ITP001 Axioms: ITP048^5.ax
%------------------------------------------------------------------------------
% File : ITP048^5 : TPTP v8.2.0. 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 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------