ITP001 Axioms: ITP048^7.ax
%------------------------------------------------------------------------------
% File : ITP048^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : inftree.ax [Gau19]
% : HL4048^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 58 ( 11 unt; 29 typ; 0 def)
% Number of atoms : 97 ( 26 equ; 2 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 942 ( 2 ~; 4 |; 19 &; 888 @)
% ( 12 <=>; 17 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 12 avg; 888 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 226 ( 226 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 28 usr; 1 con; 0-8 aty)
% Number of variables : 279 ( 35 ^ 187 !; 13 ?; 279 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Einftree_2Einftree,type,
tyop_2Einftree_2Einftree: $tType > $tType > $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2EHD,type,
c_2Elist_2EHD:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Elist_2ETL,type,
c_2Elist_2ETL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Einftree_2Efrom__inftree,type,
c_2Einftree_2Efrom__inftree:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Einftree_2EiLf,type,
c_2Einftree_2EiLf:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( A_27a > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) ).
thf(c_2Einftree_2EiNd,type,
c_2Einftree_2EiNd:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( A_27b > ( A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) ).
thf(c_2Einftree_2Einftree__CASE,type,
c_2Einftree_2Einftree__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) > ( A_27a > A_27d ) > ( A_27b > ( A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) > A_27d ) > A_27d ) ).
thf(c_2Einftree_2Einftree__rec,type,
c_2Einftree_2Einftree__rec:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27b > A_27a ) > ( A_27c > ( A_27d > A_27a ) > A_27a ) > ( tyop_2Einftree_2Einftree @ A_27b @ A_27c @ A_27d ) > A_27a ) ).
thf(c_2Einftree_2Eis__tree,type,
c_2Einftree_2Eis__tree:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Einftree_2Erelrec,type,
c_2Einftree_2Erelrec:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b ) > ( A_27c > ( A_27d > A_27b ) > A_27b ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) > A_27b > $o ) ).
thf(c_2Einftree_2Eto__inftree,type,
c_2Einftree_2Eto__inftree:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Einftree_2Eis__tree__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d )
= ( ^ [V0a0: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( c_2Ebool_2E_21 @ ( ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > $o )
@ ^ [V1is__tree_27: ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) )
@ ^ [V2a0: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V3a: A_27a] :
( c_2Emin_2E_3D @ ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) @ V2a0
@ ^ [V4p: tyop_2Elist_2Elist @ A_27d] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V3a ) ) )
@ ( c_2Ebool_2E_3F @ ( A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) )
@ ^ [V5f: A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( c_2Ebool_2E_3F @ A_27b
@ ^ [V6b: A_27b] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) @ V2a0
@ ^ [V7p: tyop_2Elist_2Elist @ A_27d] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27d ) @ V7p @ ( c_2Elist_2ENIL @ A_27d ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V6b ) @ ( V5f @ ( c_2Elist_2EHD @ A_27d @ V7p ) @ ( c_2Elist_2ETL @ A_27d @ V7p ) ) ) )
@ ( c_2Ebool_2E_21 @ A_27d
@ ^ [V8d: A_27d] : ( V1is__tree_27 @ ( V5f @ V8d ) ) ) ) ) ) )
@ ( V1is__tree_27 @ V2a0 ) ) )
@ ( V1is__tree_27 @ V0a0 ) ) ) ) ) ).
thf(thm_2Einftree_2Einftree__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
? [V0rep: ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] : ( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d ) @ ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d ) @ V0rep ) ).
thf(thm_2Einftree_2Einftree__bijections,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
( ! [V0a: tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27d] :
( ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27d @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27d @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ V1r )
<=> ( ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27d @ ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27d @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Einftree_2EiLf__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0a: A_27a] :
( ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V0a )
= ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27c
@ ^ [V1p: tyop_2Elist_2Elist @ A_27c] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V0a ) ) ) ).
thf(thm_2Einftree_2EiNd__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27b,V1f: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c )] :
( ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V0b @ V1f )
= ( c_2Einftree_2Eto__inftree @ A_27a @ A_27b @ A_27c
@ ^ [V2p: tyop_2Elist_2Elist @ A_27c] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27c ) @ V2p @ ( c_2Elist_2ENIL @ A_27c ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V0b ) @ ( c_2Einftree_2Efrom__inftree @ A_27a @ A_27b @ A_27c @ ( V1f @ ( c_2Elist_2EHD @ A_27c @ V2p ) ) @ ( c_2Elist_2ETL @ A_27c @ V2p ) ) ) ) ) ).
thf(thm_2Einftree_2Erelrec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d )
= ( ^ [V0a0: A_27a > A_27b,V1a1: A_27c > ( A_27d > A_27b ) > A_27b,V2a2: tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d,V3a3: A_27b] :
( c_2Ebool_2E_21 @ ( ( A_27a > A_27b ) > ( A_27c > ( A_27d > A_27b ) > A_27b ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) > A_27b > $o )
@ ^ [V4relrec_27: ( A_27a > A_27b ) > ( A_27c > ( A_27d > A_27b ) > A_27b ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) > A_27b > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( A_27a > A_27b )
@ ^ [V5a0: A_27a > A_27b] :
( c_2Ebool_2E_21 @ ( A_27c > ( A_27d > A_27b ) > A_27b )
@ ^ [V6a1: A_27c > ( A_27d > A_27b ) > A_27b] :
( c_2Ebool_2E_21 @ ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d )
@ ^ [V7a2: tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V8a3: A_27b] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V9a: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ V7a2 @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d @ V9a ) ) @ ( c_2Emin_2E_3D @ A_27b @ V8a3 @ ( V5a0 @ V9a ) ) ) )
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V10b: A_27c] :
( c_2Ebool_2E_3F @ ( A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) )
@ ^ [V11df: A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d )] :
( c_2Ebool_2E_3F @ ( A_27d > A_27b )
@ ^ [V12g: A_27d > A_27b] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) @ V7a2 @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d @ V10b @ V11df ) )
@ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27b @ V8a3 @ ( V6a1 @ V10b @ V12g ) )
@ ( c_2Ebool_2E_21 @ A_27d
@ ^ [V13d: A_27d] : ( V4relrec_27 @ V5a0 @ V6a1 @ ( V11df @ V13d ) @ ( V12g @ V13d ) ) ) ) ) ) ) ) )
@ ( V4relrec_27 @ V5a0 @ V6a1 @ V7a2 @ V8a3 ) ) ) ) ) )
@ ( V4relrec_27 @ V0a0 @ V1a1 @ V2a2 @ V3a3 ) ) ) ) ) ).
thf(thm_2Einftree_2Einftree__rec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0lf: A_27b > A_27a,V1nd: A_27c > ( A_27d > A_27a ) > A_27a,V2t: tyop_2Einftree_2Einftree @ A_27b @ A_27c @ A_27d] :
( ( c_2Einftree_2Einftree__rec @ A_27a @ A_27b @ A_27c @ A_27d @ V0lf @ V1nd @ V2t )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V3r: A_27a] : ( c_2Einftree_2Erelrec @ A_27b @ A_27a @ A_27c @ A_27d @ V0lf @ V1nd @ V2t @ V3r ) ) ) ).
thf(thm_2Einftree_2Einftree__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0a: A_27a,V1f: A_27a > A_27d,V2f1: A_27b > ( A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) > A_27d] :
( ( c_2Einftree_2Einftree__CASE @ A_27a @ A_27b @ A_27c @ A_27d @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V0a ) @ V1f @ V2f1 )
= ( V1f @ V0a ) )
& ! [V3b: A_27b,V4d: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ),V5f: A_27a > A_27d,V6f1: A_27b > ( A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) > A_27d] :
( ( c_2Einftree_2Einftree__CASE @ A_27a @ A_27b @ A_27c @ A_27d @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V3b @ V4d ) @ V5f @ V6f1 )
= ( V6f1 @ V3b @ V4d ) ) ) ).
thf(thm_2Einftree_2Eis__tree__rules,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
( ! [V0a: A_27a] :
( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d
@ ^ [V1p: tyop_2Elist_2Elist @ A_27d] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V0a ) )
& ! [V2f: A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ),V3b: A_27b] :
( ! [V4d: A_27d] : ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ ( V2f @ V4d ) )
=> ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d
@ ^ [V5p: tyop_2Elist_2Elist @ A_27d] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27d ) @ V5p @ ( c_2Elist_2ENIL @ A_27d ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3b ) @ ( V2f @ ( c_2Elist_2EHD @ A_27d @ V5p ) @ ( c_2Elist_2ETL @ A_27d @ V5p ) ) ) ) ) ) ).
thf(thm_2Einftree_2Eis__tree__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,V0is__tree_27: ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > $o] :
( ( ! [V1a: A_27a] :
( V0is__tree_27
@ ^ [V2p: tyop_2Elist_2Elist @ A_27d] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V1a ) )
& ! [V3f: A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ),V4b: A_27b] :
( ! [V5d: A_27d] : ( V0is__tree_27 @ ( V3f @ V5d ) )
=> ( V0is__tree_27
@ ^ [V6p: tyop_2Elist_2Elist @ A_27d] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27d ) @ V6p @ ( c_2Elist_2ENIL @ A_27d ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V4b ) @ ( V3f @ ( c_2Elist_2EHD @ A_27d @ V6p ) @ ( c_2Elist_2ETL @ A_27d @ V6p ) ) ) ) ) )
=> ! [V7a0: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ V7a0 )
=> ( V0is__tree_27 @ V7a0 ) ) ) ).
thf(thm_2Einftree_2Eis__tree__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,V0is__tree_27: ( ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > $o] :
( ( ! [V1a: A_27a] :
( V0is__tree_27
@ ^ [V2p: tyop_2Elist_2Elist @ A_27d] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V1a ) )
& ! [V3f: A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ),V4b: A_27b] :
( ! [V5d: A_27d] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ ( V3f @ V5d ) )
& ( V0is__tree_27 @ ( V3f @ V5d ) ) )
=> ( V0is__tree_27
@ ^ [V6p: tyop_2Elist_2Elist @ A_27d] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27d ) @ V6p @ ( c_2Elist_2ENIL @ A_27d ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V4b ) @ ( V3f @ ( c_2Elist_2EHD @ A_27d @ V6p ) @ ( c_2Elist_2ETL @ A_27d @ V6p ) ) ) ) ) )
=> ! [V7a0: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ V7a0 )
=> ( V0is__tree_27 @ V7a0 ) ) ) ).
thf(thm_2Einftree_2Eis__tree__cases,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,V0a0: ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b )] :
( ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ V0a0 )
<=> ( ? [V1a: A_27a] :
( V0a0
= ( ^ [V2p: tyop_2Elist_2Elist @ A_27d] : ( c_2Esum_2EINL @ A_27a @ A_27b @ V1a ) ) )
| ? [V3f: A_27d > ( tyop_2Elist_2Elist @ A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ),V4b: A_27b] :
( ( V0a0
= ( ^ [V5p: tyop_2Elist_2Elist @ A_27d] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27d ) @ V5p @ ( c_2Elist_2ENIL @ A_27d ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V4b ) @ ( V3f @ ( c_2Elist_2EHD @ A_27d @ V5p ) @ ( c_2Elist_2ETL @ A_27d @ V5p ) ) ) ) )
& ! [V6d: A_27d] : ( c_2Einftree_2Eis__tree @ A_27a @ A_27b @ A_27d @ ( V3f @ V6d ) ) ) ) ) ).
thf(thm_2Einftree_2EiNd__is__tree,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27a,V1f: A_27b > ( tyop_2Einftree_2Einftree @ A_27c @ A_27a @ A_27b )] :
( c_2Einftree_2Eis__tree @ A_27c @ A_27a @ A_27b
@ ^ [V2p: tyop_2Elist_2Elist @ A_27b] : ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27c @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27b ) @ V2p @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( c_2Esum_2EINR @ A_27c @ A_27a @ V0b ) @ ( c_2Einftree_2Efrom__inftree @ A_27c @ A_27a @ A_27b @ ( V1f @ ( c_2Elist_2EHD @ A_27b @ V2p ) ) @ ( c_2Elist_2ETL @ A_27b @ V2p ) ) ) ) ).
thf(thm_2Einftree_2Einftree__11,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ),V1f1: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ),V2b2: A_27b,V3b1: A_27b,V4a2: A_27a,V5a1: A_27a] :
( ( ( ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V5a1 )
= ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V4a2 ) )
<=> ( V5a1 = V4a2 ) )
& ( ( ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V3b1 @ V1f1 )
= ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V2b2 @ V0f2 ) )
<=> ( ( V3b1 = V2b2 )
& ( V1f1 = V0f2 ) ) ) ) ).
thf(thm_2Einftree_2Einftree__distinct,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ),V1b: A_27b,V2a: A_27a] :
( (~)
@ ( ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V2a )
= ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V1b @ V0f ) ) ) ).
thf(thm_2Einftree_2Einftree__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) > $o] :
( ( ! [V1a: A_27a] : ( V0P @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V1a ) )
& ! [V2b: A_27b,V3f: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c )] :
( ! [V4d: A_27c] : ( V0P @ ( V3f @ V4d ) )
=> ( V0P @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V2b @ V3f ) ) ) )
=> ! [V5t: tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c] : ( V0P @ V5t ) ) ).
thf(thm_2Einftree_2Erelrec__rules,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0lf: A_27a > A_27b,V1nd: A_27c > ( A_27d > A_27b ) > A_27b,V2a: A_27a] : ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V0lf @ V1nd @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d @ V2a ) @ ( V0lf @ V2a ) )
& ! [V3lf: A_27a > A_27b,V4nd: A_27c > ( A_27d > A_27b ) > A_27b,V5b: A_27c,V6df: A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ),V7g: A_27d > A_27b] :
( ! [V8d: A_27d] : ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V3lf @ V4nd @ ( V6df @ V8d ) @ ( V7g @ V8d ) )
=> ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V3lf @ V4nd @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d @ V5b @ V6df ) @ ( V4nd @ V5b @ V7g ) ) ) ) ).
thf(thm_2Einftree_2Erelrec__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0relrec_27: ( A_27a > A_27b ) > ( A_27c > ( A_27d > A_27b ) > A_27b ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) > A_27b > $o] :
( ( ! [V1lf: A_27a > A_27b,V2nd: A_27c > ( A_27d > A_27b ) > A_27b,V3a: A_27a] : ( V0relrec_27 @ V1lf @ V2nd @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d @ V3a ) @ ( V1lf @ V3a ) )
& ! [V4lf: A_27a > A_27b,V5nd: A_27c > ( A_27d > A_27b ) > A_27b,V6b: A_27c,V7df: A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ),V8g: A_27d > A_27b] :
( ! [V9d: A_27d] : ( V0relrec_27 @ V4lf @ V5nd @ ( V7df @ V9d ) @ ( V8g @ V9d ) )
=> ( V0relrec_27 @ V4lf @ V5nd @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d @ V6b @ V7df ) @ ( V5nd @ V6b @ V8g ) ) ) )
=> ! [V10a0: A_27a > A_27b,V11a1: A_27c > ( A_27d > A_27b ) > A_27b,V12a2: tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d,V13a3: A_27b] :
( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V10a0 @ V11a1 @ V12a2 @ V13a3 )
=> ( V0relrec_27 @ V10a0 @ V11a1 @ V12a2 @ V13a3 ) ) ) ).
thf(thm_2Einftree_2Erelrec__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0relrec_27: ( A_27a > A_27b ) > ( A_27c > ( A_27d > A_27b ) > A_27b ) > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ) > A_27b > $o] :
( ( ! [V1lf: A_27a > A_27b,V2nd: A_27c > ( A_27d > A_27b ) > A_27b,V3a: A_27a] : ( V0relrec_27 @ V1lf @ V2nd @ ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d @ V3a ) @ ( V1lf @ V3a ) )
& ! [V4lf: A_27a > A_27b,V5nd: A_27c > ( A_27d > A_27b ) > A_27b,V6b: A_27c,V7df: A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ),V8g: A_27d > A_27b] :
( ! [V9d: A_27d] :
( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V4lf @ V5nd @ ( V7df @ V9d ) @ ( V8g @ V9d ) )
& ( V0relrec_27 @ V4lf @ V5nd @ ( V7df @ V9d ) @ ( V8g @ V9d ) ) )
=> ( V0relrec_27 @ V4lf @ V5nd @ ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d @ V6b @ V7df ) @ ( V5nd @ V6b @ V8g ) ) ) )
=> ! [V10a0: A_27a > A_27b,V11a1: A_27c > ( A_27d > A_27b ) > A_27b,V12a2: tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d,V13a3: A_27b] :
( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V10a0 @ V11a1 @ V12a2 @ V13a3 )
=> ( V0relrec_27 @ V10a0 @ V11a1 @ V12a2 @ V13a3 ) ) ) ).
thf(thm_2Einftree_2Erelrec__cases,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0a0: A_27a > A_27b,V1a1: A_27c > ( A_27d > A_27b ) > A_27b,V2a2: tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d,V3a3: A_27b] :
( ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V0a0 @ V1a1 @ V2a2 @ V3a3 )
<=> ( ? [V4a: A_27a] :
( ( V2a2
= ( c_2Einftree_2EiLf @ A_27a @ A_27c @ A_27d @ V4a ) )
& ( V3a3
= ( V0a0 @ V4a ) ) )
| ? [V5b: A_27c,V6df: A_27d > ( tyop_2Einftree_2Einftree @ A_27a @ A_27c @ A_27d ),V7g: A_27d > A_27b] :
( ( V2a2
= ( c_2Einftree_2EiNd @ A_27a @ A_27c @ A_27d @ V5b @ V6df ) )
& ( V3a3
= ( V1a1 @ V5b @ V7g ) )
& ! [V8d: A_27d] : ( c_2Einftree_2Erelrec @ A_27a @ A_27b @ A_27c @ A_27d @ V0a0 @ V1a1 @ ( V6df @ V8d ) @ ( V7g @ V8d ) ) ) ) ) ).
thf(thm_2Einftree_2Einftree__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0lf: A_27a > A_27d,V1nd: A_27b > ( A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) ) > ( A_27c > A_27d ) > A_27d] :
? [V2f: ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) > A_27d] :
( ! [V3a: A_27a] :
( ( V2f @ ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V3a ) )
= ( V0lf @ V3a ) )
& ! [V4b: A_27b,V5d: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c )] :
( ( V2f @ ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V4b @ V5d ) )
= ( V1nd @ V4b @ V5d @ ( c_2Ecombin_2Eo @ A_27c @ A_27d @ ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c ) @ V2f @ V5d ) ) ) ) ).
thf(thm_2Einftree_2Einftree__nchotomy,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t: tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c] :
( ? [V1a: A_27a] :
( V0t
= ( c_2Einftree_2EiLf @ A_27a @ A_27b @ A_27c @ V1a ) )
| ? [V2b: A_27b,V3d: A_27c > ( tyop_2Einftree_2Einftree @ A_27a @ A_27b @ A_27c )] :
( V0t
= ( c_2Einftree_2EiNd @ A_27a @ A_27b @ A_27c @ V2b @ V3d ) ) ) ).
%------------------------------------------------------------------------------