiom,
    ! [A_27a: del,A_27b: del,A_27d: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) )
         => ( ( ap @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27d ) @ ( ap @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27d ) @ V0a ) )
            = V0a ) )
      & ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Elist_2Elist @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) )
         => ( ( p @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ V1r ) )
          <=> ( ( ap @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27d ) @ ( ap @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27d ) @ V1r ) )
              = V1r ) ) ) ) ).

thf(ax_thm_2Einftree_2EiLf__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
      ( ( mem @ V0a @ A_27a )
     => ( ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V0a )
        = ( ap @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27c )
          @ ( lam @ ( ty_2Elist_2Elist @ A_27c )
            @ ^ [V1p: $i] : ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0a ) ) ) ) ) ).

thf(ax_thm_2Einftree_2EiNd__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0b: $i] :
      ( ( mem @ V0b @ A_27b )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
         => ( ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V0b ) @ V1f )
            = ( ap @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27c )
              @ ( lam @ ( ty_2Elist_2Elist @ A_27c )
                @ ^ [V2p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27c ) ) @ V2p ) @ ( c_2Elist_2ENIL @ A_27c ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V0b ) ) @ ( ap @ ( ap @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27c ) @ ( ap @ V1f @ ( ap @ ( c_2Elist_2EHD @ A_27c ) @ V2p ) ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27c ) @ V2p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2EiNd__is__tree,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0b: $i] :
      ( ( mem @ V0b @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27b @ ( ty_2Einftree_2Einftree @ A_27c @ A_27a @ A_27b ) ) )
         => ( p
            @ ( ap @ ( c_2Einftree_2Eis__tree @ A_27c @ A_27a @ A_27b )
              @ ( lam @ ( ty_2Elist_2Elist @ A_27b )
                @ ^ [V2p: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27c @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27b ) ) @ V2p ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27c @ A_27a ) @ V0b ) ) @ ( ap @ ( ap @ ( c_2Einftree_2Efrom__inftree @ A_27c @ A_27a @ A_27b ) @ ( ap @ V1f @ ( ap @ ( c_2Elist_2EHD @ A_27b ) @ V2p ) ) ) @ ( ap @ ( c_2Elist_2ETL @ A_27b ) @ V2p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Einftree__11,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0a1: $i] :
      ( ( mem @ V0a1 @ A_27a )
     => ! [V1a2: $i] :
          ( ( mem @ V1a2 @ A_27a )
         => ! [V2b1: $i] :
              ( ( mem @ V2b1 @ A_27b )
             => ! [V3f1: $i] :
                  ( ( mem @ V3f1 @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
                 => ! [V4b2: $i] :
                      ( ( mem @ V4b2 @ A_27b )
                     => ! [V5f2: $i] :
                          ( ( mem @ V5f2 @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
                         => ( ( ( ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V0a1 )
                                = ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V1a2 ) )
                            <=> ( V0a1 = V1a2 ) )
                            & ( ( ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V2b1 ) @ V3f1 )
                                = ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V4b2 ) @ V5f2 ) )
                            <=> ( ( V2b1 = V4b2 )
                                & ( V3f1 = V5f2 ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Einftree__distinct,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
      ( ( mem @ V0a @ A_27a )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ A_27b )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
             => ( ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V0a )
               != ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V1b ) @ V2f ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Einftree__ind,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) @ bool ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ A_27a )
             => ( p @ ( ap @ V0P @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V1a ) ) ) )
          & ! [V2b: $i] :
              ( ( mem @ V2b @ A_27b )
             => ! [V3f: $i] :
                  ( ( mem @ V3f @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
                 => ( ! [V4d: $i] :
                        ( ( mem @ V4d @ A_27c )
                       => ( p @ ( ap @ V0P @ ( ap @ V3f @ V4d ) ) ) )
                   => ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V2b ) @ V3f ) ) ) ) ) ) )
       => ! [V5t: $i] :
            ( ( mem @ V5t @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) )
           => ( p @ ( ap @ V0P @ V5t ) ) ) ) ) ).

thf(ax_thm_2Einftree_2Erelrec__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
      ( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d )
      = ( lam @ ( arr @ A_27a @ A_27b )
        @ ^ [V0a0: $i] :
            ( lam @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) )
            @ ^ [V1a1: $i] :
                ( lam @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d )
                @ ^ [V2a2: $i] :
                    ( lam @ A_27b
                    @ ^ [V3a3: $i] :
                        ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ ( arr @ A_27b @ bool ) ) ) ) )
                        @ ( lam @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ ( arr @ A_27b @ bool ) ) ) )
                          @ ^ [V4relrec_27: $i] :
                              ( ap
                              @ ( ap @ c_2Emin_2E_3D_3D_3E
                                @ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ A_27b ) )
                                  @ ( lam @ ( arr @ A_27a @ A_27b )
                                    @ ^ [V5a0: $i] :
                                        ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
                                        @ ( lam @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) )
                                          @ ^ [V6a1: $i] :
                                              ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
                                              @ ( lam @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d )
                                                @ ^ [V7a2: $i] :
                                                    ( ap @ ( c_2Ebool_2E_21 @ A_27b )
                                                    @ ( lam @ A_27b
                                                      @ ^ [V8a3: $i] :
                                                          ( ap
                                                          @ ( ap @ c_2Emin_2E_3D_3D_3E
                                                            @ ( ap
                                                              @ ( ap @ c_2Ebool_2E_5C_2F
                                                                @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                                                  @ ( lam @ A_27a
                                                                    @ ^ [V9a: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) @ V7a2 ) @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d ) @ V9a ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V8a3 ) @ ( ap @ V5a0 @ V9a ) ) ) ) ) )
                                                              @ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
                                                                @ ( lam @ A_27c
                                                                  @ ^ [V10b: $i] :
                                                                      ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) )
                                                                      @ ( lam @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
                                                                        @ ^ [V11df: $i] :
                                                                            ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27d @ A_27b ) )
                                                                            @ ( lam @ ( arr @ A_27d @ A_27b )
                                                                              @ ^ [V12g: $i] :
                                                                                  ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) @ V7a2 ) @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d ) @ V10b ) @ V11df ) ) )
                                                                                  @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V8a3 ) @ ( ap @ ( ap @ V6a1 @ V10b ) @ V12g ) ) )
                                                                                    @ ( ap @ ( c_2Ebool_2E_21 @ A_27d )
                                                                                      @ ( lam @ A_27d
                                                                                        @ ^ [V13d: $i] : ( ap @ ( ap @ ( ap @ ( ap @ V4relrec_27 @ V5a0 ) @ V6a1 ) @ ( ap @ V11df @ V13d ) ) @ ( ap @ V12g @ V13d ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                                          @ ( ap @ ( ap @ ( ap @ ( ap @ V4relrec_27 @ V5a0 ) @ V6a1 ) @ V7a2 ) @ V8a3 ) ) ) ) ) ) ) ) ) ) )
                              @ ( ap @ ( ap @ ( ap @ ( ap @ V4relrec_27 @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Erelrec__rules,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
      ( ! [V0lf: $i] :
          ( ( mem @ V0lf @ ( arr @ A_27a @ A_27b ) )
         => ! [V1nd: $i] :
              ( ( mem @ V1nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
             => ! [V2a: $i] :
                  ( ( mem @ V2a @ A_27a )
                 => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0lf ) @ V1nd ) @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d ) @ V2a ) ) @ ( ap @ V0lf @ V2a ) ) ) ) ) )
      & ! [V3lf: $i] :
          ( ( mem @ V3lf @ ( arr @ A_27a @ A_27b ) )
         => ! [V4nd: $i] :
              ( ( mem @ V4nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
             => ! [V5b: $i] :
                  ( ( mem @ V5b @ A_27c )
                 => ! [V6df: $i] :
                      ( ( mem @ V6df @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) )
                     => ! [V7g: $i] :
                          ( ( mem @ V7g @ ( arr @ A_27d @ A_27b ) )
                         => ( ! [V8d: $i] :
                                ( ( mem @ V8d @ A_27d )
                               => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V3lf ) @ V4nd ) @ ( ap @ V6df @ V8d ) ) @ ( ap @ V7g @ V8d ) ) ) )
                           => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V3lf ) @ V4nd ) @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d ) @ V5b ) @ V6df ) ) @ ( ap @ ( ap @ V4nd @ V5b ) @ V7g ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Erelrec__ind,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0relrec_27: $i] :
      ( ( mem @ V0relrec_27 @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ ( arr @ A_27b @ bool ) ) ) ) )
     => ( ( ! [V1lf: $i] :
              ( ( mem @ V1lf @ ( arr @ A_27a @ A_27b ) )
             => ! [V2nd: $i] :
                  ( ( mem @ V2nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
                 => ! [V3a: $i] :
                      ( ( mem @ V3a @ A_27a )
                     => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V1lf ) @ V2nd ) @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d ) @ V3a ) ) @ ( ap @ V1lf @ V3a ) ) ) ) ) )
          & ! [V4lf: $i] :
              ( ( mem @ V4lf @ ( arr @ A_27a @ A_27b ) )
             => ! [V5nd: $i] :
                  ( ( mem @ V5nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
                 => ! [V6b: $i] :
                      ( ( mem @ V6b @ A_27c )
                     => ! [V7df: $i] :
                          ( ( mem @ V7df @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) )
                         => ! [V8g: $i] :
                              ( ( mem @ V8g @ ( arr @ A_27d @ A_27b ) )
                             => ( ! [V9d: $i] :
                                    ( ( mem @ V9d @ A_27d )
                                   => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V4lf ) @ V5nd ) @ ( ap @ V7df @ V9d ) ) @ ( ap @ V8g @ V9d ) ) ) )
                               => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V4lf ) @ V5nd ) @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d ) @ V6b ) @ V7df ) ) @ ( ap @ ( ap @ V5nd @ V6b ) @ V8g ) ) ) ) ) ) ) ) ) )
       => ! [V10a0: $i] :
            ( ( mem @ V10a0 @ ( arr @ A_27a @ A_27b ) )
           => ! [V11a1: $i] :
                ( ( mem @ V11a1 @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
               => ! [V12a2: $i] :
                    ( ( mem @ V12a2 @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
                   => ! [V13a3: $i] :
                        ( ( mem @ V13a3 @ A_27b )
                       => ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V10a0 ) @ V11a1 ) @ V12a2 ) @ V13a3 ) )
                         => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V10a0 ) @ V11a1 ) @ V12a2 ) @ V13a3 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Erelrec__strongind,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0relrec_27: $i] :
      ( ( mem @ V0relrec_27 @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ ( arr @ A_27b @ bool ) ) ) ) )
     => ( ( ! [V1lf: $i] :
              ( ( mem @ V1lf @ ( arr @ A_27a @ A_27b ) )
             => ! [V2nd: $i] :
                  ( ( mem @ V2nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
                 => ! [V3a: $i] :
                      ( ( mem @ V3a @ A_27a )
                     => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V1lf ) @ V2nd ) @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d ) @ V3a ) ) @ ( ap @ V1lf @ V3a ) ) ) ) ) )
          & ! [V4lf: $i] :
              ( ( mem @ V4lf @ ( arr @ A_27a @ A_27b ) )
             => ! [V5nd: $i] :
                  ( ( mem @ V5nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
                 => ! [V6b: $i] :
                      ( ( mem @ V6b @ A_27c )
                     => ! [V7df: $i] :
                          ( ( mem @ V7df @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) )
                         => ! [V8g: $i] :
                              ( ( mem @ V8g @ ( arr @ A_27d @ A_27b ) )
                             => ( ! [V9d: $i] :
                                    ( ( mem @ V9d @ A_27d )
                                   => ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V4lf ) @ V5nd ) @ ( ap @ V7df @ V9d ) ) @ ( ap @ V8g @ V9d ) ) )
                                      & ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V4lf ) @ V5nd ) @ ( ap @ V7df @ V9d ) ) @ ( ap @ V8g @ V9d ) ) ) ) )
                               => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V4lf ) @ V5nd ) @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d ) @ V6b ) @ V7df ) ) @ ( ap @ ( ap @ V5nd @ V6b ) @ V8g ) ) ) ) ) ) ) ) ) )
       => ! [V10a0: $i] :
            ( ( mem @ V10a0 @ ( arr @ A_27a @ A_27b ) )
           => ! [V11a1: $i] :
                ( ( mem @ V11a1 @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
               => ! [V12a2: $i] :
                    ( ( mem @ V12a2 @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
                   => ! [V13a3: $i] :
                        ( ( mem @ V13a3 @ A_27b )
                       => ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V10a0 ) @ V11a1 ) @ V12a2 ) @ V13a3 ) )
                         => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0relrec_27 @ V10a0 ) @ V11a1 ) @ V12a2 ) @ V13a3 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Erelrec__cases,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0a0: $i] :
      ( ( mem @ V0a0 @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a1: $i] :
          ( ( mem @ V1a1 @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27b ) @ A_27b ) ) )
         => ! [V2a2: $i] :
              ( ( mem @ V2a2 @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
             => ! [V3a3: $i] :
                  ( ( mem @ V3a3 @ A_27b )
                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) )
                  <=> ( ? [V4a: $i] :
                          ( ( mem @ V4a @ A_27a )
                          & ( V2a2
                            = ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d ) @ V4a ) )
                          & ( V3a3
                            = ( ap @ V0a0 @ V4a ) ) )
                      | ? [V5b: $i] :
                          ( ( mem @ V5b @ A_27c )
                          & ? [V6df: $i] :
                              ( ( mem @ V6df @ ( arr @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) ) )
                              & ? [V7g: $i] :
                                  ( ( mem @ V7g @ ( arr @ A_27d @ A_27b ) )
                                  & ( V2a2
                                    = ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d ) @ V5b ) @ V6df ) )
                                  & ( V3a3
                                    = ( ap @ ( ap @ V1a1 @ V5b ) @ V7g ) )
                                  & ! [V8d: $i] :
                                      ( ( mem @ V8d @ A_27d )
                                     => ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0a0 ) @ V1a1 ) @ ( ap @ V6df @ V8d ) ) @ ( ap @ V7g @ V8d ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Einftree_2Einftree__rec__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0lf: $i] :
      ( ( mem @ V0lf @ ( arr @ A_27b @ A_27a ) )
     => ! [V1nd: $i] :
          ( ( mem @ V1nd @ ( arr @ A_27c @ ( arr @ ( arr @ A_27d @ A_27a ) @ A_27a ) ) )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( ty_2Einftree_2Einftree @ A_27b @ A_27c @ A_27d ) )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Einftree__rec @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0lf ) @ V1nd ) @ V2t )
                = ( ap @ ( c_2Emin_2E_40 @ A_27a )
                  @ ( lam @ A_27a
                    @ ^ [V3r: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Erelrec @ A_27b @ A_27a @ A_27c @ A_27d ) @ V0lf ) @ V1nd ) @ V2t ) @ V3r ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Einftree__Axiom,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0lf: $i] :
      ( ( mem @ V0lf @ ( arr @ A_27a @ A_27d ) )
     => ! [V1nd: $i] :
          ( ( mem @ V1nd @ ( arr @ A_27b @ ( arr @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ ( arr @ ( arr @ A_27c @ A_27d ) @ A_27d ) ) ) )
         => ? [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) @ A_27d ) )
              & ! [V3a: $i] :
                  ( ( mem @ V3a @ A_27a )
                 => ( ( ap @ V2f @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V3a ) )
                    = ( ap @ V0lf @ V3a ) ) )
              & ! [V4b: $i] :
                  ( ( mem @ V4b @ A_27b )
                 => ! [V5d: $i] :
                      ( ( mem @ V5d @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
                     => ( ( ap @ V2f @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V4b ) @ V5d ) )
                        = ( ap @ ( ap @ ( ap @ V1nd @ V4b ) @ V5d ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27d @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ V2f ) @ V5d ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Einftree_2Einftree__case__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ A_27a )
         => ! [V1f: $i] :
              ( ( mem @ V1f @ ( arr @ A_27a @ A_27d ) )
             => ! [V2f1: $i] :
                  ( ( mem @ V2f1 @ ( arr @ A_27b @ ( arr @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ A_27d ) ) )
                 => ( ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Einftree__CASE @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V0a ) ) @ V1f ) @ V2f1 )
                    = ( ap @ V1f @ V0a ) ) ) ) )
      & ! [V3b: $i] :
          ( ( mem @ V3b @ A_27b )
         => ! [V4d: $i] :
              ( ( mem @ V4d @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
             => ! [V5f: $i] :
                  ( ( mem @ V5f @ ( arr @ A_27a @ A_27d ) )
                 => ! [V6f1: $i] :
                      ( ( mem @ V6f1 @ ( arr @ A_27b @ ( arr @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) @ A_27d ) ) )
                     => ( ( ap @ ( ap @ ( ap @ ( c_2Einftree_2Einftree__CASE @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V3b ) @ V4d ) ) @ V5f ) @ V6f1 )
                        = ( ap @ ( ap @ V6f1 @ V3b ) @ V4d ) ) ) ) ) ) ) ).

thf(conj_thm_2Einftree_2Einftree__nchotomy,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0t: $i] :
      ( ( mem @ V0t @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) )
     => ( ? [V1a: $i] :
            ( ( mem @ V1a @ A_27a )
            & ( V0t
              = ( ap @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c ) @ V1a ) ) )
        | ? [V2b: $i] :
            ( ( mem @ V2b @ A_27b )
            & ? [V3d: $i] :
                ( ( mem @ V3d @ ( arr @ A_27c @ ( ty_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) )
                & ( V0t
                  = ( ap @ ( ap @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c ) @ V2b ) @ V3d ) ) ) ) ) ) ).

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