ITP001 Axioms: ITP066^5.ax
%------------------------------------------------------------------------------
% File : ITP066^5 : 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 : lbtree^2.ax [Gau20]
% : HL4066^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 89 ( 6 unt; 18 typ; 0 def)
% Number of atoms : 1906 ( 75 equ; 0 cnn)
% Maximal formula atoms : 78 ( 21 avg)
% Number of connectives : 3778 ( 1 ~; 12 |; 119 &;3391 @)
% ( 20 <=>; 235 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 15 avg;3391 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 21 ( 21 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 58 con; 0-2 aty)
% Number of variables : 418 ( 64 ^ 302 !; 52 ?; 418 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Elbtree_2Elbtree,type,
ty_2Elbtree_2Elbtree: del > del ).
thf(tp_c_2Elbtree_2ELf,type,
c_2Elbtree_2ELf: del > $i ).
thf(mem_c_2Elbtree_2ELf,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2ELf @ A_27a ) @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) ).
thf(tp_c_2Elbtree_2ELfrep,type,
c_2Elbtree_2ELfrep: del > del > $i ).
thf(mem_c_2Elbtree_2ELfrep,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elbtree_2ELfrep @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ).
thf(tp_c_2Elbtree_2ENd,type,
c_2Elbtree_2ENd: del > $i ).
thf(mem_c_2Elbtree_2ENd,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2ENd @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) ) ) ) ).
thf(tp_c_2Elbtree_2ENdrep,type,
c_2Elbtree_2ENdrep: del > $i ).
thf(mem_c_2Elbtree_2ENdrep,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2ENdrep @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Elbtree_2Ebf__flatten,type,
c_2Elbtree_2Ebf__flatten: del > $i ).
thf(mem_c_2Elbtree_2Ebf__flatten,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ty_2Ellist_2Ellist @ A_27a ) ) ) ).
thf(tp_c_2Elbtree_2Edepth,type,
c_2Elbtree_2Edepth: del > $i ).
thf(mem_c_2Elbtree_2Edepth,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Edepth @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ).
thf(tp_c_2Elbtree_2Efinite,type,
c_2Elbtree_2Efinite: del > $i ).
thf(mem_c_2Elbtree_2Efinite,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Efinite @ A_27a ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) ).
thf(tp_c_2Elbtree_2Eis__lbtree,type,
c_2Elbtree_2Eis__lbtree: del > $i ).
thf(mem_c_2Elbtree_2Eis__lbtree,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Eis__lbtree @ A_27a ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) ) ).
thf(tp_c_2Elbtree_2Eis__mmindex,type,
c_2Elbtree_2Eis__mmindex: del > $i ).
thf(mem_c_2Elbtree_2Eis__mmindex,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Eis__mmindex @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ).
thf(tp_c_2Elbtree_2Elbtree__abs,type,
c_2Elbtree_2Elbtree__abs: del > $i ).
thf(mem_c_2Elbtree_2Elbtree__abs,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Elbtree__abs @ A_27a ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) ) ).
thf(tp_c_2Elbtree_2Elbtree__case,type,
c_2Elbtree_2Elbtree__case: del > del > $i ).
thf(mem_c_2Elbtree_2Elbtree__case,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elbtree_2Elbtree__case @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27b @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27a ) ) ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27a ) ) ) ) ).
thf(tp_c_2Elbtree_2Elbtree__rep,type,
c_2Elbtree_2Elbtree__rep: del > $i ).
thf(mem_c_2Elbtree_2Elbtree__rep,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Elbtree__rep @ A_27a ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Elbtree_2Emap,type,
c_2Elbtree_2Emap: del > del > $i ).
thf(mem_c_2Elbtree_2Emap,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elbtree_2Emap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) ) ) ).
thf(tp_c_2Elbtree_2Emem,type,
c_2Elbtree_2Emem: del > $i ).
thf(mem_c_2Elbtree_2Emem,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Emem @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Elbtree_2Emindepth,type,
c_2Elbtree_2Emindepth: del > $i ).
thf(mem_c_2Elbtree_2Emindepth,axiom,
! [A_27a: del] : ( mem @ ( c_2Elbtree_2Emindepth @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Elbtree_2Eoptmin,type,
c_2Elbtree_2Eoptmin: $i ).
thf(mem_c_2Elbtree_2Eoptmin,axiom,
mem @ c_2Elbtree_2Eoptmin @ ( arr @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Elbtree_2Epath__follow,type,
c_2Elbtree_2Epath__follow: del > del > $i ).
thf(mem_c_2Elbtree_2Epath__follow,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elbtree_2Epath__follow @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) ) ) @ ( arr @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2ELfrep__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Elbtree_2ELfrep @ A_27a @ A_27b )
= ( lam @ A_27a
@ ^ [V0l: $i] : ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(ax_thm_2Elbtree_2ENdrep__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENdrep @ A_27a ) @ V0a ) @ V1t1 ) @ V2t2 )
= ( lam @ ( ty_2Elist_2Elist @ bool )
@ ^ [V3l: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ bool @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V3l ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0a ) )
@ ( lam @ bool
@ ^ [V4v: $i] :
( lam @ ( ty_2Elist_2Elist @ bool )
@ ^ [V5xs: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V4v ) @ ( ap @ V1t1 @ V5xs ) ) @ ( ap @ V2t2 @ V5xs ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Eis__lbtree__def,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Eis__lbtree @ A_27a ) @ V0t ) )
<=> ? [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ V1P @ V2t ) )
=> ( ( V2t
= ( c_2Elbtree_2ELfrep @ ( ty_2Elist_2Elist @ bool ) @ A_27a ) )
| ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4t1: $i] :
( ( mem @ V4t1 @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
& ? [V5t2: $i] :
( ( mem @ V5t2 @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
& ( p @ ( ap @ V1P @ V4t1 ) )
& ( p @ ( ap @ V1P @ V5t2 ) )
& ( V2t
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENdrep @ A_27a ) @ V3a ) @ V4t1 ) @ V5t2 ) ) ) ) ) ) ) )
& ( p @ ( ap @ V1P @ V0t ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Elbtree__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( c_2Elbtree_2Eis__lbtree @ A_27a ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Elbtree_2Elbtree__absrep,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ap @ ( c_2Elbtree_2Elbtree__abs @ A_27a ) @ ( ap @ ( c_2Elbtree_2Elbtree__rep @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Elist_2Elist @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Eis__lbtree @ A_27a ) @ V1r ) )
<=> ( ( ap @ ( c_2Elbtree_2Elbtree__rep @ A_27a ) @ ( ap @ ( c_2Elbtree_2Elbtree__abs @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Elbtree_2Epath__follow__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Epath__follow @ A_27a @ A_27b ) @ V0g ) @ V1x ) @ ( c_2Elist_2ENIL @ bool ) )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) ) @ ( ap @ V0g @ V1x ) ) ) ) )
& ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ! [V4h: $i] :
( ( mem @ V4h @ bool )
=> ! [V5t: tp__c_ty_2Elist_2Elist_o] :
( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Epath__follow @ A_27a @ A_27b ) @ V2g ) @ V3x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ bool ) @ V4h ) @ ( inj__c_ty_2Elist_2Elist_o @ V5t ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V2g @ V3x ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) )
@ ^ [V6v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ A_27a ) @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ V6v )
@ ( lam @ A_27a
@ ^ [V7a: $i] :
( lam @ ( ty_2Epair_2Eprod @ A_27b @ A_27b )
@ ^ [V8v2: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ A_27a ) @ A_27b @ A_27b ) @ V8v2 )
@ ( lam @ A_27b
@ ^ [V9y: $i] :
( lam @ A_27b
@ ^ [V10z: $i] : ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Epath__follow @ A_27a @ A_27b ) @ V2g ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V4h ) @ V9y ) @ V10z ) ) @ ( inj__c_ty_2Elist_2Elist_o @ V5t ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2ELf__def,axiom,
! [A_27a: del] :
( ( c_2Elbtree_2ELf @ A_27a )
= ( ap @ ( c_2Elbtree_2Elbtree__abs @ A_27a ) @ ( c_2Elbtree_2ELfrep @ ( ty_2Elist_2Elist @ bool ) @ A_27a ) ) ) ).
thf(ax_thm_2Elbtree_2ENd__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a ) @ V1t1 ) @ V2t2 )
= ( ap @ ( c_2Elbtree_2Elbtree__abs @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENdrep @ A_27a ) @ V0a ) @ ( ap @ ( c_2Elbtree_2Elbtree__rep @ A_27a ) @ V1t1 ) ) @ ( ap @ ( c_2Elbtree_2Elbtree__rep @ A_27a ) @ V2t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Elbtree__cases,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( V0t
= ( c_2Elbtree_2ELf @ A_27a ) )
| ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V0t
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2ELf__NOT__Nd,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( c_2Elbtree_2ELf @ A_27a )
!= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a ) @ V1t1 ) @ V2t2 ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2ENd__11,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2u1: $i] :
( ( mem @ V2u1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3a2: $i] :
( ( mem @ V3a2 @ A_27a )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V5u2: $i] :
( ( mem @ V5u2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a1 ) @ V1t1 ) @ V2u1 )
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V3a2 ) @ V4t2 ) @ V5u2 ) )
<=> ( ( V0a1 = V3a2 )
& ( V1t1 = V4t2 )
& ( V2u1 = V5u2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Elbtree__ue__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ A_27a @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ^ [V1g: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ ( ap @ V1g @ V2x ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ ( ap @ V0f @ V2x ) ) @ ( c_2Elbtree_2ELf @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ^ [V3v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V3v )
@ ( lam @ A_27b
@ ^ [V4b: $i] :
( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V5v2: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27a @ A_27a ) @ V5v2 )
@ ( lam @ A_27a
@ ^ [V6y: $i] :
( lam @ A_27a
@ ^ [V7z: $i] : ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V4b ) @ ( ap @ V1g @ V6y ) ) @ ( ap @ V1g @ V7z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Elbtree__case__def,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27a ) ) ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Elbtree__case @ A_27a @ A_27b ) @ V0e ) @ V1f ) @ V2t )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ V2t ) @ ( c_2Elbtree_2ELf @ A_27b ) ) ) @ V0e )
@ ( ap
@ ( ap
@ ( ap @ V1f
@ ( ap @ ( c_2Emin_2E_40 @ A_27b )
@ ( lam @ A_27b
@ ^ [V3a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V4t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V5t2: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ V2t ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V3a ) @ V4t1 ) @ V5t2 ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V6t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V7a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V8t2: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ V2t ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V7a ) @ V6t1 ) @ V8t2 ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V9t2: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V10a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27b )
@ ^ [V11t1: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27b ) ) @ V2t ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V10a ) @ V11t1 ) @ V9t2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Elbtree__case__thm,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27b ) @ A_27a ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Elbtree__case @ A_27a @ A_27b ) @ V0e ) @ V1f ) @ ( c_2Elbtree_2ELf @ A_27b ) )
= V0e )
& ( ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Elbtree__case @ A_27a @ A_27b ) @ V0e ) @ V1f ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V2a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( ap @ V1f @ V2a ) @ V3t1 ) @ V4t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Elbtree__bisimulation,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( V0t = V1u )
<=> ? [V2R: $i] :
( ( mem @ V2R @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) )
& ( p @ ( ap @ ( ap @ V2R @ V0t ) @ V1u ) )
& ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V2R @ V3t ) @ V4u ) )
=> ( ( ( V3t
= ( c_2Elbtree_2ELf @ A_27a ) )
& ( V4u
= ( c_2Elbtree_2ELf @ A_27a ) ) )
| ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ? [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V7u1: $i] :
( ( mem @ V7u1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V9u2: $i] :
( ( mem @ V9u2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( p @ ( ap @ ( ap @ V2R @ V6t1 ) @ V7u1 ) )
& ( p @ ( ap @ ( ap @ V2R @ V8t2 ) @ V9u2 ) )
& ( V3t
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5a ) @ V6t1 ) @ V8t2 ) )
& ( V4u
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5a ) @ V7u1 ) @ V9u2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Elbtree__strong__bisimulation,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( V0t = V1u )
<=> ? [V2R: $i] :
( ( mem @ V2R @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) )
& ( p @ ( ap @ ( ap @ V2R @ V0t ) @ V1u ) )
& ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V2R @ V3t ) @ V4u ) )
=> ( ( V3t = V4u )
| ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ? [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V7u1: $i] :
( ( mem @ V7u1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V9u2: $i] :
( ( mem @ V9u2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( p @ ( ap @ ( ap @ V2R @ V6t1 ) @ V7u1 ) )
& ( p @ ( ap @ ( ap @ V2R @ V8t2 ) @ V9u2 ) )
& ( V3t
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5a ) @ V6t1 ) @ V8t2 ) )
& ( V4u
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5a ) @ V7u1 ) @ V9u2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Emem__def,axiom,
! [A_27a: del] :
( ( c_2Elbtree_2Emem @ A_27a )
= ( lam @ A_27a
@ ^ [V0a0: $i] :
( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V1a1: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) )
@ ^ [V2mem_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V4a1: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V5t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V6t2: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V4a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V3a0 ) @ V5t1 ) @ V6t2 ) ) ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7b: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V8t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V9t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V4a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V7b ) @ V8t1 ) @ V9t2 ) ) ) @ ( ap @ ( ap @ V2mem_27 @ V3a0 ) @ V8t1 ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V10b: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V11t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V12t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V4a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V10b ) @ V11t1 ) @ V12t2 ) ) ) @ ( ap @ ( ap @ V2mem_27 @ V3a0 ) @ V12t2 ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ V2mem_27 @ V3a0 ) @ V4a1 ) ) ) ) ) ) )
@ ( ap @ ( ap @ V2mem_27 @ V0a0 ) @ V1a1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__rules,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a ) @ V1t1 ) @ V2t2 ) ) ) ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27a )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V3a ) @ V5t1 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V3a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V4b ) @ V5t1 ) @ V6t2 ) ) ) ) ) ) ) )
& ! [V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ! [V8b: $i] :
( ( mem @ V8b @ A_27a )
=> ! [V9t1: $i] :
( ( mem @ V9t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V7a ) @ V10t2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V7a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V8b ) @ V9t1 ) @ V10t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__ind,axiom,
! [A_27a: del,V0mem_27: $i] :
( ( mem @ V0mem_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) ) ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27a )
=> ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0mem_27 @ V4a ) @ V6t1 ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V4a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5b ) @ V6t1 ) @ V7t2 ) ) ) ) ) ) ) )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27a )
=> ! [V10t1: $i] :
( ( mem @ V10t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V11t2: $i] :
( ( mem @ V11t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0mem_27 @ V8a ) @ V11t2 ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V8a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V9b ) @ V10t1 ) @ V11t2 ) ) ) ) ) ) ) ) )
=> ! [V12a0: $i] :
( ( mem @ V12a0 @ A_27a )
=> ! [V13a1: $i] :
( ( mem @ V13a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V12a0 ) @ V13a1 ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V12a0 ) @ V13a1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__strongind,axiom,
! [A_27a: del,V0mem_27: $i] :
( ( mem @ V0mem_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) ) ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27a )
=> ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V4a ) @ V6t1 ) )
& ( p @ ( ap @ ( ap @ V0mem_27 @ V4a ) @ V6t1 ) ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V4a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5b ) @ V6t1 ) @ V7t2 ) ) ) ) ) ) ) )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27a )
=> ! [V10t1: $i] :
( ( mem @ V10t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V11t2: $i] :
( ( mem @ V11t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V8a ) @ V11t2 ) )
& ( p @ ( ap @ ( ap @ V0mem_27 @ V8a ) @ V11t2 ) ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V8a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V9b ) @ V10t1 ) @ V11t2 ) ) ) ) ) ) ) ) )
=> ! [V12a0: $i] :
( ( mem @ V12a0 @ A_27a )
=> ! [V13a1: $i] :
( ( mem @ V13a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V12a0 ) @ V13a1 ) )
=> ( p @ ( ap @ ( ap @ V0mem_27 @ V12a0 ) @ V13a1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__cases,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a0 ) @ V1a1 ) )
<=> ( ? [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a0 ) @ V2t1 ) @ V3t2 ) ) ) )
| ? [V4b: $i] :
( ( mem @ V4b @ A_27a )
& ? [V5t1: $i] :
( ( mem @ V5t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V4b ) @ V5t1 ) @ V6t2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a0 ) @ V5t1 ) ) ) ) )
| ? [V7b: $i] :
( ( mem @ V7b @ A_27a )
& ? [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V7b ) @ V8t1 ) @ V9t2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a0 ) @ V9t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__thm,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a ) @ ( c_2Elbtree_2ELf @ A_27a ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1b ) @ V2t1 ) @ V3t2 ) ) )
<=> ( ( V0a = V1b )
| ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a ) @ V2t1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0a ) @ V3t2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Emap__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elbtree_2ELf @ A_27a ) )
= ( c_2Elbtree_2ELf @ A_27b ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ ( ap @ V0f @ V1a ) ) @ ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27a @ A_27b ) @ V0f ) @ V2t1 ) ) @ ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27a @ A_27b ) @ V0f ) @ V3t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emap__eq__Lf,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1t )
= ( c_2Elbtree_2ELf @ A_27a ) )
<=> ( V1t
= ( c_2Elbtree_2ELf @ A_27b ) ) )
& ( ( ( c_2Elbtree_2ELf @ A_27a )
= ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1t ) )
<=> ( V1t
= ( c_2Elbtree_2ELf @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emap__eq__Nd,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1t )
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V2a ) @ V3t1 ) @ V4t2 ) )
<=> ? [V5a_27: $i] :
( ( mem @ V5a_27 @ A_27b )
& ? [V6t1_27: $i] :
( ( mem @ V6t1_27 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
& ? [V7t2_27: $i] :
( ( mem @ V7t2_27 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
& ( V1t
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V5a_27 ) @ V6t1_27 ) @ V7t2_27 ) )
& ( V2a
= ( ap @ V0f @ V5a_27 ) )
& ( V3t1
= ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V6t1_27 ) )
& ( V4t2
= ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V7t2_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Efinite__def,axiom,
! [A_27a: del] :
( ( c_2Elbtree_2Efinite @ A_27a )
= ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V0a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool )
@ ^ [V1finite_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V2a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V2a0 ) @ ( c_2Elbtree_2ELf @ A_27a ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V4t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V5t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V2a0 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V3a ) @ V4t1 ) @ V5t2 ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1finite_27 @ V4t1 ) ) @ ( ap @ V1finite_27 @ V5t2 ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V1finite_27 @ V2a0 ) ) ) ) )
@ ( ap @ V1finite_27 @ V0a0 ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__rules,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ ( c_2Elbtree_2ELf @ A_27a ) ) )
& ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V1t1 ) )
& ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V2t2 ) ) )
=> ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a ) @ V1t1 ) @ V2t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__ind,axiom,
! [A_27a: del,V0finite_27: $i] :
( ( mem @ V0finite_27 @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0finite_27 @ ( c_2Elbtree_2ELf @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ V0finite_27 @ V2t1 ) )
& ( p @ ( ap @ V0finite_27 @ V3t2 ) ) )
=> ( p @ ( ap @ V0finite_27 @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) ) ) ) ) ) )
=> ! [V4a0: $i] :
( ( mem @ V4a0 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V4a0 ) )
=> ( p @ ( ap @ V0finite_27 @ V4a0 ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__strongind,axiom,
! [A_27a: del,V0finite_27: $i] :
( ( mem @ V0finite_27 @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0finite_27 @ ( c_2Elbtree_2ELf @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V2t1 ) )
& ( p @ ( ap @ V0finite_27 @ V2t1 ) )
& ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V3t2 ) )
& ( p @ ( ap @ V0finite_27 @ V3t2 ) ) )
=> ( p @ ( ap @ V0finite_27 @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) ) ) ) ) ) )
=> ! [V4a0: $i] :
( ( mem @ V4a0 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V4a0 ) )
=> ( p @ ( ap @ V0finite_27 @ V4a0 ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__cases,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V0a0 ) )
<=> ( ( V0a0
= ( c_2Elbtree_2ELf @ A_27a ) )
| ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V0a0
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) )
& ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V2t1 ) )
& ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ V3t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__thm,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27b )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ ( c_2Elbtree_2ELf @ A_27a ) ) )
<=> $true )
& ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27b ) @ V0a ) @ V1t1 ) @ V2t2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27b ) @ V1t1 ) )
& ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27b ) @ V2t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Efinite__map,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27a ) @ ( ap @ ( ap @ ( c_2Elbtree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) )
<=> ( p @ ( ap @ ( c_2Elbtree_2Efinite @ A_27b ) @ V1t ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Ebf__flatten__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( c_2Elist_2ENIL @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
= ( c_2Ellist_2ELNIL @ A_27a ) )
& ! [V0ts: $i] :
( ( mem @ V0ts @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( c_2Elbtree_2ELf @ A_27a ) ) @ V0ts ) )
= ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ V0ts ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V4ts: $i] :
( ( mem @ V4ts @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) @ V4ts ) )
= ( ap @ ( ap @ ( c_2Ellist_2ELCONS @ A_27a ) @ V1a ) @ ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V4ts ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V2t1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V3t2 ) @ ( c_2Elist_2ENIL @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Ebf__flatten__eq__lnil,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ V0l )
= ( c_2Ellist_2ELNIL @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( c_2Elbtree_2ELf @ A_27a ) ) ) @ V0l ) ) ) ) ).
thf(conj_thm_2Elbtree_2Ebf__flatten__append,axiom,
! [A_27a: del,V0l2: $i] :
( ( mem @ V0l2 @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( c_2Elbtree_2ELf @ A_27a ) ) ) @ V1l1 ) )
=> ( ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V1l1 ) @ V0l2 ) )
= ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ V0l2 ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2EEXISTS__FIRST,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ V1l ) )
=> ? [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ? [V4l2: $i] :
( ( mem @ V4l2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1l
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ V4l2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V2l1 ) )
& ( p @ ( ap @ V0P @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Eexists__bf__flatten,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1tlist: $i] :
( ( mem @ V1tlist @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ellist_2Eexists @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ V1tlist ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) ) @ V1tlist ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Edepth__def,axiom,
! [A_27a: del] :
( ( c_2Elbtree_2Edepth @ A_27a )
= ( lam @ A_27a
@ ^ [V0a0: $i] :
( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V1a1: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V2a2: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ^ [V3depth_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V5a1: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6a2: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V7t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V8t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V5a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V4a0 ) @ V7t1 ) @ V8t2 ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V6a2 ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9m: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V10a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V11t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V12t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V5a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V10a ) @ V11t1 ) @ V12t2 ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V6a2 ) @ ( ap @ c_2Enum_2ESUC @ V9m ) ) ) @ ( ap @ ( ap @ ( ap @ V3depth_27 @ V4a0 ) @ V11t1 ) @ V9m ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13m: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V14a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V15t1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
@ ( lam @ ( ty_2Elbtree_2Elbtree @ A_27a )
@ ^ [V16t2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ V5a1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V14a ) @ V15t1 ) @ V16t2 ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V6a2 ) @ ( ap @ c_2Enum_2ESUC @ V13m ) ) ) @ ( ap @ ( ap @ ( ap @ V3depth_27 @ V4a0 ) @ V16t2 ) @ V13m ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ V3depth_27 @ V4a0 ) @ V5a1 ) @ V6a2 ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ V3depth_27 @ V0a0 ) @ V1a1 ) @ V2a2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Edepth__rules,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0x ) @ V1t1 ) @ V2t2 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V4x ) @ V6t1 ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V5a ) @ V6t1 ) @ V7t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) ) ) ) )
& ! [V8m: tp__ty_2Enum_2Enum,V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ! [V10a: $i] :
( ( mem @ V10a @ A_27a )
=> ! [V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V12t2: $i] :
( ( mem @ V12t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V9x ) @ V12t2 ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V9x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V10a ) @ V11t1 ) @ V12t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V8m ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Edepth__ind,axiom,
! [A_27a: del,V0depth_27: $i] :
( ( mem @ V0depth_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V1x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1x ) @ V2t1 ) @ V3t2 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V5x ) @ V7t1 ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V5x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V6a ) @ V7t1 ) @ V8t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ) )
& ! [V9m: tp__ty_2Enum_2Enum,V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ! [V12t1: $i] :
( ( mem @ V12t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V10x ) @ V13t2 ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V10x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V11a ) @ V12t1 ) @ V13t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V9m ) ) ) ) ) ) ) ) ) )
=> ! [V14a0: $i] :
( ( mem @ V14a0 @ A_27a )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V16a2: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V14a0 ) @ V15a1 ) @ ( inj__ty_2Enum_2Enum @ V16a2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V14a0 ) @ V15a1 ) @ ( inj__ty_2Enum_2Enum @ V16a2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Edepth__strongind,axiom,
! [A_27a: del,V0depth_27: $i] :
( ( mem @ V0depth_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elbtree_2Elbtree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V1x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1x ) @ V2t1 ) @ V3t2 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V5x ) @ V7t1 ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V5x ) @ V7t1 ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V5x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V6a ) @ V7t1 ) @ V8t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ) )
& ! [V9m: tp__ty_2Enum_2Enum,V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ! [V12t1: $i] :
( ( mem @ V12t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V10x ) @ V13t2 ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V10x ) @ V13t2 ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V10x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V11a ) @ V12t1 ) @ V13t2 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V9m ) ) ) ) ) ) ) ) ) )
=> ! [V14a0: $i] :
( ( mem @ V14a0 @ A_27a )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V16a2: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V14a0 ) @ V15a1 ) @ ( inj__ty_2Enum_2Enum @ V16a2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0depth_27 @ V14a0 ) @ V15a1 ) @ ( inj__ty_2Enum_2Enum @ V16a2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Edepth__cases,axiom,
! [A_27a: del,V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2a2: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0a0 ) @ V1a1 ) @ ( inj__ty_2Enum_2Enum @ V2a2 ) ) )
<=> ( ? [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V0a0 ) @ V3t1 ) @ V4t2 ) )
& ( V2a2 = fo__c_2Enum_2E0 ) ) )
| ? [V5m: tp__ty_2Enum_2Enum,V6a: $i] :
( ( mem @ V6a @ A_27a )
& ? [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V6a ) @ V7t1 ) @ V8t2 ) )
& ( V2a2
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0a0 ) @ V7t1 ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) )
| ? [V9m: tp__ty_2Enum_2Enum,V10a: $i] :
( ( mem @ V10a @ A_27a )
& ? [V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ? [V12t2: $i] :
( ( mem @ V12t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
& ( V1a1
= ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V10a ) @ V11t1 ) @ V12t2 ) )
& ( V2a2
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V9m ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0a0 ) @ V12t2 ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__depth,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) @ V1t ) )
=> ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0x ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Edepth__mem,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0x ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) @ V1t ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Emindepth__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ V1t ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) @ V1t ) )
@ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum )
@ ( ap @ c_2Ewhile_2ELEAST
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0x ) @ V1t ) @ V2n ) ) ) ) )
@ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Eoptmin__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
& ! [V1x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
& ! [V2y: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
& ! [V3x: tp__ty_2Enum_2Enum,V4y: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) ) )
=> ! [V5v: tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum,V6v1: tp__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ V5v ) ) @ ( inj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ V6v1 ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Eoptmin__def,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elbtree_2Eoptmin @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elbtree_2Eoptmin @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elbtree_2Eoptmin @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elbtree_2Eoptmin @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emindepth__thm,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ ( c_2Elbtree_2ELf @ A_27a ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2ENd @ A_27a ) @ V1a ) @ V2t1 ) @ V3t2 ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) @ V1a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ c_2Enum_2ESUC ) @ ( ap @ ( ap @ c_2Elbtree_2Eoptmin @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ V2t1 ) ) @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ V3t2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__mindepth,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) @ V1t ) )
=> ? [V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ V1t ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emindepth__depth,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elbtree_2Elbtree @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elbtree_2Emindepth @ A_27a ) @ V0x ) @ V1t ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Edepth @ A_27a ) @ V0x ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(ax_thm_2Elbtree_2Eis__mmindex__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Eis__mmindex @ A_27a ) @ V0f ) @ V1l ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
& ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1l ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V1l ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) )
| ? [V5d_27: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V1l ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5d_27 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V5d_27 ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V5d_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emmindex__EXISTS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEXISTS @ A_27a )
@ ( lam @ A_27a
@ ^ [V2e: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) @ ( ap @ V0f @ V2e ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ V3n ) ) ) ) ) )
@ V1l ) )
=> ? [V4i: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Eis__mmindex @ A_27a ) @ V0f ) @ V1l ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emmindex__unique,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Eis__mmindex @ A_27a ) @ V0f ) @ V1l ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
=> ! [V4j: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elbtree_2Eis__mmindex @ A_27a ) @ V0f ) @ V1l ) @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
<=> ( ( V4j = V2i )
& ( V5n = V3m ) ) ) ) ) ) ).
thf(conj_thm_2Elbtree_2Emem__bf__flatten,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1tlist: $i] :
( ( mem @ V1tlist @ ( ty_2Elist_2Elist @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ellist_2Eexists @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ ( ap @ ( c_2Elbtree_2Ebf__flatten @ A_27a ) @ V1tlist ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ ( ty_2Elbtree_2Elbtree @ A_27a ) ) @ ( ap @ ( c_2Elbtree_2Emem @ A_27a ) @ V0x ) ) @ V1tlist ) ) ) ) ) ).
%------------------------------------------------------------------------------