ITP001 Axioms: ITP066_7.ax
%------------------------------------------------------------------------------
% File : ITP066_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 : lbtree.ax [Gau19]
% : HL4066_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 253 ( 75 unt; 116 typ; 0 def)
% Number of atoms : 348 ( 175 equ)
% Maximal formula atoms : 11 ( 1 avg)
% Number of connectives : 215 ( 4 ~; 20 |; 85 &)
% ( 39 <=>; 67 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 104 ( 57 >; 47 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 113 ( 113 usr; 13 con; 0-5 aty)
% Number of variables : 786 ( 592 !; 81 ?; 786 :)
% ( 113 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Elbtree_2Elbtree,type,
tyop_2Elbtree_2Elbtree: $tType > $tType ).
tff(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
tff(tyop_2Ellist_2Ellist,type,
tyop_2Ellist_2Ellist: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3C_3D_2E0,type,
c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3C_3D_2E2,type,
c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E1,type,
c_2Emin_2E_3D_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Elist_2EAPPEND_2E0,type,
c_2Elist_2EAPPEND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2EAPPEND_2E2,type,
c_2Elist_2EAPPEND_2E2:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Elist_2ECONS_2E0,type,
c_2Elist_2ECONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2ECONS_2E2,type,
c_2Elist_2ECONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Elist_2EEL_2E0,type,
c_2Elist_2EEL_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a)) ).
tff(c_2Elist_2EEL_2E2,type,
c_2Elist_2EEL_2E2:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * tyop_2Elist_2Elist(A_27a) ) > A_27a ) ).
tff(c_2Elist_2EEVERY_2E0,type,
c_2Elist_2EEVERY_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Elist_2EEVERY_2E2,type,
c_2Elist_2EEVERY_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elist_2EEXISTS_2E0,type,
c_2Elist_2EEXISTS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Elist_2EEXISTS_2E2,type,
c_2Elist_2EEXISTS_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epair_2EFST_2E0,type,
c_2Epair_2EFST_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).
tff(c_2Ellist_2ELCONS_2E0,type,
c_2Ellist_2ELCONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Ellist_2Ellist(A_27a))) ).
tff(c_2Ellist_2ELCONS_2E2,type,
c_2Ellist_2ELCONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Ellist_2Ellist(A_27a) ) > tyop_2Ellist_2Ellist(A_27a) ) ).
tff(c_2Ewhile_2ELEAST_2E0,type,
c_2Ewhile_2ELEAST_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).
tff(c_2Ewhile_2ELEAST_2E1,type,
c_2Ewhile_2ELEAST_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ).
tff(c_2Elist_2ELENGTH_2E0,type,
c_2Elist_2ELENGTH_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum) ).
tff(c_2Elist_2ELENGTH_2E1,type,
c_2Elist_2ELENGTH_2E1:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > tyop_2Enum_2Enum ) ).
tff(c_2Ellist_2ELNIL_2E0,type,
c_2Ellist_2ELNIL_2E0:
!>[A_27a: $tType] : tyop_2Ellist_2Ellist(A_27a) ).
tff(c_2Elbtree_2ELf_2E0,type,
c_2Elbtree_2ELf_2E0:
!>[A_27a: $tType] : tyop_2Elbtree_2Elbtree(A_27a) ).
tff(c_2Elbtree_2ELfrep_2E0,type,
c_2Elbtree_2ELfrep_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)) ).
tff(c_2Elbtree_2ELfrep_2E1,type,
c_2Elbtree_2ELfrep_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > tyop_2Eoption_2Eoption(A_27b) ) ).
tff(c_2Earithmetic_2EMIN_2E0,type,
c_2Earithmetic_2EMIN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMIN_2E2,type,
c_2Earithmetic_2EMIN_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Elist_2ENIL_2E0,type,
c_2Elist_2ENIL_2E0:
!>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).
tff(c_2Eoption_2ENONE_2E0,type,
c_2Eoption_2ENONE_2E0:
!>[A_27a: $tType] : tyop_2Eoption_2Eoption(A_27a) ).
tff(c_2Elbtree_2ENd_2E0,type,
c_2Elbtree_2ENd_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27a)))) ).
tff(c_2Elbtree_2ENd_2E3,type,
c_2Elbtree_2ENd_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elbtree_2Elbtree(A_27a) * tyop_2Elbtree_2Elbtree(A_27a) ) > tyop_2Elbtree_2Elbtree(A_27a) ) ).
tff(c_2Elbtree_2ENdrep_2E0,type,
c_2Elbtree_2ENdrep_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))))) ).
tff(c_2Elbtree_2ENdrep_2E3,type,
c_2Elbtree_2ENdrep_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) * tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) ) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) ) ).
tff(c_2Elbtree_2ENdrep_2E4,type,
c_2Elbtree_2ENdrep_2E4:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) * tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) * tyop_2Elist_2Elist(tyop_2Emin_2Ebool) ) > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Eoption_2EOPTION__MAP_2E0,type,
c_2Eoption_2EOPTION__MAP_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b))) ).
tff(c_2Eoption_2EOPTION__MAP_2E2,type,
c_2Eoption_2EOPTION__MAP_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Eoption_2Eoption(A_27a) ) > tyop_2Eoption_2Eoption(A_27b) ) ).
tff(c_2Eoption_2ESOME_2E0,type,
c_2Eoption_2ESOME_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)) ).
tff(c_2Eoption_2ESOME_2E1,type,
c_2Eoption_2ESOME_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Elbtree_2Ebf__flatten_2E0,type,
c_2Elbtree_2Ebf__flatten_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Ellist_2Ellist(A_27a)) ).
tff(c_2Elbtree_2Ebf__flatten_2E1,type,
c_2Elbtree_2Ebf__flatten_2E1:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)) > tyop_2Ellist_2Ellist(A_27a) ) ).
tff(c_2Elbtree_2Edepth_2E0,type,
c_2Elbtree_2Edepth_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))) ).
tff(c_2Elbtree_2Edepth_2E3,type,
c_2Elbtree_2Edepth_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elbtree_2Elbtree(A_27a) * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ellist_2Eexists_2E0,type,
c_2Ellist_2Eexists_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ellist_2Eexists_2E2,type,
c_2Ellist_2Eexists_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Ellist_2Ellist(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elbtree_2Efinite_2E0,type,
c_2Elbtree_2Efinite_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool) ).
tff(c_2Elbtree_2Efinite_2E1,type,
c_2Elbtree_2Efinite_2E1:
!>[A_27a: $tType] : ( tyop_2Elbtree_2Elbtree(A_27a) > tyop_2Emin_2Ebool ) ).
tff(c_2Elbtree_2Eis__lbtree_2E0,type,
c_2Elbtree_2Eis__lbtree_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool) ).
tff(c_2Elbtree_2Eis__lbtree_2E1,type,
c_2Elbtree_2Eis__lbtree_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) > tyop_2Emin_2Ebool ) ).
tff(c_2Elbtree_2Eis__mmindex_2E0,type,
c_2Elbtree_2Eis__mmindex_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))) ).
tff(c_2Elbtree_2Eis__mmindex_2E4,type,
c_2Elbtree_2Eis__mmindex_2E4:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)) * tyop_2Elist_2Elist(A_27a) * tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elbtree_2Elbtree__abs_2E0,type,
c_2Elbtree_2Elbtree__abs_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Elbtree_2Elbtree(A_27a)) ).
tff(c_2Elbtree_2Elbtree__abs_2E1,type,
c_2Elbtree_2Elbtree__abs_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) > tyop_2Elbtree_2Elbtree(A_27a) ) ).
tff(c_2Elbtree_2Elbtree__case_2E0,type,
c_2Elbtree_2Elbtree__case_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))) ).
tff(c_2Elbtree_2Elbtree__case_2E3,type,
c_2Elbtree_2Elbtree__case_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))) * tyop_2Elbtree_2Elbtree(A_27b) ) > A_27a ) ).
tff(c_2Elbtree_2Elbtree__rep_2E0,type,
c_2Elbtree_2Elbtree__rep_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))) ).
tff(c_2Elbtree_2Elbtree__rep_2E1,type,
c_2Elbtree_2Elbtree__rep_2E1:
!>[A_27a: $tType] : ( tyop_2Elbtree_2Elbtree(A_27a) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)) ) ).
tff(c_2Elist_2Elist__CASE_2E0,type,
c_2Elist_2Elist__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)),A_27b))) ).
tff(c_2Elist_2Elist__CASE_2E3,type,
c_2Elist_2Elist__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)) ) > A_27b ) ).
tff(c_2Elbtree_2Emap_2E0,type,
c_2Elbtree_2Emap_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27b))) ).
tff(c_2Elbtree_2Emap_2E2,type,
c_2Elbtree_2Emap_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Elbtree_2Elbtree(A_27a) ) > tyop_2Elbtree_2Elbtree(A_27b) ) ).
tff(c_2Elbtree_2Emem_2E0,type,
c_2Elbtree_2Emem_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Elbtree_2Emem_2E1,type,
c_2Elbtree_2Emem_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool) ) ).
tff(c_2Elbtree_2Emem_2E2,type,
c_2Elbtree_2Emem_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elbtree_2Elbtree(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elbtree_2Emindepth_2E0,type,
c_2Elbtree_2Emindepth_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))) ).
tff(c_2Elbtree_2Emindepth_2E2,type,
c_2Elbtree_2Emindepth_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elbtree_2Elbtree(A_27a) ) > tyop_2Eoption_2Eoption(tyop_2Enum_2Enum) ) ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Eoption_2Eoption__CASE_2E0,type,
c_2Eoption_2Eoption__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b))) ).
tff(c_2Eoption_2Eoption__CASE_2E3,type,
c_2Eoption_2Eoption__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) ) > A_27b ) ).
tff(c_2Elbtree_2Eoptmin_2E0,type,
c_2Elbtree_2Eoptmin_2E0: tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))) ).
tff(c_2Elbtree_2Eoptmin_2E2,type,
c_2Elbtree_2Eoptmin_2E2: ( tyop_2Eoption_2Eoption(tyop_2Enum_2Enum) * tyop_2Eoption_2Eoption(tyop_2Enum_2Enum) ) > tyop_2Eoption_2Eoption(tyop_2Enum_2Enum) ).
tff(c_2Epair_2Epair__CASE_2E0,type,
c_2Epair_2Epair__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),A_27a)) ).
tff(c_2Epair_2Epair__CASE_2E2,type,
c_2Epair_2Epair__CASE_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod(A_27b,A_27c) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)) ) > A_27a ) ).
tff(c_2Elbtree_2Epath__follow_2E0,type,
c_2Elbtree_2Epath__follow_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))) ).
tff(c_2Elbtree_2Epath__follow_2E3,type,
c_2Elbtree_2Epath__follow_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))) * A_27b * tyop_2Elist_2Elist(tyop_2Emin_2Ebool) ) > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_3D_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Emin_2E_3D_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Emin_2E_3D_2E1_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Emin_2E_3D_2E1(tyop_2Elbtree_2Elbtree(A_27a),X0_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Elbtree_2Elbtree_28A_27b_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27b,c_2Emin_2E_40_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27b_29,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Elbtree_2Elbtree(A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),tyop_2Elbtree_2Elbtree(A_27b),c_2Emin_2E_40_2E0(tyop_2Elbtree_2Elbtree(A_27b)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EAPPEND_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2EAPPEND_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),X1_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elist_2EAPPEND_2E2(tyop_2Elbtree_2Elbtree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))),c_2Elist_2EAPPEND_2E0(tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ebool_2ECOND_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ebool_2ECOND_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27b,X2_2E0: A_27b] : ( c_2Ebool_2ECOND_2E3(A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),c_2Ebool_2ECOND_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Eoption_2Eoption(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28tyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X2_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),c_2Ebool_2ECOND_2E0(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ECONS_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2ECONS_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( c_2Elist_2ECONS_2E2(tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Elist_2Elist(tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Elist_2Elist(tyop_2Emin_2Ebool)),c_2Elist_2ECONS_2E0(tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elbtree_2Elbtree(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elist_2ECONS_2E2(tyop_2Elbtree_2Elbtree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))),c_2Elist_2ECONS_2E0(tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EEL_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EEL_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a),c_2Elist_2EEL_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EEVERY_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EEVERY_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Elist_2EEVERY_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EEVERY_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elist_2EEVERY_2E2(tyop_2Elbtree_2Elbtree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Emin_2Ebool),c_2Elist_2EEVERY_2E0(tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EEXISTS_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EEXISTS_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Elist_2EEXISTS_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EEXISTS_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),X1_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elist_2EEXISTS_2E2(tyop_2Elbtree_2Elbtree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Emin_2Ebool),c_2Elist_2EEXISTS_2E0(tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ellist_2ELCONS_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Ellist_2Ellist(A_27a)] : ( c_2Ellist_2ELCONS_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ellist_2Ellist(A_27a),tyop_2Ellist_2Ellist(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Ellist_2Ellist(A_27a)),c_2Ellist_2ELCONS_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ewhile_2ELEAST_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2ELEAST_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ELENGTH_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2ELfrep_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a] : ( c_2Elbtree_2ELfrep_2E1(A_27a,A_27b,X0_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27b),c_2Elbtree_2ELfrep_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2EMIN_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMIN_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMIN_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2ENd_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elbtree_2Elbtree(A_27a),X2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2ENd_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27a),app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27a))),c_2Elbtree_2ENd_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2ENd_2E3_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Elbtree_2Elbtree(A_27b),X2_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( c_2Elbtree_2ENd_2E3(A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Elbtree_2Elbtree(A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Elbtree_2Elbtree(A_27b))),c_2Elbtree_2ENd_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2ENdrep_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),X2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] : ( c_2Elbtree_2ENdrep_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))),c_2Elbtree_2ENdrep_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq4_2Ec_2Elbtree_2ENdrep_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),X2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),X3_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( c_2Elbtree_2ENdrep_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))),c_2Elbtree_2ENdrep_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq2_2Ec_2Eoption_2EOPTION__MAP_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)] : ( c_2Eoption_2EOPTION__MAP_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Eoption_2EOPTION__MAP_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eoption_2EOPTION__MAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Epair_2Eprod_28A_27b_2CA_27b_29_29_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),A_27a),X1_2E0: tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))] : ( c_2Eoption_2EOPTION__MAP_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),tyop_2Eoption_2Eoption(A_27a)),c_2Eoption_2EOPTION__MAP_2E0(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Eoption_2ESOME_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28tyop_2Emin_2Ebool_29_2Ctyop_2Eoption_2Eoption_28A_27a_29_29_20mono_2Etyop_2Elbtree_2Elbtree_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Elbtree_2Elbtree(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Elbtree_2Elbtree(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Elbtree_2Ebf__flatten_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),tyop_2Ellist_2Ellist(A_27a),c_2Elbtree_2Ebf__flatten_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2Edepth_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elbtree_2Elbtree(A_27a),X2_2E0: tyop_2Enum_2Enum] : ( c_2Elbtree_2Edepth_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Elbtree_2Edepth_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ellist_2Eexists_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Ellist_2Ellist(A_27a)] : ( c_2Ellist_2Eexists_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ellist_2Ellist(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Emin_2Ebool),c_2Ellist_2Eexists_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Efinite_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Efinite_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,c_2Elbtree_2Efinite_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Efinite_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( c_2Elbtree_2Efinite_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool,c_2Elbtree_2Efinite_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Eis__lbtree_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] : ( c_2Elbtree_2Eis__lbtree_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,c_2Elbtree_2Eis__lbtree_2E0(A_27a),X0_2E0) ) ).
tff(arityeq4_2Ec_2Elbtree_2Eis__mmindex_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),X1_2E0: tyop_2Elist_2Elist(A_27a),X2_2E0: tyop_2Enum_2Enum,X3_2E0: tyop_2Enum_2Enum] : ( c_2Elbtree_2Eis__mmindex_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Elbtree_2Eis__mmindex_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Elbtree__abs_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] : ( c_2Elbtree_2Elbtree__abs_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2Elbtree__abs_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2Elbtree__case_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),X2_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( c_2Elbtree_2Elbtree__case_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a)),c_2Elbtree_2Elbtree__case_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Elbtree__rep_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Elbtree__rep_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),c_2Elbtree_2Elbtree__rep_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Elist_2Elist__CASE_2E3_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Elist_2Elist__CASE_2E3(tyop_2Emin_2Ebool,tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a))),c_2Elist_2Elist__CASE_2E0(tyop_2Emin_2Ebool,tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elbtree_2Emap_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Emap_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Elbtree_2Elbtree(A_27b)),c_2Elbtree_2Emap_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elbtree_2Emap_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X1_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( c_2Elbtree_2Emap_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Elbtree_2Elbtree(A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Elbtree_2Elbtree(A_27a)),c_2Elbtree_2Emap_2E0(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Elbtree_2Emem_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Elbtree_2Emem_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),c_2Elbtree_2Emem_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Elbtree_2Emem_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Emem_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),c_2Elbtree_2Emem_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Elbtree_2Emindepth_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Emindepth_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Elbtree_2Emindepth_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Epair_2Eprod_28A_27b_2CA_27b_29_29_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a))] : ( c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),tyop_2Eoption_2Eoption(A_27a))),c_2Eoption_2Eoption__CASE_2E0(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Epair_2Eprod_28A_27a_2CA_27a_29_29_20mono_2Etyop_2Elbtree_2Elbtree_28A_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a))),X1_2E0: tyop_2Elbtree_2Elbtree(A_27b),X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b))] : ( c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Elbtree_2Elbtree(A_27b)),app_2E2(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a))),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Elbtree_2Elbtree(A_27b))),c_2Eoption_2Eoption__CASE_2E0(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elbtree_2Eoptmin_2E2,axiom,
! [X0_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)] : ( c_2Elbtree_2Eoptmin_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Elbtree_2Eoptmin_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27b_29_20mono_2EA_27a_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27a,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))),tyop_2Elbtree_2Elbtree(A_27b)),c_2Epair_2Epair__CASE_2E0(tyop_2Elbtree_2Elbtree(A_27b),A_27a,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29_20mono_2EA_27a_20mono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Eoption_2Eoption(A_27a),A_27a,tyop_2Epair_2Eprod(A_27b,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a)),c_2Epair_2Epair__CASE_2E0(tyop_2Eoption_2Eoption(A_27a),A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29_20mono_2EA_27b_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Eoption_2Eoption(A_27a),A_27b,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a)),c_2Epair_2Epair__CASE_2E0(tyop_2Eoption_2Eoption(A_27a),A_27b,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Elbtree_2Elbtree_28A_27b_29_20mono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27a_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b)))] : ( c_2Epair_2Epair__CASE_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27b,tyop_2Epair_2Eprod(A_27a,A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b))),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b))),tyop_2Elbtree_2Elbtree(A_27b)),c_2Epair_2Epair__CASE_2E0(tyop_2Elbtree_2Elbtree(A_27b),A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Elbtree_2Epath__follow_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),X1_2E0: A_27b,X2_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( c_2Elbtree_2Epath__follow_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),c_2Elbtree_2Epath__follow_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Elbtree_2ELfrep__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a] : ( c_2Elbtree_2ELfrep_2E1(A_27a,A_27b,V0x_2E0) = c_2Eoption_2ENONE_2E0(A_27b) ) ).
tff(thm_2Elbtree_2ENdrep__def,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))))] :
( ! [V1t1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),V2t2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),V4v_2E0: tyop_2Emin_2Ebool,V5xs_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0,V1t1_2E0),V2t2_2E0),V4v_2E0),V5xs_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),V4v_2E0,app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a),V1t1_2E0,V5xs_2E0),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a),V2t2_2E0,V5xs_2E0)) )
=> ! [V0a_2E0: A_27a,V1t1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),V2t2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),V3x_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( c_2Elbtree_2ENdrep_2E4(A_27a,V0a_2E0,V1t1_2E0,V2t2_2E0,V3x_2E0) = c_2Elist_2Elist__CASE_2E3(tyop_2Emin_2Ebool,tyop_2Eoption_2Eoption(A_27a),V3x_2E0,c_2Eoption_2ESOME_2E1(A_27a,V0a_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0,V1t1_2E0),V2t2_2E0)) ) ) ).
tff(thm_2Elbtree_2Eis__lbtree__def,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Elbtree_2Eis__lbtree_2E1(A_27a,V0t_2E0))
<=> ? [V1P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool)] :
( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1P_2E0,V2t_2E0))
=> ( ( V2t_2E0 = c_2Elbtree_2ELfrep_2E0(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),A_27a) )
| ? [V3a_2E0: A_27a,V4t1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),V5t2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1P_2E0,V4t1_2E0))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1P_2E0,V5t2_2E0))
& ( V2t_2E0 = c_2Elbtree_2ENdrep_2E3(A_27a,V3a_2E0,V4t1_2E0,V5t2_2E0) ) ) ) )
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1P_2E0,V0t_2E0)) ) ) ).
tff(thm_2Elbtree_2Elbtree__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a)),tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2Eis__lbtree_2E0(A_27a),V0rep_2E0)) ).
tff(thm_2Elbtree_2Elbtree__absrep,axiom,
! [A_27a: $tType] :
( ! [V0a_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Elbtree__abs_2E1(A_27a,c_2Elbtree_2Elbtree__rep_2E1(A_27a,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Elbtree_2Eis__lbtree_2E1(A_27a,V1r_2E0))
<=> ( c_2Elbtree_2Elbtree__rep_2E1(A_27a,c_2Elbtree_2Elbtree__abs_2E1(A_27a,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Elbtree_2Epath__follow__def,axiom,
! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a)))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)))))] :
( ! [V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),V4h_2E0: tyop_2Emin_2Ebool,V5t_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool),V9y_2E0: A_27b,V10z_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))))),F2_2E0,V2g_2E0),V4h_2E0),V5t_2E0),V9y_2E0),V10z_2E0) = c_2Elbtree_2Epath__follow_2E3(A_27a,A_27b,V2g_2E0,c_2Ebool_2ECOND_2E3(A_27b,V4h_2E0,V9y_2E0,V10z_2E0),V5t_2E0) )
=> ( ! [V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),V4h_2E0: tyop_2Emin_2Ebool,V5t_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool),V7a_2E0: A_27a,V8v2_2E0: tyop_2Epair_2Eprod(A_27b,A_27b)] : ( app_2E2(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))))),F1_2E0,V2g_2E0),V4h_2E0),V5t_2E0),V7a_2E0),V8v2_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Eoption_2Eoption(A_27a),A_27b,A_27b,V8v2_2E0,app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))))),F2_2E0,V2g_2E0),V4h_2E0),V5t_2E0)) )
=> ( ! [V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),V4h_2E0: tyop_2Emin_2Ebool,V5t_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool),V6v_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0,V2g_2E0),V4h_2E0),V5t_2E0),V6v_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Eoption_2Eoption(A_27a),A_27a,tyop_2Epair_2Eprod(A_27b,A_27b),V6v_2E0,app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27b),tyop_2Eoption_2Eoption(A_27a))))),F1_2E0,V2g_2E0),V4h_2E0),V5t_2E0)) )
=> ( ! [V0g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),V1x_2E0: A_27b] : ( c_2Elbtree_2Epath__follow_2E3(A_27a,A_27b,V0g_2E0,V1x_2E0,c_2Elist_2ENIL_2E0(tyop_2Emin_2Ebool)) = c_2Eoption_2EOPTION__MAP_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),A_27a,c_2Epair_2EFST_2E0(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),app_2E2(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),V0g_2E0,V1x_2E0)) )
& ! [V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),V3x_2E0: A_27b,V4h_2E0: tyop_2Emin_2Ebool,V5t_2E0: tyop_2Elist_2Elist(tyop_2Emin_2Ebool)] : ( c_2Elbtree_2Epath__follow_2E3(A_27a,A_27b,V2g_2E0,V3x_2E0,c_2Elist_2ECONS_2E2(tyop_2Emin_2Ebool,V4h_2E0,V5t_2E0)) = c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b))),V2g_2E0,V3x_2E0),c_2Eoption_2ENONE_2E0(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,A_27b)),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0,V2g_2E0),V4h_2E0),V5t_2E0)) ) ) ) ) ) ).
tff(thm_2Elbtree_2ELf__def,axiom,
! [A_27a: $tType] : ( c_2Elbtree_2ELf_2E0(A_27a) = c_2Elbtree_2Elbtree__abs_2E1(A_27a,c_2Elbtree_2ELfrep_2E0(tyop_2Elist_2Elist(tyop_2Emin_2Ebool),A_27a)) ) ).
tff(thm_2Elbtree_2ENd__def,axiom,
! [A_27a: $tType,V0a_2E0: A_27a,V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2ENd_2E3(A_27a,V0a_2E0,V1t1_2E0,V2t2_2E0) = c_2Elbtree_2Elbtree__abs_2E1(A_27a,c_2Elbtree_2ENdrep_2E3(A_27a,V0a_2E0,c_2Elbtree_2Elbtree__rep_2E1(A_27a,V1t1_2E0),c_2Elbtree_2Elbtree__rep_2E1(A_27a,V2t2_2E0))) ) ).
tff(thm_2Elbtree_2Elbtree__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool)),F3_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ! [V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool,F0_2E0,V2t_2E0))
<=> ( V2t_2E0 = c_2Elbtree_2ELf_2E0(A_27b) ) )
=> ( ! [V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27b)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),F2_2E0,V2t_2E0),V6t1_2E0))
<=> ? [V7a_2E0: A_27b,V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( V2t_2E0 = c_2Elbtree_2ENd_2E3(A_27b,V7a_2E0,V6t1_2E0,V8t2_2E0) ) )
=> ( ! [V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V9t2_2E0: tyop_2Elbtree_2Elbtree(A_27b)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),F3_2E0,V2t_2E0),V9t2_2E0))
<=> ? [V10a_2E0: A_27b,V11t1_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( V2t_2E0 = c_2Elbtree_2ENd_2E3(A_27b,V10a_2E0,V11t1_2E0,V9t2_2E0) ) )
=> ( ! [V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V3a_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V2t_2E0),V3a_2E0))
<=> ? [V4t1_2E0: tyop_2Elbtree_2Elbtree(A_27b),V5t2_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( V2t_2E0 = c_2Elbtree_2ENd_2E3(A_27b,V3a_2E0,V4t1_2E0,V5t2_2E0) ) )
=> ! [V0e_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b)] : ( c_2Elbtree_2Elbtree__case_2E3(A_27a,A_27b,V0e_2E0,V1f_2E0,V2t_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool,F0_2E0,V2t_2E0),V0e_2E0,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27a,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a)),V1f_2E0,c_2Emin_2E_40_2E1(A_27b,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V2t_2E0))),c_2Emin_2E_40_2E1(tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),F2_2E0,V2t_2E0))),c_2Emin_2E_40_2E1(tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Ebool),F3_2E0,V2t_2E0)))) ) ) ) ) ) ).
tff(thm_2Elbtree_2Emem__def,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1x_27_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V0x_2E0,V1x_27_2E0))
<=> ! [V2mem_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool))] :
( ! [V3a0_2E0: A_27a,V4a1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( ? [V5t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V6t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( V4a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V3a0_2E0,V5t1_2E0,V6t2_2E0) )
| ? [V7b_2E0: A_27a,V8t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V9t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V4a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V7b_2E0,V8t1_2E0,V9t2_2E0) )
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2mem_27_2E0,V3a0_2E0),V8t1_2E0)) )
| ? [V10b_2E0: A_27a,V11t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V12t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V4a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V10b_2E0,V11t1_2E0,V12t2_2E0) )
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2mem_27_2E0,V3a0_2E0),V12t2_2E0)) ) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2mem_27_2E0,V3a0_2E0),V4a1_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2mem_27_2E0,V0x_2E0),V1x_27_2E0)) ) ) ).
tff(thm_2Elbtree_2Emap__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ( c_2Elbtree_2Emap_2E2(A_27a,A_27b,V0f_2E0,c_2Elbtree_2ELf_2E0(A_27a)) = c_2Elbtree_2ELf_2E0(A_27b) )
& ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Emap_2E2(A_27a,A_27b,V0f_2E0,c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0)) = c_2Elbtree_2ENd_2E3(A_27b,app_2E2(A_27a,A_27b,V0f_2E0,V1a_2E0),c_2Elbtree_2Emap_2E2(A_27a,A_27b,V0f_2E0,V2t1_2E0),c_2Elbtree_2Emap_2E2(A_27a,A_27b,V0f_2E0,V3t2_2E0)) ) ) ).
tff(thm_2Elbtree_2Efinite__def,axiom,
! [A_27a: $tType,V0x_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Efinite_2E1(A_27a,V0x_2E0))
<=> ! [V1finite_27_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool)] :
( ! [V2a0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( ( V2a0_2E0 = c_2Elbtree_2ELf_2E0(A_27a) )
| ? [V3a_2E0: A_27a,V4t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V5t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V2a0_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V3a_2E0,V4t1_2E0,V5t2_2E0) )
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V1finite_27_2E0,V4t1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V1finite_27_2E0,V5t2_2E0)) ) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V1finite_27_2E0,V2a0_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V1finite_27_2E0,V0x_2E0)) ) ) ).
tff(thm_2Elbtree_2Ebf__flatten__def,axiom,
! [A_27a: $tType] :
( ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,c_2Elist_2ENIL_2E0(tyop_2Elbtree_2Elbtree(A_27a))) = c_2Ellist_2ELNIL_2E0(A_27a) )
& ! [V0ts_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,c_2Elist_2ECONS_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2ELf_2E0(A_27a),V0ts_2E0)) = c_2Elbtree_2Ebf__flatten_2E1(A_27a,V0ts_2E0) )
& ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V4ts_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,c_2Elist_2ECONS_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0),V4ts_2E0)) = c_2Ellist_2ELCONS_2E2(A_27a,V1a_2E0,c_2Elbtree_2Ebf__flatten_2E1(A_27a,c_2Elist_2EAPPEND_2E2(tyop_2Elbtree_2Elbtree(A_27a),V4ts_2E0,c_2Elist_2ECONS_2E2(tyop_2Elbtree_2Elbtree(A_27a),V2t1_2E0,c_2Elist_2ECONS_2E2(tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0,c_2Elist_2ENIL_2E0(tyop_2Elbtree_2Elbtree(A_27a))))))) ) ) ).
tff(thm_2Elbtree_2Edepth__def,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1x_27_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2x_27_27_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,V1x_27_2E0,V2x_27_27_2E0))
<=> ! [V3depth_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
( ! [V4a0_2E0: A_27a,V5a1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V6a2_2E0: tyop_2Enum_2Enum] :
( ( ? [V7t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V5a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V4a0_2E0,V7t1_2E0,V8t2_2E0) )
& ( V6a2_2E0 = c_2Enum_2E0_2E0 ) )
| ? [V9m_2E0: tyop_2Enum_2Enum,V10a_2E0: A_27a,V11t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V12t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V5a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V10a_2E0,V11t1_2E0,V12t2_2E0) )
& ( V6a2_2E0 = c_2Enum_2ESUC_2E1(V9m_2E0) )
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V3depth_27_2E0,V4a0_2E0),V11t1_2E0),V9m_2E0)) )
| ? [V13m_2E0: tyop_2Enum_2Enum,V14a_2E0: A_27a,V15t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V16t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V5a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V14a_2E0,V15t1_2E0,V16t2_2E0) )
& ( V6a2_2E0 = c_2Enum_2ESUC_2E1(V13m_2E0) )
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V3depth_27_2E0,V4a0_2E0),V16t2_2E0),V13m_2E0)) ) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V3depth_27_2E0,V4a0_2E0),V5a1_2E0),V6a2_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V3depth_27_2E0,V0x_2E0),V1x_27_2E0),V2x_27_27_2E0)) ) ) ).
tff(thm_2Elbtree_2Emindepth__def,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
( ! [V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0x_2E0),V1t_2E0),V2n_2E0) = c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,V1t_2E0,V2n_2E0) )
=> ! [V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,V1t_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Elbtree_2Emem_2E2(A_27a,V0x_2E0,V1t_2E0),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0x_2E0),V1t_2E0))),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)) ) ) ).
tff(thm_2Elbtree_2Eis__mmindex__def,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),V1l_2E0: tyop_2Elist_2Elist(A_27a),V2n_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Eis__mmindex_2E4(A_27a,V0f_2E0,V1l_2E0,V2n_2E0,V3d_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,c_2Elist_2ELENGTH_2E1(A_27a,V1l_2E0)))
& ( app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),V0f_2E0,c_2Elist_2EEL_2E2(A_27a,V2n_2E0,V1l_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V3d_2E0) )
& ! [V4i_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4i_2E0,c_2Elist_2ELENGTH_2E1(A_27a,V1l_2E0)))
=> ( ( app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),V0f_2E0,c_2Elist_2EEL_2E2(A_27a,V4i_2E0,V1l_2E0)) = c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum) )
| ? [V5d_27_2E0: tyop_2Enum_2Enum] :
( ( app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),V0f_2E0,c_2Elist_2EEL_2E2(A_27a,V4i_2E0,V1l_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V5d_27_2E0) )
& p(c_2Earithmetic_2E_3C_3D_2E2(V3d_2E0,V5d_27_2E0))
& ( p(c_2Eprim__rec_2E_3C_2E2(V4i_2E0,V2n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V3d_2E0,V5d_27_2E0)) ) ) ) ) ) ) ).
tff(thm_2Elbtree_2Elbtree__cases,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V0t_2E0 = c_2Elbtree_2ELf_2E0(A_27a) )
| ? [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( V0t_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0) ) ) ).
tff(thm_2Elbtree_2ELf__NOT__Nd,axiom,
! [A_27a: $tType,V0t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2a_2E0: A_27a] : ( c_2Elbtree_2ELf_2E0(A_27a) != c_2Elbtree_2ENd_2E3(A_27a,V2a_2E0,V1t1_2E0,V0t2_2E0) ) ).
tff(thm_2Elbtree_2ENd__11,axiom,
! [A_27a: $tType,V0u2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1u1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V4a2_2E0: A_27a,V5a1_2E0: A_27a] :
( ( c_2Elbtree_2ENd_2E3(A_27a,V5a1_2E0,V3t1_2E0,V1u1_2E0) = c_2Elbtree_2ENd_2E3(A_27a,V4a2_2E0,V2t2_2E0,V0u2_2E0) )
<=> ( ( V5a1_2E0 = V4a2_2E0 )
& ( V3t1_2E0 = V2t2_2E0 )
& ( V1u1_2E0 = V0u2_2E0 ) ) ) ).
tff(thm_2Elbtree_2Elbtree__ue__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,F3_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool))] :
( ! [V4b_2E0: A_27b,V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),V6y_2E0: A_27a,V7z_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Elbtree_2Elbtree(A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)))),F3_2E0,V4b_2E0),V1g_2E0),V6y_2E0),V7z_2E0) = c_2Elbtree_2ENd_2E3(A_27b,V4b_2E0,app_2E2(A_27a,tyop_2Elbtree_2Elbtree(A_27b),V1g_2E0,V6y_2E0),app_2E2(A_27a,tyop_2Elbtree_2Elbtree(A_27b),V1g_2E0,V7z_2E0)) )
=> ( ! [V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),V4b_2E0: A_27b,V5v2_2E0: tyop_2Epair_2Eprod(A_27a,A_27a)] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b))),F2_2E0,V1g_2E0),V4b_2E0),V5v2_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27a,A_27a,V5v2_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)))),F3_2E0,V4b_2E0),V1g_2E0)) )
=> ( ! [V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),V3v_2E0: tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a))] : ( app_2E2(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),F1_2E0,V1g_2E0),V3v_2E0) = c_2Epair_2Epair__CASE_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27b,tyop_2Epair_2Eprod(A_27a,A_27a),V3v_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Elbtree_2Elbtree(A_27b))),F2_2E0,V1g_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)))),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool),F0_2E0,V0f_2E0),V1g_2E0))
<=> ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Elbtree_2Elbtree(A_27b),V1g_2E0,V2x_2E0) = c_2Eoption_2Eoption__CASE_2E3(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b),app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a))),V0f_2E0,V2x_2E0),c_2Elbtree_2ELf_2E0(A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)),tyop_2Elbtree_2Elbtree(A_27b)),F1_2E0,V1g_2E0)) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a))))] : p(c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27b,tyop_2Epair_2Eprod(A_27a,A_27a)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Elbtree_2Elbtree(A_27b)),tyop_2Emin_2Ebool),F0_2E0,V0f_2E0))) ) ) ) ) ).
tff(thm_2Elbtree_2Elbtree__case__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0t2_2E0: tyop_2Elbtree_2Elbtree(A_27b),V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27b),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a))),V3e_2E0: A_27a,V4a_2E0: A_27b] :
( ( c_2Elbtree_2Elbtree__case_2E3(A_27a,A_27b,V3e_2E0,V2f_2E0,c_2Elbtree_2ELf_2E0(A_27b)) = V3e_2E0 )
& ( c_2Elbtree_2Elbtree__case_2E3(A_27a,A_27b,V3e_2E0,V2f_2E0,c_2Elbtree_2ENd_2E3(A_27b,V4a_2E0,V1t1_2E0,V0t2_2E0)) = app_2E2(tyop_2Elbtree_2Elbtree(A_27b),A_27a,app_2E2(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27b),A_27a)),V2f_2E0,V4a_2E0),V1t1_2E0),V0t2_2E0) ) ) ).
tff(thm_2Elbtree_2Elbtree__bisimulation,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1u_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V0t_2E0 = V1u_2E0 )
<=> ? [V2R_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V0t_2E0),V1u_2E0))
& ! [V3t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V4u_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V3t_2E0),V4u_2E0))
=> ( ( ( V3t_2E0 = c_2Elbtree_2ELf_2E0(A_27a) )
& ( V4u_2E0 = c_2Elbtree_2ELf_2E0(A_27a) ) )
| ? [V5a_2E0: A_27a,V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V7u1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V9u2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V6t1_2E0),V7u1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V8t2_2E0),V9u2_2E0))
& ( V3t_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V5a_2E0,V6t1_2E0,V8t2_2E0) )
& ( V4u_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V5a_2E0,V7u1_2E0,V9u2_2E0) ) ) ) ) ) ) ).
tff(thm_2Elbtree_2Elbtree__strong__bisimulation,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1u_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V0t_2E0 = V1u_2E0 )
<=> ? [V2R_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V0t_2E0),V1u_2E0))
& ! [V3t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V4u_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V3t_2E0),V4u_2E0))
=> ( ( V3t_2E0 = V4u_2E0 )
| ? [V5a_2E0: A_27a,V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V7u1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V9u2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V6t1_2E0),V7u1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V2R_2E0,V8t2_2E0),V9u2_2E0))
& ( V3t_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V5a_2E0,V6t1_2E0,V8t2_2E0) )
& ( V4u_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V5a_2E0,V7u1_2E0,V9u2_2E0) ) ) ) ) ) ) ).
tff(thm_2Elbtree_2Emem__rules,axiom,
! [A_27a: $tType] :
( ! [V0a_2E0: A_27a,V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(c_2Elbtree_2Emem_2E2(A_27a,V0a_2E0,c_2Elbtree_2ENd_2E3(A_27a,V0a_2E0,V1t1_2E0,V2t2_2E0)))
& ! [V3a_2E0: A_27a,V4b_2E0: A_27a,V5t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V6t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,V5t1_2E0))
=> p(c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,c_2Elbtree_2ENd_2E3(A_27a,V4b_2E0,V5t1_2E0,V6t2_2E0))) )
& ! [V7a_2E0: A_27a,V8b_2E0: A_27a,V9t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V10t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V7a_2E0,V10t2_2E0))
=> p(c_2Elbtree_2Emem_2E2(A_27a,V7a_2E0,c_2Elbtree_2ENd_2E3(A_27a,V8b_2E0,V9t1_2E0,V10t2_2E0))) ) ) ).
tff(thm_2Elbtree_2Emem__ind,axiom,
! [A_27a: $tType,V0mem_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool))] :
( ( ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V1a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0)))
& ! [V4a_2E0: A_27a,V5b_2E0: A_27a,V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V7t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V4a_2E0),V6t1_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V4a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V5b_2E0,V6t1_2E0,V7t2_2E0))) )
& ! [V8a_2E0: A_27a,V9b_2E0: A_27a,V10t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V11t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V8a_2E0),V11t2_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V8a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V9b_2E0,V10t1_2E0,V11t2_2E0))) ) )
=> ! [V12a0_2E0: A_27a,V13a1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V12a0_2E0,V13a1_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V12a0_2E0),V13a1_2E0)) ) ) ).
tff(thm_2Elbtree_2Emem__strongind,axiom,
! [A_27a: $tType,V0mem_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool))] :
( ( ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V1a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0)))
& ! [V4a_2E0: A_27a,V5b_2E0: A_27a,V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V7t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Emem_2E2(A_27a,V4a_2E0,V6t1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V4a_2E0),V6t1_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V4a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V5b_2E0,V6t1_2E0,V7t2_2E0))) )
& ! [V8a_2E0: A_27a,V9b_2E0: A_27a,V10t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V11t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Emem_2E2(A_27a,V8a_2E0,V11t2_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V8a_2E0),V11t2_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V8a_2E0),c_2Elbtree_2ENd_2E3(A_27a,V9b_2E0,V10t1_2E0,V11t2_2E0))) ) )
=> ! [V12a0_2E0: A_27a,V13a1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V12a0_2E0,V13a1_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool),V0mem_27_2E0,V12a0_2E0),V13a1_2E0)) ) ) ).
tff(thm_2Elbtree_2Emem__cases,axiom,
! [A_27a: $tType,V0a0_2E0: A_27a,V1a1_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V0a0_2E0,V1a1_2E0))
<=> ( ? [V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V0a0_2E0,V2t1_2E0,V3t2_2E0) )
| ? [V4b_2E0: A_27a,V5t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V6t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V4b_2E0,V5t1_2E0,V6t2_2E0) )
& p(c_2Elbtree_2Emem_2E2(A_27a,V0a0_2E0,V5t1_2E0)) )
| ? [V7b_2E0: A_27a,V8t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V9t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V7b_2E0,V8t1_2E0,V9t2_2E0) )
& p(c_2Elbtree_2Emem_2E2(A_27a,V0a0_2E0,V9t2_2E0)) ) ) ) ).
tff(thm_2Elbtree_2Emem__thm,axiom,
! [A_27a: $tType,V0t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2b_2E0: A_27a,V3a_2E0: A_27a] :
( ( c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,c_2Elbtree_2ELf_2E0(A_27a)) = c_2Ebool_2EF_2E0 )
& ( p(c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,c_2Elbtree_2ENd_2E3(A_27a,V2b_2E0,V1t1_2E0,V0t2_2E0)))
<=> ( ( V3a_2E0 = V2b_2E0 )
| p(c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,V1t1_2E0))
| p(c_2Elbtree_2Emem_2E2(A_27a,V3a_2E0,V0t2_2E0)) ) ) ) ).
tff(thm_2Elbtree_2Emap__eq__Lf,axiom,
! [A_27a: $tType,A_27b: $tType,V0t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
( ( ( c_2Elbtree_2Emap_2E2(A_27b,A_27a,V1f_2E0,V0t_2E0) = c_2Elbtree_2ELf_2E0(A_27a) )
<=> ( V0t_2E0 = c_2Elbtree_2ELf_2E0(A_27b) ) )
& ( ( c_2Elbtree_2ELf_2E0(A_27a) = c_2Elbtree_2Emap_2E2(A_27b,A_27a,V1f_2E0,V0t_2E0) )
<=> ( V0t_2E0 = c_2Elbtree_2ELf_2E0(A_27b) ) ) ) ).
tff(thm_2Elbtree_2Emap__eq__Nd,axiom,
! [A_27a: $tType,A_27b: $tType,V0t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V3f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V4a_2E0: A_27a] :
( ( c_2Elbtree_2Emap_2E2(A_27b,A_27a,V3f_2E0,V2t_2E0) = c_2Elbtree_2ENd_2E3(A_27a,V4a_2E0,V1t1_2E0,V0t2_2E0) )
<=> ? [V5a_27_2E0: A_27b,V6t1_27_2E0: tyop_2Elbtree_2Elbtree(A_27b),V7t2_27_2E0: tyop_2Elbtree_2Elbtree(A_27b)] :
( ( V2t_2E0 = c_2Elbtree_2ENd_2E3(A_27b,V5a_27_2E0,V6t1_27_2E0,V7t2_27_2E0) )
& ( V4a_2E0 = app_2E2(A_27b,A_27a,V3f_2E0,V5a_27_2E0) )
& ( V1t1_2E0 = c_2Elbtree_2Emap_2E2(A_27b,A_27a,V3f_2E0,V6t1_27_2E0) )
& ( V0t2_2E0 = c_2Elbtree_2Emap_2E2(A_27b,A_27a,V3f_2E0,V7t2_27_2E0) ) ) ) ).
tff(thm_2Elbtree_2Efinite__rules,axiom,
! [A_27a: $tType] :
( p(c_2Elbtree_2Efinite_2E1(A_27a,c_2Elbtree_2ELf_2E0(A_27a)))
& ! [V0a_2E0: A_27a,V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Efinite_2E1(A_27a,V1t1_2E0))
& p(c_2Elbtree_2Efinite_2E1(A_27a,V2t2_2E0)) )
=> p(c_2Elbtree_2Efinite_2E1(A_27a,c_2Elbtree_2ENd_2E3(A_27a,V0a_2E0,V1t1_2E0,V2t2_2E0))) ) ) ).
tff(thm_2Elbtree_2Efinite__ind,axiom,
! [A_27a: $tType,V0finite_27_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,c_2Elbtree_2ELf_2E0(A_27a)))
& ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V2t1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V3t2_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0))) ) )
=> ! [V4a0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Efinite_2E1(A_27a,V4a0_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V4a0_2E0)) ) ) ).
tff(thm_2Elbtree_2Efinite__strongind,axiom,
! [A_27a: $tType,V0finite_27_2E0: tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,c_2Elbtree_2ELf_2E0(A_27a)))
& ! [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Efinite_2E1(A_27a,V2t1_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V2t1_2E0))
& p(c_2Elbtree_2Efinite_2E1(A_27a,V3t2_2E0))
& p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V3t2_2E0)) )
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0))) ) )
=> ! [V4a0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Efinite_2E1(A_27a,V4a0_2E0))
=> p(app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Ebool,V0finite_27_2E0,V4a0_2E0)) ) ) ).
tff(thm_2Elbtree_2Efinite__cases,axiom,
! [A_27a: $tType,V0a0_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Efinite_2E1(A_27a,V0a0_2E0))
<=> ( ( V0a0_2E0 = c_2Elbtree_2ELf_2E0(A_27a) )
| ? [V1a_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V0a0_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V1a_2E0,V2t1_2E0,V3t2_2E0) )
& p(c_2Elbtree_2Efinite_2E1(A_27a,V2t1_2E0))
& p(c_2Elbtree_2Efinite_2E1(A_27a,V3t2_2E0)) ) ) ) ).
tff(thm_2Elbtree_2Efinite__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0t2_2E0: tyop_2Elbtree_2Elbtree(A_27b),V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27b),V2a_2E0: A_27b] :
( ( c_2Elbtree_2Efinite_2E1(A_27a,c_2Elbtree_2ELf_2E0(A_27a)) = c_2Ebool_2ET_2E0 )
& ( p(c_2Elbtree_2Efinite_2E1(A_27b,c_2Elbtree_2ENd_2E3(A_27b,V2a_2E0,V1t1_2E0,V0t2_2E0)))
<=> ( p(c_2Elbtree_2Efinite_2E1(A_27b,V1t1_2E0))
& p(c_2Elbtree_2Efinite_2E1(A_27b,V0t2_2E0)) ) ) ) ).
tff(thm_2Elbtree_2Efinite__map,axiom,
! [A_27a: $tType,A_27b: $tType,V0t_2E0: tyop_2Elbtree_2Elbtree(A_27b),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] : ( c_2Elbtree_2Efinite_2E1(A_27a,c_2Elbtree_2Emap_2E2(A_27b,A_27a,V1f_2E0,V0t_2E0)) = c_2Elbtree_2Efinite_2E1(A_27b,V0t_2E0) ) ).
tff(thm_2Elbtree_2Ebf__flatten__eq__lnil,axiom,
! [A_27a: $tType,V0l_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] :
( ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,V0l_2E0) = c_2Ellist_2ELNIL_2E0(A_27a) )
<=> p(c_2Elist_2EEVERY_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Emin_2E_3D_2E1(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2ELf_2E0(A_27a)),V0l_2E0)) ) ).
tff(thm_2Elbtree_2Ebf__flatten__append,axiom,
! [A_27a: $tType,V0l2_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a)),V1l1_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] :
( p(c_2Elist_2EEVERY_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Emin_2E_3D_2E1(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2ELf_2E0(A_27a)),V1l1_2E0))
=> ( c_2Elbtree_2Ebf__flatten_2E1(A_27a,c_2Elist_2EAPPEND_2E2(tyop_2Elbtree_2Elbtree(A_27a),V1l1_2E0,V0l2_2E0)) = c_2Elbtree_2Ebf__flatten_2E1(A_27a,V0l2_2E0) ) ) ).
tff(thm_2Elbtree_2EEXISTS__FIRST,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Elist_2Elist(A_27a)] :
( p(c_2Elist_2EEXISTS_2E2(A_27a,V0P_2E0,V1l_2E0))
=> ? [V2l1_2E0: tyop_2Elist_2Elist(A_27a),V3x_2E0: A_27a,V4l2_2E0: tyop_2Elist_2Elist(A_27a)] :
( ( V1l_2E0 = c_2Elist_2EAPPEND_2E2(A_27a,V2l1_2E0,c_2Elist_2ECONS_2E2(A_27a,V3x_2E0,V4l2_2E0)) )
& p(c_2Elist_2EEVERY_2E2(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,V0P_2E0),V2l1_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) ) ) ).
tff(thm_2Elbtree_2Eexists__bf__flatten,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1tlist_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] :
( p(c_2Ellist_2Eexists_2E2(A_27a,c_2Emin_2E_3D_2E1(A_27a,V0x_2E0),c_2Elbtree_2Ebf__flatten_2E1(A_27a,V1tlist_2E0)))
=> p(c_2Elist_2EEXISTS_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2Emem_2E1(A_27a,V0x_2E0),V1tlist_2E0)) ) ).
tff(thm_2Elbtree_2Edepth__rules,axiom,
! [A_27a: $tType] :
( ! [V0x_2E0: A_27a,V1t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,c_2Elbtree_2ENd_2E3(A_27a,V0x_2E0,V1t1_2E0,V2t2_2E0),c_2Enum_2E0_2E0))
& ! [V3m_2E0: tyop_2Enum_2Enum,V4x_2E0: A_27a,V5a_2E0: A_27a,V6t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V7t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V4x_2E0,V6t1_2E0,V3m_2E0))
=> p(c_2Elbtree_2Edepth_2E3(A_27a,V4x_2E0,c_2Elbtree_2ENd_2E3(A_27a,V5a_2E0,V6t1_2E0,V7t2_2E0),c_2Enum_2ESUC_2E1(V3m_2E0))) )
& ! [V8m_2E0: tyop_2Enum_2Enum,V9x_2E0: A_27a,V10a_2E0: A_27a,V11t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V12t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V9x_2E0,V12t2_2E0,V8m_2E0))
=> p(c_2Elbtree_2Edepth_2E3(A_27a,V9x_2E0,c_2Elbtree_2ENd_2E3(A_27a,V10a_2E0,V11t1_2E0,V12t2_2E0),c_2Enum_2ESUC_2E1(V8m_2E0))) ) ) ).
tff(thm_2Elbtree_2Edepth__ind,axiom,
! [A_27a: $tType,V0depth_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
( ( ! [V1x_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V1x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V1x_2E0,V2t1_2E0,V3t2_2E0)),c_2Enum_2E0_2E0))
& ! [V4m_2E0: tyop_2Enum_2Enum,V5x_2E0: A_27a,V6a_2E0: A_27a,V7t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V5x_2E0),V7t1_2E0),V4m_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V5x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V6a_2E0,V7t1_2E0,V8t2_2E0)),c_2Enum_2ESUC_2E1(V4m_2E0))) )
& ! [V9m_2E0: tyop_2Enum_2Enum,V10x_2E0: A_27a,V11a_2E0: A_27a,V12t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V13t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V10x_2E0),V13t2_2E0),V9m_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V10x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V11a_2E0,V12t1_2E0,V13t2_2E0)),c_2Enum_2ESUC_2E1(V9m_2E0))) ) )
=> ! [V14a0_2E0: A_27a,V15a1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V16a2_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V14a0_2E0,V15a1_2E0,V16a2_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V14a0_2E0),V15a1_2E0),V16a2_2E0)) ) ) ).
tff(thm_2Elbtree_2Edepth__strongind,axiom,
! [A_27a: $tType,V0depth_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
( ( ! [V1x_2E0: A_27a,V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V1x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V1x_2E0,V2t1_2E0,V3t2_2E0)),c_2Enum_2E0_2E0))
& ! [V4m_2E0: tyop_2Enum_2Enum,V5x_2E0: A_27a,V6a_2E0: A_27a,V7t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Edepth_2E3(A_27a,V5x_2E0,V7t1_2E0,V4m_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V5x_2E0),V7t1_2E0),V4m_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V5x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V6a_2E0,V7t1_2E0,V8t2_2E0)),c_2Enum_2ESUC_2E1(V4m_2E0))) )
& ! [V9m_2E0: tyop_2Enum_2Enum,V10x_2E0: A_27a,V11a_2E0: A_27a,V12t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V13t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( p(c_2Elbtree_2Edepth_2E3(A_27a,V10x_2E0,V13t2_2E0,V9m_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V10x_2E0),V13t2_2E0),V9m_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V10x_2E0),c_2Elbtree_2ENd_2E3(A_27a,V11a_2E0,V12t1_2E0,V13t2_2E0)),c_2Enum_2ESUC_2E1(V9m_2E0))) ) )
=> ! [V14a0_2E0: A_27a,V15a1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V16a2_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V14a0_2E0,V15a1_2E0,V16a2_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elbtree_2Elbtree(A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),V0depth_27_2E0,V14a0_2E0),V15a1_2E0),V16a2_2E0)) ) ) ).
tff(thm_2Elbtree_2Edepth__cases,axiom,
! [A_27a: $tType,V0a0_2E0: A_27a,V1a1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2a2_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V0a0_2E0,V1a1_2E0,V2a2_2E0))
<=> ( ? [V3t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V4t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V0a0_2E0,V3t1_2E0,V4t2_2E0) )
& ( V2a2_2E0 = c_2Enum_2E0_2E0 ) )
| ? [V5m_2E0: tyop_2Enum_2Enum,V6a_2E0: A_27a,V7t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V8t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V6a_2E0,V7t1_2E0,V8t2_2E0) )
& ( V2a2_2E0 = c_2Enum_2ESUC_2E1(V5m_2E0) )
& p(c_2Elbtree_2Edepth_2E3(A_27a,V0a0_2E0,V7t1_2E0,V5m_2E0)) )
| ? [V9m_2E0: tyop_2Enum_2Enum,V10a_2E0: A_27a,V11t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V12t2_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( ( V1a1_2E0 = c_2Elbtree_2ENd_2E3(A_27a,V10a_2E0,V11t1_2E0,V12t2_2E0) )
& ( V2a2_2E0 = c_2Enum_2ESUC_2E1(V9m_2E0) )
& p(c_2Elbtree_2Edepth_2E3(A_27a,V0a0_2E0,V12t2_2E0,V9m_2E0)) ) ) ) ).
tff(thm_2Elbtree_2Emem__depth,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V0x_2E0,V1t_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,V1t_2E0,V2n_2E0)) ) ).
tff(thm_2Elbtree_2Edepth__mem,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,V1t_2E0,V2n_2E0))
=> p(c_2Elbtree_2Emem_2E2(A_27a,V0x_2E0,V1t_2E0)) ) ).
tff(thm_2Elbtree_2Eoptmin__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool))] :
( ( p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V0P_2E0,c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)))
& ! [V1x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V0P_2E0,c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V1x_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)))
& ! [V2y_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V0P_2E0,c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V2y_2E0)))
& ! [V3x_2E0: tyop_2Enum_2Enum,V4y_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V0P_2E0,c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V3x_2E0)),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V4y_2E0))) )
=> ! [V5v_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),V6v1_2E0: tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)] : p(app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),V0P_2E0,V5v_2E0),V6v1_2E0)) ) ).
tff(thm_2Elbtree_2Eoptmin__def,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] :
( ( c_2Elbtree_2Eoptmin_2E2(c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)) = c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum) )
& ( c_2Elbtree_2Eoptmin_2E2(c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V1x_2E0),c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V1x_2E0) )
& ( c_2Elbtree_2Eoptmin_2E2(c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0y_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0y_2E0) )
& ( c_2Elbtree_2Eoptmin_2E2(c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V1x_2E0),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V0y_2E0)) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,c_2Earithmetic_2EMIN_2E2(V1x_2E0,V0y_2E0)) ) ) ).
tff(thm_2Elbtree_2Emindepth__thm,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: A_27a,V3a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V3a_2E0))
<=> ( V0x_2E0 = V3a_2E0 ) )
=> ! [V0x_2E0: A_27a,V1t2_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2t1_2E0: tyop_2Elbtree_2Elbtree(A_27a),V3a_2E0: A_27a] :
( ( c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,c_2Elbtree_2ELf_2E0(A_27a)) = c_2Eoption_2ENONE_2E0(tyop_2Enum_2Enum) )
& ( c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,c_2Elbtree_2ENd_2E3(A_27a,V3a_2E0,V2t1_2E0,V1t2_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V3a_2E0),c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),c_2Eoption_2EOPTION__MAP_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,c_2Elbtree_2Eoptmin_2E2(c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,V2t1_2E0),c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,V1t2_2E0)))) ) ) ) ).
tff(thm_2Elbtree_2Emem__mindepth,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a)] :
( p(c_2Elbtree_2Emem_2E2(A_27a,V0x_2E0,V1t_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] : ( c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,V1t_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V2n_2E0) ) ) ).
tff(thm_2Elbtree_2Emindepth__depth,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1t_2E0: tyop_2Elbtree_2Elbtree(A_27a),V2n_2E0: tyop_2Enum_2Enum] :
( ( c_2Elbtree_2Emindepth_2E2(A_27a,V0x_2E0,V1t_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V2n_2E0) )
=> p(c_2Elbtree_2Edepth_2E3(A_27a,V0x_2E0,V1t_2E0,V2n_2E0)) ) ).
tff(thm_2Elbtree_2Emmindex__EXISTS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),V2e_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1f_2E0),V2e_2E0))
<=> ? [V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),V1f_2E0,V2e_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Enum_2Enum,V3n_2E0) ) )
=> ! [V0l_2E0: tyop_2Elist_2Elist(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))] :
( p(c_2Elist_2EEXISTS_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1f_2E0),V0l_2E0))
=> ? [V4i_2E0: tyop_2Enum_2Enum,V5m_2E0: tyop_2Enum_2Enum] : p(c_2Elbtree_2Eis__mmindex_2E4(A_27a,V1f_2E0,V0l_2E0,V4i_2E0,V5m_2E0)) ) ) ).
tff(thm_2Elbtree_2Emmindex__unique,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Enum_2Enum,V1l_2E0: tyop_2Elist_2Elist(A_27a),V2i_2E0: tyop_2Enum_2Enum,V3f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))] :
( p(c_2Elbtree_2Eis__mmindex_2E4(A_27a,V3f_2E0,V1l_2E0,V2i_2E0,V0m_2E0))
=> ! [V4j_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Elbtree_2Eis__mmindex_2E4(A_27a,V3f_2E0,V1l_2E0,V4j_2E0,V5n_2E0))
<=> ( ( V4j_2E0 = V2i_2E0 )
& ( V5n_2E0 = V0m_2E0 ) ) ) ) ).
tff(thm_2Elbtree_2Emem__bf__flatten,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1tlist_2E0: tyop_2Elist_2Elist(tyop_2Elbtree_2Elbtree(A_27a))] : ( c_2Ellist_2Eexists_2E2(A_27a,c_2Emin_2E_3D_2E1(A_27a,V0x_2E0),c_2Elbtree_2Ebf__flatten_2E1(A_27a,V1tlist_2E0)) = c_2Elist_2EEXISTS_2E2(tyop_2Elbtree_2Elbtree(A_27a),c_2Elbtree_2Emem_2E1(A_27a,V0x_2E0),V1tlist_2E0) ) ).
%------------------------------------------------------------------------------