ITP001 Axioms: ITP102^7.ax
%------------------------------------------------------------------------------
% File : ITP102^7 : 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 : sptree.ax [Gau19]
% : HL4102^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 295 ( 87 unt; 118 typ; 0 def)
% Number of atoms : 684 ( 352 equ; 31 cnn)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 6166 ( 31 ~; 10 |; 232 &;5794 @)
% ( 40 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg;5794 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 437 ( 437 >; 0 *; 0 +; 0 <<)
% Number of symbols : 117 ( 115 usr; 6 con; 0-7 aty)
% Number of variables : 1236 ( 152 ^ 954 !; 17 ?;1236 :)
% ( 113 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esptree_2Espt,type,
tyop_2Esptree_2Espt: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Ealist_2EALOOKUP,type,
c_2Ealist_2EALOOKUP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) > A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Esptree_2EBN,type,
c_2Esptree_2EBN:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Esptree_2EBS,type,
c_2Esptree_2EBS:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Elist_2EEL,type,
c_2Elist_2EEL:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Eind__type_2EFCONS,type,
c_2Eind__type_2EFCONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Esptree_2ELN,type,
c_2Esptree_2ELN:
!>[A_27a: $tType] : ( tyop_2Esptree_2Espt @ A_27a ) ).
thf(c_2Esptree_2ELS,type,
c_2Esptree_2ELS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Esptree_2Edelete,type,
c_2Esptree_2Edelete:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Edifference,type,
c_2Esptree_2Edifference:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27b ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Edomain,type,
c_2Esptree_2Edomain:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Esptree_2Efilter__v,type,
c_2Esptree_2Efilter__v:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Efoldi,type,
c_2Esptree_2Efoldi:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Enum_2Enum > A_27b > A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a > ( tyop_2Esptree_2Espt @ A_27b ) > A_27a ) ).
thf(c_2Esptree_2EfromAList,type,
c_2Esptree_2EfromAList:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2EfromList,type,
c_2Esptree_2EfromList:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Einsert,type,
c_2Esptree_2Einsert:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Einter,type,
c_2Esptree_2Einter:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27b ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Einter__eq,type,
c_2Esptree_2Einter__eq:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Elist_2Elist__CASE,type,
c_2Elist_2Elist__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27b > ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Esptree_2Elist__insert,type,
c_2Esptree_2Elist__insert: ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) > ( tyop_2Esptree_2Espt @ tyop_2Eone_2Eone ) > ( tyop_2Esptree_2Espt @ tyop_2Eone_2Eone ) ).
thf(c_2Esptree_2Elist__to__num__set,type,
c_2Esptree_2Elist__to__num__set: ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) > ( tyop_2Esptree_2Espt @ tyop_2Eone_2Eone ) ).
thf(c_2Esptree_2Elookup,type,
c_2Esptree_2Elookup:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Esptree_2Elrnext,type,
c_2Esptree_2Elrnext: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Esptree_2Emap,type,
c_2Esptree_2Emap:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > ( tyop_2Esptree_2Espt @ A_27b ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Emapi,type,
c_2Esptree_2Emapi:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Enum_2Enum > A_27b > A_27a ) > ( tyop_2Esptree_2Espt @ A_27b ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Emapi0,type,
c_2Esptree_2Emapi0:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Enum_2Enum > A_27b > A_27a ) > tyop_2Enum_2Enum > ( tyop_2Esptree_2Espt @ A_27b ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Emk__BN,type,
c_2Esptree_2Emk__BN:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Emk__BS,type,
c_2Esptree_2Emk__BS:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Emk__wf,type,
c_2Esptree_2Emk__wf:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Eone_2Eone,type,
c_2Eone_2Eone: tyop_2Eone_2Eone ).
thf(c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > ( A_27b > A_27c > A_27a ) > A_27a ) ).
thf(c_2Esptree_2Esize,type,
c_2Esptree_2Esize:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Esptree_2Espt__CASE,type,
c_2Esptree_2Espt__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > A_27b > ( A_27a > A_27b ) > ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b ) > ( ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Esptree_2Espt__acc,type,
c_2Esptree_2Espt__acc: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Esptree_2Espt__center,type,
c_2Esptree_2Espt__center:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Esptree_2Espt__fold,type,
c_2Esptree_2Espt__fold:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27b ) > A_27a ) ).
thf(c_2Esptree_2Espt__left,type,
c_2Esptree_2Espt__left:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Espt__right,type,
c_2Esptree_2Espt__right:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Espt__size,type,
c_2Esptree_2Espt__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Esptree_2Espt @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Esptree_2Esubspt,type,
c_2Esptree_2Esubspt:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > $o ) ).
thf(c_2Esptree_2EtoAList,type,
c_2Esptree_2EtoAList:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) ) ).
thf(c_2Esptree_2EtoList,type,
c_2Esptree_2EtoList:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Esptree_2EtoListA,type,
c_2Esptree_2EtoListA:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Esptree_2Eunion,type,
c_2Esptree_2Eunion:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) ).
thf(c_2Esptree_2Ewf,type,
c_2Esptree_2Ewf:
!>[A_27a: $tType] : ( ( tyop_2Esptree_2Espt @ A_27a ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Esptree_2Efilter__v__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > $o] :
( ( c_2Esptree_2Efilter__v @ A_27a @ V0f @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1f: A_27a > $o,V2x: A_27a] :
( ( c_2Esptree_2Efilter__v @ A_27a @ V1f @ ( c_2Esptree_2ELS @ A_27a @ V2x ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( V1f @ V2x ) @ ( c_2Esptree_2ELS @ A_27a @ V2x ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V3f: A_27a > $o,V4l: tyop_2Esptree_2Espt @ A_27a,V5r: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Efilter__v @ A_27a @ V3f @ ( c_2Esptree_2EBN @ A_27a @ V4l @ V5r ) )
= ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Efilter__v @ A_27a @ V3f @ V4l ) @ ( c_2Esptree_2Efilter__v @ A_27a @ V3f @ V5r ) ) )
& ! [V6f: A_27a > $o,V7l: tyop_2Esptree_2Espt @ A_27a,V8x: A_27a,V9r: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Efilter__v @ A_27a @ V6f @ ( c_2Esptree_2EBS @ A_27a @ V7l @ V8x @ V9r ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( V6f @ V8x ) @ ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Efilter__v @ A_27a @ V6f @ V7l ) @ V8x @ ( c_2Esptree_2Efilter__v @ A_27a @ V6f @ V9r ) ) @ ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Efilter__v @ A_27a @ V6f @ V7l ) @ ( c_2Esptree_2Efilter__v @ A_27a @ V6f @ V9r ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__center__primitive__def,axiom,
! [A_27a: $tType] :
( ( c_2Esptree_2Espt__center @ A_27a )
= ( c_2Erelation_2EWFREC @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > $o )
@ ^ [V0R: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > $o] : ( c_2Erelation_2EWF @ ( tyop_2Esptree_2Espt @ A_27a ) @ V0R ) )
@ ^ [V1spt__center: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ),V2a: tyop_2Esptree_2Espt @ A_27a] :
( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V2a @ ( c_2Ecombin_2EI @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ^ [V3x: A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a @ V3x ) )
@ ^ [V4v7: tyop_2Esptree_2Espt @ A_27a,V5v8: tyop_2Esptree_2Espt @ A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ^ [V6t1: tyop_2Esptree_2Espt @ A_27a,V7x_27: A_27a,V8t2: tyop_2Esptree_2Espt @ A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a @ V7x_27 ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__right__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Espt__right @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: A_27a] :
( ( c_2Esptree_2Espt__right @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0x ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__right @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
= V2t2 )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4x: A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__right @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V3t1 @ V4x @ V5t2 ) )
= V5t2 ) ) ).
thf(thm_2Esptree_2Espt__left__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Espt__left @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: A_27a] :
( ( c_2Esptree_2Espt__left @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0x ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__left @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
= V1t1 )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4x: A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__left @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V3t1 @ V4x @ V5t2 ) )
= V3t1 ) ) ).
thf(thm_2Esptree_2Emap__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27b > A_27a] :
( ( c_2Esptree_2Emap @ A_27a @ A_27b @ V0f @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1f: A_27b > A_27a,V2a: A_27b] :
( ( c_2Esptree_2Emap @ A_27a @ A_27b @ V1f @ ( c_2Esptree_2ELS @ A_27b @ V2a ) )
= ( c_2Esptree_2ELS @ A_27a @ ( V1f @ V2a ) ) )
& ! [V3f: A_27b > A_27a,V4t1: tyop_2Esptree_2Espt @ A_27b,V5t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Emap @ A_27a @ A_27b @ V3f @ ( c_2Esptree_2EBN @ A_27b @ V4t1 @ V5t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Emap @ A_27a @ A_27b @ V3f @ V4t1 ) @ ( c_2Esptree_2Emap @ A_27a @ A_27b @ V3f @ V5t2 ) ) )
& ! [V6f: A_27b > A_27a,V7t1: tyop_2Esptree_2Espt @ A_27b,V8a: A_27b,V9t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Emap @ A_27a @ A_27b @ V6f @ ( c_2Esptree_2EBS @ A_27b @ V7t1 @ V8a @ V9t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Emap @ A_27a @ A_27b @ V6f @ V7t1 ) @ ( V6f @ V8a ) @ ( c_2Esptree_2Emap @ A_27a @ A_27b @ V6f @ V9t2 ) ) ) ) ).
thf(thm_2Esptree_2EfromAList__primitive__def,axiom,
! [A_27a: $tType] :
( ( c_2Esptree_2EfromAList @ A_27a )
= ( c_2Erelation_2EWFREC @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) @ ( tyop_2Esptree_2Espt @ A_27a )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > $o )
@ ^ [V0R: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) @ V0R )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V1y: A_27a] :
( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum] :
( c_2Ebool_2E_21 @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) )
@ ^ [V3xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( V0R @ V3xs @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2x @ V1y ) @ V3xs ) ) ) ) ) ) )
@ ^ [V4fromAList: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Esptree_2Espt @ A_27a ),V5a: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( c_2Elist_2Elist__CASE @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Esptree_2Espt @ A_27a ) @ V5a @ ( c_2Ecombin_2EI @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ^ [V6v: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a,V7xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( c_2Epair_2Epair__CASE @ ( tyop_2Esptree_2Espt @ A_27a ) @ tyop_2Enum_2Enum @ A_27a @ V6v
@ ^ [V8x: tyop_2Enum_2Enum,V9y: A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Esptree_2Einsert @ A_27a @ V8x @ V9y @ ( V4fromAList @ V7xs ) ) ) ) ) ) ) ).
thf(thm_2Esptree_2Emk__wf__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: A_27a] :
( ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0x ) )
= ( c_2Esptree_2ELS @ A_27a @ V0x ) )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
= ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Emk__wf @ A_27a @ V1t1 ) @ ( c_2Esptree_2Emk__wf @ A_27a @ V2t2 ) ) )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4x: A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V3t1 @ V4x @ V5t2 ) )
= ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Emk__wf @ A_27a @ V3t1 ) @ V4x @ ( c_2Esptree_2Emk__wf @ A_27a @ V5t2 ) ) ) ) ).
thf(thm_2Esptree_2EtoList__def,axiom,
! [A_27a: $tType,V0m: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2EtoList @ A_27a @ V0m )
= ( c_2Esptree_2EtoListA @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) @ V0m ) ) ).
thf(thm_2Esptree_2EtoListA__def,axiom,
! [A_27a: $tType] :
( ! [V0acc: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2EtoListA @ A_27a @ V0acc @ ( c_2Esptree_2ELN @ A_27a ) )
= V0acc )
& ! [V1acc: tyop_2Elist_2Elist @ A_27a,V2a: A_27a] :
( ( c_2Esptree_2EtoListA @ A_27a @ V1acc @ ( c_2Esptree_2ELS @ A_27a @ V2a ) )
= ( c_2Elist_2ECONS @ A_27a @ V2a @ V1acc ) )
& ! [V3acc: tyop_2Elist_2Elist @ A_27a,V4t1: tyop_2Esptree_2Espt @ A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2EtoListA @ A_27a @ V3acc @ ( c_2Esptree_2EBN @ A_27a @ V4t1 @ V5t2 ) )
= ( c_2Esptree_2EtoListA @ A_27a @ ( c_2Esptree_2EtoListA @ A_27a @ V3acc @ V5t2 ) @ V4t1 ) )
& ! [V6acc: tyop_2Elist_2Elist @ A_27a,V7t1: tyop_2Esptree_2Espt @ A_27a,V8a: A_27a,V9t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2EtoListA @ A_27a @ V6acc @ ( c_2Esptree_2EBS @ A_27a @ V7t1 @ V8a @ V9t2 ) )
= ( c_2Esptree_2EtoListA @ A_27a @ ( c_2Elist_2ECONS @ A_27a @ V8a @ ( c_2Esptree_2EtoListA @ A_27a @ V6acc @ V9t2 ) ) @ V7t1 ) ) ) ).
thf(thm_2Esptree_2EtoAList__def,axiom,
! [A_27a: $tType] :
( ( c_2Esptree_2EtoAList @ A_27a )
= ( c_2Esptree_2Efoldi @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) @ A_27a
@ ^ [V0k: tyop_2Enum_2Enum,V1v: A_27a,V2a: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V0k @ V1v ) @ V2a )
@ c_2Enum_2E0
@ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(thm_2Esptree_2Emapi__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Enum_2Enum > A_27b > A_27a,V1pt: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Emapi @ A_27a @ A_27b @ V0f @ V1pt )
= ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V0f @ c_2Enum_2E0 @ V1pt ) ) ).
thf(thm_2Esptree_2Emapi0__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: tyop_2Enum_2Enum > A_27b > A_27a,V1i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V0f @ V1i @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V2f: tyop_2Enum_2Enum > A_27b > A_27a,V3i: tyop_2Enum_2Enum,V4a: A_27b] :
( ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V2f @ V3i @ ( c_2Esptree_2ELS @ A_27b @ V4a ) )
= ( c_2Esptree_2ELS @ A_27a @ ( V2f @ V3i @ V4a ) ) )
& ! [V5f: tyop_2Enum_2Enum > A_27b > A_27a,V6i: tyop_2Enum_2Enum,V7t1: tyop_2Esptree_2Espt @ A_27b,V8t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V5f @ V6i @ ( c_2Esptree_2EBN @ A_27b @ V7t1 @ V8t2 ) )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a )
@ ^ [V9inc: tyop_2Enum_2Enum] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V5f @ ( c_2Earithmetic_2E_2B @ V6i @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V9inc ) ) @ V7t1 ) @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V5f @ ( c_2Earithmetic_2E_2B @ V6i @ V9inc ) @ V8t2 ) )
@ ( c_2Esptree_2Elrnext @ V6i ) ) )
& ! [V10f: tyop_2Enum_2Enum > A_27b > A_27a,V11i: tyop_2Enum_2Enum,V12t1: tyop_2Esptree_2Espt @ A_27b,V13a: A_27b,V14t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V10f @ V11i @ ( c_2Esptree_2EBS @ A_27b @ V12t1 @ V13a @ V14t2 ) )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a )
@ ^ [V15inc: tyop_2Enum_2Enum] : ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V10f @ ( c_2Earithmetic_2E_2B @ V11i @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V15inc ) ) @ V12t1 ) @ ( V10f @ V11i @ V13a ) @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b @ V10f @ ( c_2Earithmetic_2E_2B @ V11i @ V15inc ) @ V14t2 ) )
@ ( c_2Esptree_2Elrnext @ V11i ) ) ) ) ).
thf(thm_2Esptree_2Efoldi__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: tyop_2Enum_2Enum > A_27b > A_27a > A_27a,V1i: tyop_2Enum_2Enum,V2acc: A_27a] :
( ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V0f @ V1i @ V2acc @ ( c_2Esptree_2ELN @ A_27b ) )
= V2acc )
& ! [V3f: tyop_2Enum_2Enum > A_27b > A_27a > A_27a,V4i: tyop_2Enum_2Enum,V5acc: A_27a,V6a: A_27b] :
( ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V3f @ V4i @ V5acc @ ( c_2Esptree_2ELS @ A_27b @ V6a ) )
= ( V3f @ V4i @ V6a @ V5acc ) )
& ! [V7f: tyop_2Enum_2Enum > A_27b > A_27a > A_27a,V8i: tyop_2Enum_2Enum,V9acc: A_27a,V10t1: tyop_2Esptree_2Espt @ A_27b,V11t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V7f @ V8i @ V9acc @ ( c_2Esptree_2EBN @ A_27b @ V10t1 @ V11t2 ) )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ A_27a
@ ^ [V12inc: tyop_2Enum_2Enum] : ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V7f @ ( c_2Earithmetic_2E_2B @ V8i @ V12inc ) @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V7f @ ( c_2Earithmetic_2E_2B @ V8i @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V12inc ) ) @ V9acc @ V10t1 ) @ V11t2 )
@ ( c_2Esptree_2Elrnext @ V8i ) ) )
& ! [V13f: tyop_2Enum_2Enum > A_27b > A_27a > A_27a,V14i: tyop_2Enum_2Enum,V15acc: A_27a,V16t1: tyop_2Esptree_2Espt @ A_27b,V17a: A_27b,V18t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V13f @ V14i @ V15acc @ ( c_2Esptree_2EBS @ A_27b @ V16t1 @ V17a @ V18t2 ) )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ A_27a
@ ^ [V19inc: tyop_2Enum_2Enum] : ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V13f @ ( c_2Earithmetic_2E_2B @ V14i @ V19inc ) @ ( V13f @ V14i @ V17a @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b @ V13f @ ( c_2Earithmetic_2E_2B @ V14i @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V19inc ) ) @ V15acc @ V16t1 ) ) @ V18t2 )
@ ( c_2Esptree_2Elrnext @ V14i ) ) ) ) ).
thf(thm_2Esptree_2Espt__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ A_27a )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ ( tyop_2Esptree_2Espt @ A_27a )
@ ^ [V1a0_27: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o )
@ ^ [V2_27spt_27: ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0_27: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( c_2Eind__type_2ECONSTR @ A_27a @ c_2Enum_2E0 @ ( c_2Ebool_2EARB @ A_27a )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V5a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( ^ [V6a: A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ V6a
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) )
@ V5a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V8a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V9a1: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( ^ [V10a0: tyop_2Eind__type_2Erecspace @ A_27a,V11a1: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ A_27a )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V10a0
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V11a1
@ ^ [V12n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V8a0
@ V9a1 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27spt_27 @ V8a0 ) @ ( V2_27spt_27 @ V9a1 ) ) ) ) )
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V13a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V14a1: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V15a2: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0_27
@ ( ^ [V16a0: tyop_2Eind__type_2Erecspace @ A_27a,V17a1: A_27a,V18a2: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ) @ V17a1
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V16a0
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V18a2
@ ^ [V19n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V13a0
@ V14a1
@ V15a2 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27spt_27 @ V13a0 ) @ ( V2_27spt_27 @ V15a2 ) ) ) ) ) ) ) ) )
@ ( V2_27spt_27 @ V3a0_27 ) ) )
@ ( V2_27spt_27 @ V1a0_27 ) ) )
@ V0rep ) ).
thf(thm_2Esptree_2Espt__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v: A_27b,V1f: A_27a > A_27b,V2f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V3f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ ( c_2Esptree_2ELN @ A_27a ) @ V0v @ V1f @ V2f1 @ V3f2 )
= V0v )
& ! [V4a: A_27a,V5v: A_27b,V6f: A_27a > A_27b,V7f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V8f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ ( c_2Esptree_2ELS @ A_27a @ V4a ) @ V5v @ V6f @ V7f1 @ V8f2 )
= ( V6f @ V4a ) )
& ! [V9a0: tyop_2Esptree_2Espt @ A_27a,V10a1: tyop_2Esptree_2Espt @ A_27a,V11v: A_27b,V12f: A_27a > A_27b,V13f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V14f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ ( c_2Esptree_2EBN @ A_27a @ V9a0 @ V10a1 ) @ V11v @ V12f @ V13f1 @ V14f2 )
= ( V13f1 @ V9a0 @ V10a1 ) )
& ! [V15a0: tyop_2Esptree_2Espt @ A_27a,V16a1: A_27a,V17a2: tyop_2Esptree_2Espt @ A_27a,V18v: A_27b,V19f: A_27a > A_27b,V20f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V21f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ ( c_2Esptree_2EBS @ A_27a @ V15a0 @ V16a1 @ V17a2 ) @ V18v @ V19f @ V20f1 @ V21f2 )
= ( V21f2 @ V15a0 @ V16a1 @ V17a2 ) ) ) ).
thf(thm_2Esptree_2Espt__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__size @ A_27a @ V0f @ ( c_2Esptree_2ELN @ A_27a ) )
= c_2Enum_2E0 )
& ! [V1f: A_27a > tyop_2Enum_2Enum,V2a: A_27a] :
( ( c_2Esptree_2Espt__size @ A_27a @ V1f @ ( c_2Esptree_2ELS @ A_27a @ V2a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V1f @ V2a ) ) )
& ! [V3f: A_27a > tyop_2Enum_2Enum,V4a0: tyop_2Esptree_2Espt @ A_27a,V5a1: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__size @ A_27a @ V3f @ ( c_2Esptree_2EBN @ A_27a @ V4a0 @ V5a1 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Esptree_2Espt__size @ A_27a @ V3f @ V4a0 ) @ ( c_2Esptree_2Espt__size @ A_27a @ V3f @ V5a1 ) ) ) )
& ! [V6f: A_27a > tyop_2Enum_2Enum,V7a0: tyop_2Esptree_2Espt @ A_27a,V8a1: A_27a,V9a2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Espt__size @ A_27a @ V6f @ ( c_2Esptree_2EBS @ A_27a @ V7a0 @ V8a1 @ V9a2 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Esptree_2Espt__size @ A_27a @ V6f @ V7a0 ) @ ( c_2Earithmetic_2E_2B @ ( V6f @ V8a1 ) @ ( c_2Esptree_2Espt__size @ A_27a @ V6f @ V9a2 ) ) ) ) ) ) ).
thf(thm_2Esptree_2Ewf__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= c_2Ebool_2ET )
& ! [V0a: A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0a ) )
= c_2Ebool_2ET )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
<=> ( ( c_2Esptree_2Ewf @ A_27a @ V1t1 )
& ( c_2Esptree_2Ewf @ A_27a @ V2t2 )
& ( (~)
@ ( ( V1t1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V2t2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4a: A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V3t1 @ V4a @ V5t2 ) )
<=> ( ( c_2Esptree_2Ewf @ A_27a @ V3t1 )
& ( c_2Esptree_2Ewf @ A_27a @ V5t2 )
& ( (~)
@ ( ( V3t1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V5t2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ) ).
thf(thm_2Esptree_2Edelete__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Edelete @ A_27a @ V0k @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1k: tyop_2Enum_2Enum,V2a: A_27a] :
( ( c_2Esptree_2Edelete @ A_27a @ V1k @ ( c_2Esptree_2ELS @ A_27a @ V2a ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ c_2Enum_2E0 ) @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELS @ A_27a @ V2a ) ) )
& ! [V3k: tyop_2Enum_2Enum,V4t1: tyop_2Esptree_2Espt @ A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edelete @ A_27a @ V3k @ ( c_2Esptree_2EBN @ A_27a @ V4t1 @ V5t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3k @ c_2Enum_2E0 ) @ ( c_2Esptree_2EBN @ A_27a @ V4t1 @ V5t2 ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V3k ) @ ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V4t1 ) @ V5t2 ) @ ( c_2Esptree_2Emk__BN @ A_27a @ V4t1 @ ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V5t2 ) ) ) ) )
& ! [V6k: tyop_2Enum_2Enum,V7t1: tyop_2Esptree_2Espt @ A_27a,V8a: A_27a,V9t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edelete @ A_27a @ V6k @ ( c_2Esptree_2EBS @ A_27a @ V7t1 @ V8a @ V9t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V6k @ c_2Enum_2E0 ) @ ( c_2Esptree_2EBN @ A_27a @ V7t1 @ V9t2 ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V6k ) @ ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V7t1 ) @ V8a @ V9t2 ) @ ( c_2Esptree_2Emk__BS @ A_27a @ V7t1 @ V8a @ ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V9t2 ) ) ) ) ) ) ).
thf(thm_2Esptree_2EfromList__def,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2EfromList @ A_27a @ V0l )
= ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a )
@ ( c_2Elist_2EFOLDL @ A_27a @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) )
@ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( A_27a > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) ) )
@ ^ [V1i: tyop_2Enum_2Enum,V2t: tyop_2Esptree_2Espt @ A_27a,V3a: A_27a] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2E_2B @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Esptree_2Einsert @ A_27a @ V1i @ V3a @ V2t ) ) )
@ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) @ c_2Enum_2E0 @ ( c_2Esptree_2ELN @ A_27a ) )
@ V0l ) ) ) ).
thf(thm_2Esptree_2Esize__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= c_2Enum_2E0 )
& ! [V0a: A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0a ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Esptree_2Esize @ A_27a @ V1t1 ) @ ( c_2Esptree_2Esize @ A_27a @ V2t2 ) ) )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4a: A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V3t1 @ V4a @ V5t2 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ ( c_2Esptree_2Esize @ A_27a @ V3t1 ) @ ( c_2Esptree_2Esize @ A_27a @ V5t2 ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Esptree_2Eunion__def,axiom,
! [A_27a: $tType] :
( ! [V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= V0t )
& ! [V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ V2t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V2t @ ( c_2Esptree_2ELS @ A_27a @ V1a )
@ ^ [V3b: A_27a] : ( c_2Esptree_2ELS @ A_27a @ V1a )
@ ^ [V4t1: tyop_2Esptree_2Espt @ A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBS @ A_27a @ V4t1 @ V1a @ V5t2 )
@ ^ [V6t1_27: tyop_2Esptree_2Espt @ A_27a,V7v4: A_27a,V8t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBS @ A_27a @ V6t1_27 @ V1a @ V8t2_27 ) ) )
& ! [V9t1: tyop_2Esptree_2Espt @ A_27a,V10t2: tyop_2Esptree_2Espt @ A_27a,V11t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 ) @ V11t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V11t @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 )
@ ^ [V12a: A_27a] : ( c_2Esptree_2EBS @ A_27a @ V9t1 @ V12a @ V10t2 )
@ ^ [V13t1_27: tyop_2Esptree_2Espt @ A_27a,V14t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V9t1 @ V13t1_27 ) @ ( c_2Esptree_2Eunion @ A_27a @ V10t2 @ V14t2_27 ) )
@ ^ [V15t1_27_27: tyop_2Esptree_2Espt @ A_27a,V16a_27_27: A_27a,V17t2_27_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V9t1 @ V15t1_27_27 ) @ V16a_27_27 @ ( c_2Esptree_2Eunion @ A_27a @ V10t2 @ V17t2_27_27 ) ) ) )
& ! [V18t1: tyop_2Esptree_2Espt @ A_27a,V19a: A_27a,V20t2: tyop_2Esptree_2Espt @ A_27a,V21t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 ) @ V21t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V21t @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 )
@ ^ [V22a_27: A_27a] : ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 )
@ ^ [V23t1_27: tyop_2Esptree_2Espt @ A_27a,V24t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V18t1 @ V23t1_27 ) @ V19a @ ( c_2Esptree_2Eunion @ A_27a @ V20t2 @ V24t2_27 ) )
@ ^ [V25t1_27_27: tyop_2Esptree_2Espt @ A_27a,V26a_27_27_27: A_27a,V27t2_27_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V18t1 @ V25t1_27_27 ) @ V19a @ ( c_2Esptree_2Eunion @ A_27a @ V20t2 @ V27t2_27_27 ) ) ) ) ) ).
thf(thm_2Esptree_2Einter__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ V2t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V2t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V3b: A_27b] : ( c_2Esptree_2ELS @ A_27a @ V1a )
@ ^ [V4t1: tyop_2Esptree_2Espt @ A_27b,V5t2: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2ELN @ A_27a )
@ ^ [V6t1_27: tyop_2Esptree_2Espt @ A_27b,V7v4: A_27b,V8t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2ELS @ A_27a @ V1a ) ) )
& ! [V9t1: tyop_2Esptree_2Espt @ A_27a,V10t2: tyop_2Esptree_2Espt @ A_27a,V11t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 ) @ V11t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V11t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V12a: A_27b] : ( c_2Esptree_2ELN @ A_27a )
@ ^ [V13t1_27: tyop_2Esptree_2Espt @ A_27b,V14t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V9t1 @ V13t1_27 ) @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V10t2 @ V14t2_27 ) )
@ ^ [V15t1_27_27: tyop_2Esptree_2Espt @ A_27b,V16a_27_27: A_27b,V17t2_27_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V9t1 @ V15t1_27_27 ) @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V10t2 @ V17t2_27_27 ) ) ) )
& ! [V18t1: tyop_2Esptree_2Espt @ A_27a,V19a: A_27a,V20t2: tyop_2Esptree_2Espt @ A_27a,V21t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 ) @ V21t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V21t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V22a_27: A_27b] : ( c_2Esptree_2ELS @ A_27a @ V19a )
@ ^ [V23t1_27: tyop_2Esptree_2Espt @ A_27b,V24t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V18t1 @ V23t1_27 ) @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V20t2 @ V24t2_27 ) )
@ ^ [V25t1_27_27: tyop_2Esptree_2Espt @ A_27b,V26a_27_27_27: A_27b,V27t2_27_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V18t1 @ V25t1_27_27 ) @ V19a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V20t2 @ V27t2_27_27 ) ) ) ) ) ).
thf(thm_2Esptree_2Einter__eq__def,axiom,
! [A_27a: $tType] :
( ! [V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einter__eq @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einter__eq @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ V2t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V2t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V3b: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V1a @ V3b ) @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ^ [V4t1: tyop_2Esptree_2Espt @ A_27a,V5t2: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2ELN @ A_27a )
@ ^ [V6t1_27: tyop_2Esptree_2Espt @ A_27a,V7b_27: A_27a,V8t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V1a @ V7b_27 ) @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ! [V9t1: tyop_2Esptree_2Espt @ A_27a,V10t2: tyop_2Esptree_2Espt @ A_27a,V11t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einter__eq @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 ) @ V11t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V11t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V12a: A_27a] : ( c_2Esptree_2ELN @ A_27a )
@ ^ [V13t1_27: tyop_2Esptree_2Espt @ A_27a,V14t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter__eq @ A_27a @ V9t1 @ V13t1_27 ) @ ( c_2Esptree_2Einter__eq @ A_27a @ V10t2 @ V14t2_27 ) )
@ ^ [V15t1_27_27: tyop_2Esptree_2Espt @ A_27a,V16a_27_27: A_27a,V17t2_27_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter__eq @ A_27a @ V9t1 @ V15t1_27_27 ) @ ( c_2Esptree_2Einter__eq @ A_27a @ V10t2 @ V17t2_27_27 ) ) ) )
& ! [V18t1: tyop_2Esptree_2Espt @ A_27a,V19a: A_27a,V20t2: tyop_2Esptree_2Espt @ A_27a,V21t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einter__eq @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 ) @ V21t )
= ( c_2Esptree_2Espt__CASE @ A_27a @ ( tyop_2Esptree_2Espt @ A_27a ) @ V21t @ ( c_2Esptree_2ELN @ A_27a )
@ ^ [V22a_27: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V22a_27 @ V19a ) @ ( c_2Esptree_2ELS @ A_27a @ V19a ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ^ [V23t1_27: tyop_2Esptree_2Espt @ A_27a,V24t2_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter__eq @ A_27a @ V18t1 @ V23t1_27 ) @ ( c_2Esptree_2Einter__eq @ A_27a @ V20t2 @ V24t2_27 ) )
@ ^ [V25t1_27_27: tyop_2Esptree_2Espt @ A_27a,V26a_27_27_27: A_27a,V27t2_27_27: tyop_2Esptree_2Espt @ A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V26a_27_27_27 @ V19a ) @ ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Einter__eq @ A_27a @ V18t1 @ V25t1_27_27 ) @ V19a @ ( c_2Esptree_2Einter__eq @ A_27a @ V20t2 @ V27t2_27_27 ) ) @ ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Einter__eq @ A_27a @ V18t1 @ V25t1_27_27 ) @ ( c_2Esptree_2Einter__eq @ A_27a @ V20t2 @ V27t2_27_27 ) ) ) ) ) ) ).
thf(thm_2Esptree_2Edifference__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Edifference @ A_27a @ A_27b @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Edifference @ A_27a @ A_27b @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ V2t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V2t @ ( c_2Esptree_2ELS @ A_27a @ V1a )
@ ^ [V3b: A_27b] : ( c_2Esptree_2ELN @ A_27a )
@ ^ [V4t1: tyop_2Esptree_2Espt @ A_27b,V5t2: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2ELS @ A_27a @ V1a )
@ ^ [V6t1_27: tyop_2Esptree_2Espt @ A_27b,V7b_27: A_27b,V8t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V9t1: tyop_2Esptree_2Espt @ A_27a,V10t2: tyop_2Esptree_2Espt @ A_27a,V11t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Edifference @ A_27a @ A_27b @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 ) @ V11t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V11t @ ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 )
@ ^ [V12a: A_27b] : ( c_2Esptree_2EBN @ A_27a @ V9t1 @ V10t2 )
@ ^ [V13t1_27: tyop_2Esptree_2Espt @ A_27b,V14t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V9t1 @ V13t1_27 ) @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V10t2 @ V14t2_27 ) )
@ ^ [V15t1_27_27: tyop_2Esptree_2Espt @ A_27b,V16a_27_27: A_27b,V17t2_27_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V9t1 @ V15t1_27_27 ) @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V10t2 @ V17t2_27_27 ) ) ) )
& ! [V18t1: tyop_2Esptree_2Espt @ A_27a,V19a: A_27a,V20t2: tyop_2Esptree_2Espt @ A_27a,V21t: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Edifference @ A_27a @ A_27b @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 ) @ V21t )
= ( c_2Esptree_2Espt__CASE @ A_27b @ ( tyop_2Esptree_2Espt @ A_27a ) @ V21t @ ( c_2Esptree_2EBS @ A_27a @ V18t1 @ V19a @ V20t2 )
@ ^ [V22a_27: A_27b] : ( c_2Esptree_2EBN @ A_27a @ V18t1 @ V20t2 )
@ ^ [V23t1_27: tyop_2Esptree_2Espt @ A_27b,V24t2_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V18t1 @ V23t1_27 ) @ V19a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V20t2 @ V24t2_27 ) )
@ ^ [V25t1_27_27: tyop_2Esptree_2Espt @ A_27b,V26a_27_27_27: A_27b,V27t2_27_27: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V18t1 @ V25t1_27_27 ) @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V20t2 @ V27t2_27_27 ) ) ) ) ) ).
thf(thm_2Esptree_2Elrnext__primitive__def,axiom,
( c_2Esptree_2Elrnext
= ( c_2Erelation_2EWFREC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ( c_2Emin_2E_40 @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o )
@ ^ [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ tyop_2Enum_2Enum @ V0R )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1n @ c_2Enum_2E0 ) ) @ ( V0R @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1n ) ) ) ) )
@ ^ [V2lrnext: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] : ( c_2Ecombin_2EI @ tyop_2Enum_2Enum @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3a @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( V2lrnext @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V3a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(thm_2Esptree_2Edomain__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
& ! [V0v0: A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0v0 ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ! [V1t1: tyop_2Esptree_2Espt @ A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V2t2 ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
@ ( c_2Esptree_2Edomain @ A_27a @ V1t1 ) )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V4n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
@ ( c_2Esptree_2Edomain @ A_27a @ V2t2 ) ) ) )
& ! [V5t1: tyop_2Esptree_2Espt @ A_27a,V6v1: A_27a,V7t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V5t1 @ V6v1 @ V7t2 ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V8n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V8n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
@ ( c_2Esptree_2Edomain @ A_27a @ V5t1 ) ) )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V9n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
@ ( c_2Esptree_2Edomain @ A_27a @ V7t2 ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__fold__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27b > A_27a > A_27a,V1acc: A_27a] :
( ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V0f @ V1acc @ ( c_2Esptree_2ELN @ A_27b ) )
= V1acc )
& ! [V2f: A_27b > A_27a > A_27a,V3acc: A_27a,V4a: A_27b] :
( ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V2f @ V3acc @ ( c_2Esptree_2ELS @ A_27b @ V4a ) )
= ( V2f @ V4a @ V3acc ) )
& ! [V5f: A_27b > A_27a > A_27a,V6acc: A_27a,V7t1: tyop_2Esptree_2Espt @ A_27b,V8t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V5f @ V6acc @ ( c_2Esptree_2EBN @ A_27b @ V7t1 @ V8t2 ) )
= ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V5f @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V5f @ V6acc @ V7t1 ) @ V8t2 ) )
& ! [V9f: A_27b > A_27a > A_27a,V10acc: A_27a,V11t1: tyop_2Esptree_2Espt @ A_27b,V12a: A_27b,V13t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V9f @ V10acc @ ( c_2Esptree_2EBS @ A_27b @ V11t1 @ V12a @ V13t2 ) )
= ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V9f @ ( V9f @ V12a @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b @ V9f @ V10acc @ V11t1 ) ) @ V13t2 ) ) ) ).
thf(thm_2Esptree_2Elist__to__num__set__def,axiom,
( ( ( c_2Esptree_2Elist__to__num__set @ ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) )
= ( c_2Esptree_2ELN @ tyop_2Eone_2Eone ) )
& ! [V0n: tyop_2Enum_2Enum,V1ns: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elist__to__num__set @ ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V0n @ V1ns ) )
= ( c_2Esptree_2Einsert @ tyop_2Eone_2Eone @ V0n @ c_2Eone_2Eone @ ( c_2Esptree_2Elist__to__num__set @ V1ns ) ) ) ) ).
thf(thm_2Esptree_2Elist__insert__def,axiom,
( ! [V0t: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( c_2Esptree_2Elist__insert @ ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) @ V0t )
= V0t )
& ! [V1n: tyop_2Enum_2Enum,V2ns: tyop_2Elist_2Elist @ tyop_2Enum_2Enum,V3t: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( c_2Esptree_2Elist__insert @ ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V1n @ V2ns ) @ V3t )
= ( c_2Esptree_2Elist__insert @ V2ns @ ( c_2Esptree_2Einsert @ tyop_2Eone_2Eone @ V1n @ c_2Eone_2Eone @ V3t ) ) ) ) ).
thf(thm_2Esptree_2Elookup__fromList__outside,axiom,
! [A_27a: $tType,V0args: tyop_2Elist_2Elist @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Elist_2ELENGTH @ A_27a @ V0args ) @ V1k )
=> ( ( c_2Esptree_2Elookup @ A_27a @ V1k @ ( c_2Esptree_2EfromList @ A_27a @ V0args ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Esptree_2Esize__delete,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V0n @ V1t ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V0n @ V1t ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Esptree_2Esize @ A_27a @ V1t ) @ ( c_2Earithmetic_2E_2D @ ( c_2Esptree_2Esize @ A_27a @ V1t ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Esptree_2Edelete__fail,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V1t )
=> ( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0n @ ( c_2Esptree_2Edomain @ A_27a @ V1t ) ) )
<=> ( ( c_2Esptree_2Edelete @ A_27a @ V0n @ V1t )
= V1t ) ) ) ).
thf(thm_2Esptree_2Ewf__difference,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27b] :
( ( ( c_2Esptree_2Ewf @ A_27a @ V0t1 )
& ( c_2Esptree_2Ewf @ A_27b @ V1t2 ) )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V0t1 @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Edifference__sub,axiom,
! [A_27a: $tType,A_27b: $tType,V0b: tyop_2Esptree_2Espt @ A_27b,V1a: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V1a @ V0b )
= ( c_2Esptree_2ELN @ A_27a ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V1a ) @ ( c_2Esptree_2Edomain @ A_27b @ V0b ) ) ) ).
thf(thm_2Esptree_2Eunion__num__set__sym,axiom,
! [V0t1: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone,V1t2: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( c_2Esptree_2Eunion @ tyop_2Eone_2Eone @ V0t1 @ V1t2 )
= ( c_2Esptree_2Eunion @ tyop_2Eone_2Eone @ V1t2 @ V0t1 ) ) ).
thf(thm_2Esptree_2Enum__set__domain__eq,axiom,
! [V0t1: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone,V1t2: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( ( c_2Esptree_2Ewf @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Esptree_2Ewf @ tyop_2Eone_2Eone @ V1t2 ) )
=> ( ( ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ V0t1 )
= ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ V1t2 ) )
<=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Esptree_2Esize__domain,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ V0t )
= ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V0t ) ) ) ).
thf(thm_2Esptree_2Edomain__mapi,axiom,
! [A_27a: $tType,A_27b: $tType,V0pt: tyop_2Esptree_2Espt @ A_27b,V1f: tyop_2Enum_2Enum > A_27b > A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Emapi @ A_27a @ A_27b @ V1f @ V0pt ) )
= ( c_2Esptree_2Edomain @ A_27b @ V0pt ) ) ).
thf(thm_2Esptree_2Emapi__Alist,axiom,
! [A_27a: $tType,A_27b: $tType,V0pt: tyop_2Esptree_2Espt @ A_27b,V1f: tyop_2Enum_2Enum > A_27b > A_27a] :
( ( c_2Esptree_2Emapi @ A_27a @ A_27b @ V1f @ V0pt )
= ( c_2Esptree_2EfromAList @ A_27a
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27b ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )
@ ^ [V2kv: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27b] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ A_27b @ V2kv ) @ ( V1f @ ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ A_27b @ V2kv ) @ ( c_2Epair_2ESND @ tyop_2Enum_2Enum @ A_27b @ V2kv ) ) )
@ ( c_2Esptree_2EtoAList @ A_27b @ V0pt ) ) ) ) ).
thf(thm_2Esptree_2EMAP__foldi,axiom,
! [A_27a: $tType,A_27b: $tType,V0pt: tyop_2Esptree_2Espt @ A_27a,V1f: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) > A_27b,V2n: tyop_2Enum_2Enum,V3acc: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ A_27b @ V1f
@ ( c_2Esptree_2Efoldi @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) @ A_27a
@ ^ [V4k: tyop_2Enum_2Enum,V5v: A_27a,V6a: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V4k @ V5v ) @ V6a )
@ V2n
@ V3acc
@ V0pt ) )
= ( c_2Esptree_2Efoldi @ ( tyop_2Elist_2Elist @ A_27b ) @ A_27a
@ ^ [V7k: tyop_2Enum_2Enum,V8v: A_27a,V9a: tyop_2Elist_2Elist @ A_27b] : ( c_2Elist_2ECONS @ A_27b @ ( V1f @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V7k @ V8v ) ) @ V9a )
@ V2n
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ A_27b @ V1f @ V3acc )
@ V0pt ) ) ).
thf(thm_2Esptree_2Elookup__mk__BN,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a,V2i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2i @ ( c_2Esptree_2Emk__BN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2i @ c_2Enum_2E0 ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V2i ) @ V1t1 @ V0t2 ) ) ) ) ).
thf(thm_2Esptree_2Ewf__mapi,axiom,
! [A_27a: $tType,A_27b: $tType,V0pt: tyop_2Esptree_2Espt @ A_27b,V1f: tyop_2Enum_2Enum > A_27b > A_27a] : ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Emapi @ A_27a @ A_27b @ V1f @ V0pt ) ) ).
thf(thm_2Esptree_2Ewf__mk__BS,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a,V2a: A_27a] :
( ( ( c_2Esptree_2Ewf @ A_27a @ V1t1 )
& ( c_2Esptree_2Ewf @ A_27a @ V0t2 ) )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Emk__BS @ A_27a @ V1t1 @ V2a @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Ewf__mk__BN,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Ewf @ A_27a @ V1t1 )
& ( c_2Esptree_2Ewf @ A_27a @ V0t2 ) )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Emk__BN @ A_27a @ V1t1 @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Ewf__filter__v,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1f: A_27a > $o] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Efilter__v @ A_27a @ V1f @ V0t ) ) ) ).
thf(thm_2Esptree_2Elookup__filter__v,axiom,
! [A_27a: $tType,V0k: tyop_2Enum_2Enum,V1t: tyop_2Esptree_2Espt @ A_27a,V2f: A_27a > $o] :
( ( c_2Esptree_2Elookup @ A_27a @ V0k @ ( c_2Esptree_2Efilter__v @ A_27a @ V2f @ V1t ) )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V0k @ V1t ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V3v: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V2f @ V3v ) @ ( c_2Eoption_2ESOME @ A_27a @ V3v ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(thm_2Esptree_2Esubspt__LN,axiom,
! [A_27a: $tType,V0sp: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Esubspt @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0sp )
= c_2Ebool_2ET )
& ( ( c_2Esptree_2Esubspt @ A_27a @ V0sp @ ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( c_2Esptree_2Edomain @ A_27a @ V0sp )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Esptree_2Esubspt__trans,axiom,
! [A_27a: $tType,V0sp3: tyop_2Esptree_2Espt @ A_27a,V1sp2: tyop_2Esptree_2Espt @ A_27a,V2sp1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Esubspt @ A_27a @ V2sp1 @ V1sp2 )
& ( c_2Esptree_2Esubspt @ A_27a @ V1sp2 @ V0sp3 ) )
=> ( c_2Esptree_2Esubspt @ A_27a @ V2sp1 @ V0sp3 ) ) ).
thf(thm_2Esptree_2Esubspt__refl,axiom,
! [A_27a: $tType,V0sp: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2Esubspt @ A_27a @ V0sp @ V0sp ) ).
thf(thm_2Esptree_2Esubspt__def,axiom,
! [A_27a: $tType,V0sp1: tyop_2Esptree_2Espt @ A_27a,V1sp2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ V0sp1 @ V1sp2 )
<=> ! [V2k: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2k @ ( c_2Esptree_2Edomain @ A_27a @ V0sp1 ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2k @ ( c_2Esptree_2Edomain @ A_27a @ V1sp2 ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ V2k @ V1sp2 )
= ( c_2Esptree_2Elookup @ A_27a @ V2k @ V0sp1 ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__center__def,axiom,
! [A_27a: $tType,V0x: A_27a,V1v2: tyop_2Esptree_2Espt @ A_27a,V2v1: tyop_2Esptree_2Espt @ A_27a,V3t2: tyop_2Esptree_2Espt @ A_27a,V4t1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Espt__center @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V0x ) )
= ( c_2Eoption_2ESOME @ A_27a @ V0x ) )
& ( ( c_2Esptree_2Espt__center @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V4t1 @ V0x @ V3t2 ) )
= ( c_2Eoption_2ESOME @ A_27a @ V0x ) )
& ( ( c_2Esptree_2Espt__center @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Espt__center @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V2v1 @ V1v2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Esptree_2Espt__center__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ! [V1x: A_27a] : ( V0P @ ( c_2Esptree_2ELS @ A_27a @ V1x ) )
& ! [V2t1: tyop_2Esptree_2Espt @ A_27a,V3x: A_27a,V4t2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2EBS @ A_27a @ V2t1 @ V3x @ V4t2 ) )
& ( V0P @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V5v1: tyop_2Esptree_2Espt @ A_27a,V6v2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2EBN @ A_27a @ V5v1 @ V6v2 ) ) )
=> ! [V7v: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V7v ) ) ).
thf(thm_2Esptree_2Einsert__shadow,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1a: tyop_2Enum_2Enum,V2b: A_27a,V3c: A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V1a @ V2b @ ( c_2Esptree_2Einsert @ A_27a @ V1a @ V3c @ V0t ) )
= ( c_2Esptree_2Einsert @ A_27a @ V1a @ V2b @ V0t ) ) ).
thf(thm_2Esptree_2Einsert__insert,axiom,
! [A_27a: $tType,V0x1: tyop_2Enum_2Enum,V1x2: tyop_2Enum_2Enum,V2v1: A_27a,V3v2: A_27a,V4t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V0x1 @ V2v1 @ ( c_2Esptree_2Einsert @ A_27a @ V1x2 @ V3v2 @ V4t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0x1 @ V1x2 ) @ ( c_2Esptree_2Einsert @ A_27a @ V0x1 @ V2v1 @ V4t ) @ ( c_2Esptree_2Einsert @ A_27a @ V1x2 @ V3v2 @ ( c_2Esptree_2Einsert @ A_27a @ V0x1 @ V2v1 @ V4t ) ) ) ) ).
thf(thm_2Esptree_2Emap__fromAList,axiom,
! [A_27a: $tType,A_27b: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27b ),V1f: A_27b > A_27a] :
( ( c_2Esptree_2Emap @ A_27a @ A_27b @ V1f @ ( c_2Esptree_2EfromAList @ A_27b @ V0ls ) )
= ( c_2Esptree_2EfromAList @ A_27a
@ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27b ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )
@ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ A_27b @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )
@ ^ [V2k: tyop_2Enum_2Enum,V3v: A_27b] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2k @ ( V1f @ V3v ) ) )
@ V0ls ) ) ) ).
thf(thm_2Esptree_2Emap__insert,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: tyop_2Enum_2Enum,V2y: A_27a,V3z: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ ( c_2Esptree_2Einsert @ A_27a @ V1x @ V2y @ V3z ) )
= ( c_2Esptree_2Einsert @ A_27b @ V1x @ ( V0f @ V2y ) @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ V3z ) ) ) ).
thf(thm_2Esptree_2Emap__map__o,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1f: A_27b > A_27c,V2g: A_27a > A_27b] :
( ( c_2Esptree_2Emap @ A_27c @ A_27b @ V1f @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V2g @ V0t ) )
= ( c_2Esptree_2Emap @ A_27c @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1f @ V2g ) @ V0t ) ) ).
thf(thm_2Esptree_2Ewf__map,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1f: A_27a > A_27b] :
( ( c_2Esptree_2Ewf @ A_27b @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V1f @ V0t ) )
= ( c_2Esptree_2Ewf @ A_27a @ V0t ) ) ).
thf(thm_2Esptree_2Emap__LN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ V1t )
= ( c_2Esptree_2ELN @ A_27b ) )
<=> ( V1t
= ( c_2Esptree_2ELN @ A_27a ) ) ) ).
thf(thm_2Esptree_2Elookup__map,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: tyop_2Esptree_2Espt @ A_27a,V2x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27b @ V2x @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ V1s ) )
= ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V0f @ ( c_2Esptree_2Elookup @ A_27a @ V2x @ V1s ) ) ) ).
thf(thm_2Esptree_2Edomain__map,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27b @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ V1s ) )
= ( c_2Esptree_2Edomain @ A_27a @ V1s ) ) ).
thf(thm_2Esptree_2EtoList__map,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2EtoList @ A_27b @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ V0f @ V1s ) )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V0f @ ( c_2Esptree_2EtoList @ A_27a @ V1s ) ) ) ).
thf(thm_2Esptree_2EfromAList__append,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ),V1l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Esptree_2EfromAList @ A_27a @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ V0l1 @ V1l2 ) )
= ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2EfromAList @ A_27a @ V0l1 ) @ ( c_2Esptree_2EfromAList @ A_27a @ V1l2 ) ) ) ).
thf(thm_2Esptree_2Eunion__insert__LN,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1y: A_27a,V2t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V0x @ V1y @ ( c_2Esptree_2ELN @ A_27a ) ) @ V2t2 )
= ( c_2Esptree_2Einsert @ A_27a @ V0x @ V1y @ V2t2 ) ) ).
thf(thm_2Esptree_2EfromAList__toAList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( ( c_2Esptree_2EfromAList @ A_27a @ ( c_2Esptree_2EtoAList @ A_27a @ V0t ) )
= V0t ) ) ).
thf(thm_2Esptree_2Ewf__fromAList,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2EfromAList @ A_27a @ V0ls ) ) ).
thf(thm_2Esptree_2Elookup__fromAList__toAList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V1x @ ( c_2Esptree_2EfromAList @ A_27a @ ( c_2Esptree_2EtoAList @ A_27a @ V0t ) ) )
= ( c_2Esptree_2Elookup @ A_27a @ V1x @ V0t ) ) ).
thf(thm_2Esptree_2Edomain__fromAList,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EfromAList @ A_27a @ V0ls ) )
= ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ tyop_2Enum_2Enum @ ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ A_27a ) @ V0ls ) ) ) ).
thf(thm_2Esptree_2Elookup__fromAList,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ),V1x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V1x @ ( c_2Esptree_2EfromAList @ A_27a @ V0ls ) )
= ( c_2Ealist_2EALOOKUP @ A_27a @ tyop_2Enum_2Enum @ V0ls @ V1x ) ) ).
thf(thm_2Esptree_2EfromAList__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Esptree_2EfromAList @ A_27a @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0y: A_27a,V1xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ),V2x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2EfromAList @ A_27a @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2x @ V0y ) @ V1xs ) )
= ( c_2Esptree_2Einsert @ A_27a @ V2x @ V0y @ ( c_2Esptree_2EfromAList @ A_27a @ V1xs ) ) ) ) ).
thf(thm_2Esptree_2EfromAList__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > $o] :
( ( ( V0P @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) )
& ! [V1x: tyop_2Enum_2Enum,V2y: A_27a,V3xs: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( ( V0P @ V3xs )
=> ( V0P @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1x @ V2y ) @ V3xs ) ) ) )
=> ! [V4v: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( V0P @ V4v ) ) ).
thf(thm_2Esptree_2Edelete__compute,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a,V2t: tyop_2Esptree_2Espt @ A_27a,V3n: tyop_2Enum_2Enum,V4a: A_27a] :
( ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ V3n ) @ V2t )
= ( c_2Esptree_2Edelete @ A_27a @ V3n @ V2t ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Esptree_2ELS @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2Emk__BN @ A_27a @ V1t1 @ ( c_2Esptree_2Edelete @ A_27a @ V3n @ V0t2 ) ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2Emk__BS @ A_27a @ V1t1 @ V4a @ ( c_2Esptree_2Edelete @ A_27a @ V3n @ V0t2 ) ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Esptree_2ELS @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V3n @ V1t1 ) @ V0t2 ) )
& ( ( c_2Esptree_2Edelete @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V3n @ V1t1 ) @ V4a @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Einsert__compute,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a,V2t: tyop_2Esptree_2Espt @ A_27a,V3n: tyop_2Enum_2Enum,V4a_27: A_27a,V5a: A_27a] :
( ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ V3n ) @ V5a @ V2t )
= ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ V2t ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Enum_2E0 @ V5a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELS @ A_27a @ V5a ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Enum_2E0 @ V5a @ ( c_2Esptree_2ELS @ A_27a @ V4a_27 ) )
= ( c_2Esptree_2ELS @ A_27a @ V5a ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Enum_2E0 @ V5a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V5a @ V0t2 ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Enum_2E0 @ V5a @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a_27 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V5a @ V0t2 ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Earithmetic_2EZERO @ V5a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELS @ A_27a @ V5a ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Earithmetic_2EZERO @ V5a @ ( c_2Esptree_2ELS @ A_27a @ V4a_27 ) )
= ( c_2Esptree_2ELS @ A_27a @ V5a ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Earithmetic_2EZERO @ V5a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V5a @ V0t2 ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ c_2Earithmetic_2EZERO @ V5a @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a_27 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V5a @ V0t2 ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ V5a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ V5a @ ( c_2Esptree_2ELS @ A_27a @ V4a_27 ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V4a_27 @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ V5a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ V1t1 @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ V0t2 ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ V5a @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a_27 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a_27 @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ V0t2 ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ V5a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ V5a @ ( c_2Esptree_2ELS @ A_27a @ V4a_27 ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ ( c_2Esptree_2ELN @ A_27a ) ) @ V4a_27 @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ V5a @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ V1t1 ) @ V0t2 ) )
& ( ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ V5a @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a_27 @ V0t2 ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V3n @ V5a @ V1t1 ) @ V4a_27 @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Elookup__compute,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a,V2t: tyop_2Esptree_2Espt @ A_27a,V3n: tyop_2Enum_2Enum,V4a: A_27a] :
( ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2ENUMERAL @ V3n ) @ V2t )
= ( c_2Esptree_2Elookup @ A_27a @ V3n @ V2t ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Eoption_2ESOME @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Enum_2E0 @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Eoption_2ESOME @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Eoption_2ESOME @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ c_2Earithmetic_2EZERO @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Eoption_2ESOME @ A_27a @ V4a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2Elookup @ A_27a @ V3n @ V0t2 ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT1 @ V3n ) @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2Elookup @ A_27a @ V3n @ V0t2 ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2ELS @ A_27a @ V4a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2EBN @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Esptree_2Elookup @ A_27a @ V3n @ V1t1 ) )
& ( ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EBIT2 @ V3n ) @ ( c_2Esptree_2EBS @ A_27a @ V1t1 @ V4a @ V0t2 ) )
= ( c_2Esptree_2Elookup @ A_27a @ V3n @ V1t1 ) ) ) ).
thf(thm_2Esptree_2Einter__assoc,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27b,V2t3: tyop_2Esptree_2Espt @ A_27c] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t1 @ ( c_2Esptree_2Einter @ A_27b @ A_27c @ V1t2 @ V2t3 ) )
= ( c_2Esptree_2Einter @ A_27a @ A_27c @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t1 @ V1t2 ) @ V2t3 ) ) ).
thf(thm_2Esptree_2Eunion__assoc,axiom,
! [A_27a: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27a,V2t3: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ V0t1 @ ( c_2Esptree_2Eunion @ A_27a @ V1t2 @ V2t3 ) )
= ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V0t1 @ V1t2 ) @ V2t3 ) ) ).
thf(thm_2Esptree_2Einter__LN,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Einter @ A_27c @ A_27a @ ( c_2Esptree_2ELN @ A_27c ) @ V0t )
= ( c_2Esptree_2ELN @ A_27c ) ) ) ).
thf(thm_2Esptree_2Eunion__LN,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Eunion @ A_27a @ V0t @ ( c_2Esptree_2ELN @ A_27a ) )
= V0t )
& ( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= V0t ) ) ).
thf(thm_2Esptree_2Edelete__mk__wf,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edelete @ A_27a @ V0x @ ( c_2Esptree_2Emk__wf @ A_27a @ V1t ) )
= ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V0x @ V1t ) ) ) ).
thf(thm_2Esptree_2Einsert__mk__wf,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1v: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V0x @ V1v @ ( c_2Esptree_2Emk__wf @ A_27a @ V2t ) )
= ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V0x @ V1v @ V2t ) ) ) ).
thf(thm_2Esptree_2Einter__mk__wf,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Einter @ A_27a @ A_27b @ ( c_2Esptree_2Emk__wf @ A_27a @ V0t1 ) @ ( c_2Esptree_2Emk__wf @ A_27b @ V1t2 ) )
= ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t1 @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Eunion__mk__wf,axiom,
! [A_27a: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2Emk__wf @ A_27a @ V0t1 ) @ ( c_2Esptree_2Emk__wf @ A_27a @ V1t2 ) )
= ( c_2Esptree_2Emk__wf @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V0t1 @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Einter__eq,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27b,V2t3: tyop_2Esptree_2Espt @ A_27a,V3t4: tyop_2Esptree_2Espt @ A_27c] :
( ( ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t1 @ V1t2 )
= ( c_2Esptree_2Einter @ A_27a @ A_27c @ V2t3 @ V3t4 ) )
<=> ! [V4x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V4x @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0t1 @ V1t2 ) )
= ( c_2Esptree_2Elookup @ A_27a @ V4x @ ( c_2Esptree_2Einter @ A_27a @ A_27c @ V2t3 @ V3t4 ) ) ) ) ).
thf(thm_2Esptree_2Emk__wf__eq,axiom,
! [A_27a: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Emk__wf @ A_27a @ V0t1 )
= ( c_2Esptree_2Emk__wf @ A_27a @ V1t2 ) )
<=> ! [V2x: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2x @ V0t1 )
= ( c_2Esptree_2Elookup @ A_27a @ V2x @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Edomain__mk__wf,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Emk__wf @ A_27a @ V0t ) )
= ( c_2Esptree_2Edomain @ A_27a @ V0t ) ) ).
thf(thm_2Esptree_2Elookup__mk__wf,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V0x @ ( c_2Esptree_2Emk__wf @ A_27a @ V1t ) )
= ( c_2Esptree_2Elookup @ A_27a @ V0x @ V1t ) ) ).
thf(thm_2Esptree_2Ewf__mk__id,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( ( c_2Esptree_2Emk__wf @ A_27a @ V0t )
= V0t ) ) ).
thf(thm_2Esptree_2Ewf__mk__wf,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] : ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Emk__wf @ A_27a @ V0t ) ) ).
thf(thm_2Esptree_2Espt__eq__thm,axiom,
! [A_27a: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Ewf @ A_27a @ V0t1 )
& ( c_2Esptree_2Ewf @ A_27a @ V1t2 ) )
=> ( ( V0t1 = V1t2 )
<=> ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2n @ V0t1 )
= ( c_2Esptree_2Elookup @ A_27a @ V2n @ V1t2 ) ) ) ) ).
thf(thm_2Esptree_2EMEM__toList,axiom,
! [A_27a: $tType,V0x: A_27a,V1t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Esptree_2EtoList @ A_27a @ V1t ) ) )
<=> ? [V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2k @ V1t )
= ( c_2Eoption_2ESOME @ A_27a @ V0x ) ) ) ).
thf(thm_2Esptree_2EisEmpty__toList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( c_2Esptree_2EtoList @ A_27a @ V0t )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(thm_2Esptree_2EisEmpty__toListA,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1acc: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( c_2Esptree_2EtoListA @ A_27a @ V1acc @ V0t )
= V1acc ) ) ) ).
thf(thm_2Esptree_2EtoListA__append,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1acc: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2EtoListA @ A_27a @ V1acc @ V0t )
= ( c_2Elist_2EAPPEND @ A_27a @ ( c_2Esptree_2EtoListA @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) @ V0t ) @ V1acc ) ) ).
thf(thm_2Esptree_2Efoldi__FOLDR__toAList,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Enum_2Enum > A_27a > A_27b > A_27b,V1a: A_27b,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Efoldi @ A_27b @ A_27a @ V0f @ c_2Enum_2E0 @ V1a @ V2t )
= ( c_2Elist_2EFOLDR @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ A_27b @ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ A_27a @ ( A_27b > A_27b ) @ V0f ) @ V1a @ ( c_2Esptree_2EtoAList @ A_27a @ V2t ) ) ) ).
thf(thm_2Esptree_2EALL__DISTINCT__MAP__FST__toAList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] : ( c_2Elist_2EALL__DISTINCT @ tyop_2Enum_2Enum @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ tyop_2Enum_2Enum @ ( c_2Epair_2EFST @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Esptree_2EtoAList @ A_27a @ V0t ) ) ) ).
thf(thm_2Esptree_2Edomain__empty,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( c_2Esptree_2Edomain @ A_27a @ V0t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Esptree_2Einsert__union,axiom,
! [A_27a: $tType,V0k: tyop_2Enum_2Enum,V1v: A_27a,V2s: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1v @ V2s )
= ( c_2Esptree_2Eunion @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1v @ ( c_2Esptree_2ELN @ A_27a ) ) @ V2s ) ) ).
thf(thm_2Esptree_2EALOOKUP__toAList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1x: tyop_2Enum_2Enum] :
( ( c_2Ealist_2EALOOKUP @ A_27a @ tyop_2Enum_2Enum @ ( c_2Esptree_2EtoAList @ A_27a @ V0t ) @ V1x )
= ( c_2Esptree_2Elookup @ A_27a @ V1x @ V0t ) ) ).
thf(thm_2Esptree_2EMEM__toAList,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum,V2v: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2v ) @ ( c_2Elist_2ELIST__TO__SET @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Esptree_2EtoAList @ A_27a @ V0t ) ) )
<=> ( ( c_2Esptree_2Elookup @ A_27a @ V1k @ V0t )
= ( c_2Eoption_2ESOME @ A_27a @ V2v ) ) ) ).
thf(thm_2Esptree_2Elookup__mapi,axiom,
! [A_27a: $tType,A_27b: $tType,V0pt: tyop_2Esptree_2Espt @ A_27b,V1k: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > A_27b > A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V1k @ ( c_2Esptree_2Emapi @ A_27a @ A_27b @ V2f @ V0pt ) )
= ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a @ ( V2f @ V1k ) @ ( c_2Esptree_2Elookup @ A_27b @ V1k @ V0pt ) ) ) ).
thf(thm_2Esptree_2Elookup__mapi0,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Enum_2Enum > A_27a > A_27b,V1pt: tyop_2Esptree_2Espt @ A_27a,V2i: tyop_2Enum_2Enum,V3k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27b @ V3k @ ( c_2Esptree_2Emapi0 @ A_27b @ A_27a @ V0f @ V2i @ V1pt ) )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Esptree_2Elookup @ A_27a @ V3k @ V1pt ) @ ( c_2Eoption_2ENONE @ A_27b )
@ ^ [V4v: A_27a] : ( c_2Eoption_2ESOME @ A_27b @ ( V0f @ ( c_2Esptree_2Espt__acc @ V2i @ V3k ) @ V4v ) ) ) ) ).
thf(thm_2Esptree_2Edomain__foldi,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ V0t )
= ( c_2Esptree_2Efoldi @ ( tyop_2Enum_2Enum > $o ) @ A_27a
@ ^ [V1k: tyop_2Enum_2Enum,V2v: A_27a,V3a: tyop_2Enum_2Enum > $o] : ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V1k @ V3a )
@ c_2Enum_2E0
@ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum )
@ V0t ) ) ).
thf(thm_2Esptree_2Eset__foldi__keys,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1a: tyop_2Enum_2Enum > $o,V2i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Efoldi @ ( tyop_2Enum_2Enum > $o ) @ A_27a
@ ^ [V3k: tyop_2Enum_2Enum,V4v: A_27a,V5a: tyop_2Enum_2Enum > $o] : ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V3k @ V5a )
@ V2i
@ V1a
@ V0t )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V1a
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ V2i @ ( c_2Earithmetic_2E_2A @ ( c_2Esptree_2Elrnext @ V2i ) @ V6n ) )
@ ( c_2Esptree_2Edomain @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Esptree_2Espt__acc__0,axiom,
! [V0k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ c_2Enum_2E0 @ V0k )
= V0k ) ).
thf(thm_2Esptree_2Espt__acc__eqn,axiom,
! [V0k: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V1i @ V0k )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Esptree_2Elrnext @ V1i ) @ V0k ) @ V1i ) ) ).
thf(thm_2Esptree_2Espt__acc__thm,axiom,
! [V0k: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V1i @ V0k )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0k @ c_2Enum_2E0 ) @ V1i @ ( c_2Esptree_2Espt__acc @ ( c_2Earithmetic_2E_2B @ V1i @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ V0k ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V1i ) ) @ ( c_2Esptree_2Elrnext @ V1i ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__acc__def__compute,axiom,
( ! [V0i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V0i @ c_2Enum_2E0 )
= V0i )
& ! [V1k: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V2i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1k ) ) )
= ( c_2Esptree_2Espt__acc @ ( c_2Earithmetic_2E_2B @ V2i @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1k ) ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1k ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V3k: tyop_2Enum_2Enum,V4i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V4i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V3k ) ) )
= ( c_2Esptree_2Espt__acc @ ( c_2Earithmetic_2E_2B @ V4i @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V3k ) ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V4i ) ) @ ( c_2Esptree_2Elrnext @ V4i ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3k ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__acc__def,axiom,
( ! [V0i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V0i @ c_2Enum_2E0 )
= V0i )
& ! [V1k: tyop_2Enum_2Enum,V2i: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Espt__acc @ V2i @ ( c_2Enum_2ESUC @ V1k ) )
= ( c_2Esptree_2Espt__acc @ ( c_2Earithmetic_2E_2B @ V2i @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ ( c_2Enum_2ESUC @ V1k ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) ) @ ( c_2Earithmetic_2EDIV @ V1k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Esptree_2Espt__acc__ind,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1i: tyop_2Enum_2Enum] : ( V0P @ V1i @ c_2Enum_2E0 )
& ! [V2i: tyop_2Enum_2Enum,V3k: tyop_2Enum_2Enum] :
( ( V0P @ ( c_2Earithmetic_2E_2B @ V2i @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ ( c_2Enum_2ESUC @ V3k ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) @ ( c_2Esptree_2Elrnext @ V2i ) ) ) @ ( c_2Earithmetic_2EDIV @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V0P @ V2i @ ( c_2Enum_2ESUC @ V3k ) ) ) )
=> ! [V4v: tyop_2Enum_2Enum,V5v1: tyop_2Enum_2Enum] : ( V0P @ V4v @ V5v1 ) ) ).
thf(thm_2Esptree_2Edomain__delete,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V1k @ V0t ) )
= ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V0t ) @ V1k ) ) ).
thf(thm_2Esptree_2Elookup__delete,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k1: tyop_2Enum_2Enum,V2k2: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V1k1 @ ( c_2Esptree_2Edelete @ A_27a @ V2k2 @ V0t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k1 @ V2k2 ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V1k1 @ V0t ) ) ) ).
thf(thm_2Esptree_2Edomain__fromList,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EfromList @ A_27a @ V0l ) )
= ( c_2Epred__set_2Ecount @ ( c_2Elist_2ELENGTH @ A_27a @ V0l ) ) ) ).
thf(thm_2Esptree_2Edomain__sing,axiom,
! [A_27a: $tType,V0v: A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V1k @ V0v @ ( c_2Esptree_2ELN @ A_27a ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V1k @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Esptree_2Edomain__difference,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27b] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V0t1 @ V1t2 ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V0t1 ) @ ( c_2Esptree_2Edomain @ A_27b @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Edomain__insert,axiom,
! [A_27a: $tType,V0v: A_27a,V1t: tyop_2Esptree_2Espt @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V2k @ V0v @ V1t ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V2k @ ( c_2Esptree_2Edomain @ A_27a @ V1t ) ) ) ).
thf(thm_2Esptree_2Edomain__inter,axiom,
! [A_27a: $tType,A_27b: $tType,V0t2: tyop_2Esptree_2Espt @ A_27b,V1t1: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V1t1 @ V0t2 ) )
= ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V1t1 ) @ ( c_2Esptree_2Edomain @ A_27b @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Edomain__union,axiom,
! [A_27a: $tType,V0t2: tyop_2Esptree_2Espt @ A_27a,V1t1: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V1t1 @ V0t2 ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V1t1 ) @ ( c_2Esptree_2Edomain @ A_27a @ V0t2 ) ) ) ).
thf(thm_2Esptree_2Elookup__NONE__domain,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( ( c_2Esptree_2Elookup @ A_27a @ V1k @ V0t )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Esptree_2Edomain @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Esptree_2Elookup__inter__alt,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Enum_2Enum,V1t2: tyop_2Esptree_2Espt @ A_27b,V2t1: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V0x @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V2t1 @ V1t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Esptree_2Edomain @ A_27b @ V1t2 ) ) @ ( c_2Esptree_2Elookup @ A_27a @ V0x @ V2t1 ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Esptree_2Edomain__lookup,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Esptree_2Edomain @ A_27a @ V0t ) )
<=> ? [V2v: A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V1k @ V0t )
= ( c_2Eoption_2ESOME @ A_27a @ V2v ) ) ) ).
thf(thm_2Esptree_2Elookup__fromList,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V0n @ ( c_2Esptree_2EfromList @ A_27a @ V1l ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eprim__rec_2E_3C @ V0n @ ( c_2Elist_2ELENGTH @ A_27a @ V1l ) ) @ ( c_2Eoption_2ESOME @ A_27a @ ( c_2Elist_2EEL @ A_27a @ V0n @ V1l ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Esptree_2Esize__insert,axiom,
! [A_27a: $tType,V0k: tyop_2Enum_2Enum,V1v: A_27a,V2m: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1v @ V2m ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0k @ ( c_2Esptree_2Edomain @ A_27a @ V2m ) ) @ ( c_2Esptree_2Esize @ A_27a @ V2m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Esptree_2Esize @ A_27a @ V2m ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Esptree_2Esubspt__domain,axiom,
! [V0t1: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone,V1t2: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( c_2Esptree_2Esubspt @ tyop_2Eone_2Eone @ V0t1 @ V1t2 )
= ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ V0t1 ) @ ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ V1t2 ) ) ) ).
thf(thm_2Esptree_2Esubspt__lookup,axiom,
! [A_27a: $tType,V0t1: tyop_2Esptree_2Espt @ A_27a,V1t2: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ V0t1 @ V1t2 )
<=> ! [V2x: tyop_2Enum_2Enum,V3y: A_27a] :
( ( ( c_2Esptree_2Elookup @ A_27a @ V2x @ V0t1 )
= ( c_2Eoption_2ESOME @ A_27a @ V3y ) )
=> ( ( c_2Esptree_2Elookup @ A_27a @ V2x @ V1t2 )
= ( c_2Eoption_2ESOME @ A_27a @ V3y ) ) ) ) ).
thf(thm_2Esptree_2Esubspt__eq,axiom,
! [A_27a: $tType] :
( ! [V0t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0t )
= c_2Ebool_2ET )
& ! [V1x: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V1x ) @ V2t )
<=> ( ( c_2Esptree_2Espt__center @ A_27a @ V2t )
= ( c_2Eoption_2ESOME @ A_27a @ V1x ) ) )
& ! [V3t1: tyop_2Esptree_2Espt @ A_27a,V4t2: tyop_2Esptree_2Espt @ A_27a,V5t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V3t1 @ V4t2 ) @ V5t )
<=> ( ( c_2Esptree_2Esubspt @ A_27a @ V3t1 @ ( c_2Esptree_2Espt__left @ A_27a @ V5t ) )
& ( c_2Esptree_2Esubspt @ A_27a @ V4t2 @ ( c_2Esptree_2Espt__right @ A_27a @ V5t ) ) ) )
& ! [V6t1: tyop_2Esptree_2Espt @ A_27a,V7x: A_27a,V8t2: tyop_2Esptree_2Espt @ A_27a,V9t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Esubspt @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V6t1 @ V7x @ V8t2 ) @ V9t )
<=> ( ( ( c_2Esptree_2Espt__center @ A_27a @ V9t )
= ( c_2Eoption_2ESOME @ A_27a @ V7x ) )
& ( c_2Esptree_2Esubspt @ A_27a @ V6t1 @ ( c_2Esptree_2Espt__left @ A_27a @ V9t ) )
& ( c_2Esptree_2Esubspt @ A_27a @ V8t2 @ ( c_2Esptree_2Espt__right @ A_27a @ V9t ) ) ) ) ) ).
thf(thm_2Esptree_2Edatatype__spt,axiom,
! [A_27a: $tType,V0spt: ( tyop_2Esptree_2Espt @ A_27a ) > ( A_27a > ( tyop_2Esptree_2Espt @ A_27a ) ) > ( ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) > ( ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0spt @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELS @ A_27a ) @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2EBS @ A_27a ) ) ) ).
thf(thm_2Esptree_2Espt__11,axiom,
! [A_27a: $tType] :
( ! [V0a: A_27a,V1a_27: A_27a] :
( ( ( c_2Esptree_2ELS @ A_27a @ V0a )
= ( c_2Esptree_2ELS @ A_27a @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a0: tyop_2Esptree_2Espt @ A_27a,V3a1: tyop_2Esptree_2Espt @ A_27a,V4a0_27: tyop_2Esptree_2Espt @ A_27a,V5a1_27: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2EBN @ A_27a @ V2a0 @ V3a1 )
= ( c_2Esptree_2EBN @ A_27a @ V4a0_27 @ V5a1_27 ) )
<=> ( ( V2a0 = V4a0_27 )
& ( V3a1 = V5a1_27 ) ) )
& ! [V6a0: tyop_2Esptree_2Espt @ A_27a,V7a1: A_27a,V8a2: tyop_2Esptree_2Espt @ A_27a,V9a0_27: tyop_2Esptree_2Espt @ A_27a,V10a1_27: A_27a,V11a2_27: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2EBS @ A_27a @ V6a0 @ V7a1 @ V8a2 )
= ( c_2Esptree_2EBS @ A_27a @ V9a0_27 @ V10a1_27 @ V11a2_27 ) )
<=> ( ( V6a0 = V9a0_27 )
& ( V7a1 = V10a1_27 )
& ( V8a2 = V11a2_27 ) ) ) ) ).
thf(thm_2Esptree_2Espt__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a: A_27a] :
( (~)
@ ( ( c_2Esptree_2ELN @ A_27a )
= ( c_2Esptree_2ELS @ A_27a @ V0a ) ) )
& ! [V1a1: tyop_2Esptree_2Espt @ A_27a,V2a0: tyop_2Esptree_2Espt @ A_27a] :
( (~)
@ ( ( c_2Esptree_2ELN @ A_27a )
= ( c_2Esptree_2EBN @ A_27a @ V2a0 @ V1a1 ) ) )
& ! [V3a2: tyop_2Esptree_2Espt @ A_27a,V4a1: A_27a,V5a0: tyop_2Esptree_2Espt @ A_27a] :
( (~)
@ ( ( c_2Esptree_2ELN @ A_27a )
= ( c_2Esptree_2EBS @ A_27a @ V5a0 @ V4a1 @ V3a2 ) ) )
& ! [V6a1: tyop_2Esptree_2Espt @ A_27a,V7a0: tyop_2Esptree_2Espt @ A_27a,V8a: A_27a] :
( (~)
@ ( ( c_2Esptree_2ELS @ A_27a @ V8a )
= ( c_2Esptree_2EBN @ A_27a @ V7a0 @ V6a1 ) ) )
& ! [V9a2: tyop_2Esptree_2Espt @ A_27a,V10a1: A_27a,V11a0: tyop_2Esptree_2Espt @ A_27a,V12a: A_27a] :
( (~)
@ ( ( c_2Esptree_2ELS @ A_27a @ V12a )
= ( c_2Esptree_2EBS @ A_27a @ V11a0 @ V10a1 @ V9a2 ) ) )
& ! [V13a2: tyop_2Esptree_2Espt @ A_27a,V14a1_27: A_27a,V15a1: tyop_2Esptree_2Espt @ A_27a,V16a0_27: tyop_2Esptree_2Espt @ A_27a,V17a0: tyop_2Esptree_2Espt @ A_27a] :
( (~)
@ ( ( c_2Esptree_2EBN @ A_27a @ V17a0 @ V15a1 )
= ( c_2Esptree_2EBS @ A_27a @ V16a0_27 @ V14a1_27 @ V13a2 ) ) ) ) ).
thf(thm_2Esptree_2Espt__nchotomy,axiom,
! [A_27a: $tType,V0ss: tyop_2Esptree_2Espt @ A_27a] :
( ( V0ss
= ( c_2Esptree_2ELN @ A_27a ) )
| ? [V1a: A_27a] :
( V0ss
= ( c_2Esptree_2ELS @ A_27a @ V1a ) )
| ? [V2s: tyop_2Esptree_2Espt @ A_27a,V3s0: tyop_2Esptree_2Espt @ A_27a] :
( V0ss
= ( c_2Esptree_2EBN @ A_27a @ V2s @ V3s0 ) )
| ? [V4s: tyop_2Esptree_2Espt @ A_27a,V5a: A_27a,V6s0: tyop_2Esptree_2Espt @ A_27a] :
( V0ss
= ( c_2Esptree_2EBS @ A_27a @ V4s @ V5a @ V6s0 ) ) ) ).
thf(thm_2Esptree_2Espt__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27b,V1f1: A_27a > A_27b,V2f2: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b > A_27b > A_27b,V3f3: A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b > A_27b > A_27b] :
? [V4fn: ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( ( V4fn @ ( c_2Esptree_2ELN @ A_27a ) )
= V0f0 )
& ! [V5a: A_27a] :
( ( V4fn @ ( c_2Esptree_2ELS @ A_27a @ V5a ) )
= ( V1f1 @ V5a ) )
& ! [V6a0: tyop_2Esptree_2Espt @ A_27a,V7a1: tyop_2Esptree_2Espt @ A_27a] :
( ( V4fn @ ( c_2Esptree_2EBN @ A_27a @ V6a0 @ V7a1 ) )
= ( V2f2 @ V6a0 @ V7a1 @ ( V4fn @ V6a0 ) @ ( V4fn @ V7a1 ) ) )
& ! [V8a0: tyop_2Esptree_2Espt @ A_27a,V9a1: A_27a,V10a2: tyop_2Esptree_2Espt @ A_27a] :
( ( V4fn @ ( c_2Esptree_2EBS @ A_27a @ V8a0 @ V9a1 @ V10a2 ) )
= ( V3f3 @ V9a1 @ V8a0 @ V10a2 @ ( V4fn @ V8a0 ) @ ( V4fn @ V10a2 ) ) ) ) ).
thf(thm_2Esptree_2Espt__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1a: A_27a] : ( V0P @ ( c_2Esptree_2ELS @ A_27a @ V1a ) )
& ! [V2s: tyop_2Esptree_2Espt @ A_27a,V3s0: tyop_2Esptree_2Espt @ A_27a] :
( ( ( V0P @ V2s )
& ( V0P @ V3s0 ) )
=> ( V0P @ ( c_2Esptree_2EBN @ A_27a @ V2s @ V3s0 ) ) )
& ! [V4s: tyop_2Esptree_2Espt @ A_27a,V5s0: tyop_2Esptree_2Espt @ A_27a] :
( ( ( V0P @ V4s )
& ( V0P @ V5s0 ) )
=> ! [V6a: A_27a] : ( V0P @ ( c_2Esptree_2EBS @ A_27a @ V4s @ V6a @ V5s0 ) ) ) )
=> ! [V7s: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V7s ) ) ).
thf(thm_2Esptree_2Espt__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27: A_27b,V1f2_27: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V2f1_27: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V3f_27: A_27a > A_27b,V4M: tyop_2Esptree_2Espt @ A_27a,V5M_27: tyop_2Esptree_2Espt @ A_27a,V6v: A_27b,V7f: A_27a > A_27b,V8f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V9f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b] :
( ( ( V4M = V5M_27 )
& ( ( V5M_27
= ( c_2Esptree_2ELN @ A_27a ) )
=> ( V6v = V0v_27 ) )
& ! [V10a: A_27a] :
( ( V5M_27
= ( c_2Esptree_2ELS @ A_27a @ V10a ) )
=> ( ( V7f @ V10a )
= ( V3f_27 @ V10a ) ) )
& ! [V11a0: tyop_2Esptree_2Espt @ A_27a,V12a1: tyop_2Esptree_2Espt @ A_27a] :
( ( V5M_27
= ( c_2Esptree_2EBN @ A_27a @ V11a0 @ V12a1 ) )
=> ( ( V8f1 @ V11a0 @ V12a1 )
= ( V2f1_27 @ V11a0 @ V12a1 ) ) )
& ! [V13a0: tyop_2Esptree_2Espt @ A_27a,V14a1: A_27a,V15a2: tyop_2Esptree_2Espt @ A_27a] :
( ( V5M_27
= ( c_2Esptree_2EBS @ A_27a @ V13a0 @ V14a1 @ V15a2 ) )
=> ( ( V9f2 @ V13a0 @ V14a1 @ V15a2 )
= ( V1f2_27 @ V13a0 @ V14a1 @ V15a2 ) ) ) )
=> ( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ V4M @ V6v @ V7f @ V8f1 @ V9f2 )
= ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ V5M_27 @ V0v_27 @ V3f_27 @ V2f1_27 @ V1f2_27 ) ) ) ).
thf(thm_2Esptree_2Espt__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esptree_2Espt @ A_27a,V1v_27: A_27b,V2v: A_27b,V3f2: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V4f1: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > A_27b,V5f: A_27a > A_27b] :
( ( ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b @ V0x @ V2v @ V5f @ V4f1 @ V3f2 )
= V1v_27 )
<=> ( ( ( V0x
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V2v = V1v_27 ) )
| ? [V6a: A_27a] :
( ( V0x
= ( c_2Esptree_2ELS @ A_27a @ V6a ) )
& ( ( V5f @ V6a )
= V1v_27 ) )
| ? [V7s: tyop_2Esptree_2Espt @ A_27a,V8s0: tyop_2Esptree_2Espt @ A_27a] :
( ( V0x
= ( c_2Esptree_2EBN @ A_27a @ V7s @ V8s0 ) )
& ( ( V4f1 @ V7s @ V8s0 )
= V1v_27 ) )
| ? [V9s: tyop_2Esptree_2Espt @ A_27a,V10a: A_27a,V11s0: tyop_2Esptree_2Espt @ A_27a] :
( ( V0x
= ( c_2Esptree_2EBS @ A_27a @ V9s @ V10a @ V11s0 ) )
& ( ( V3f2 @ V9s @ V10a @ V11s0 )
= V1v_27 ) ) ) ) ).
thf(thm_2Esptree_2Elookup__ind,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ! [V1k: tyop_2Enum_2Enum] : ( V0P @ V1k @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V2k: tyop_2Enum_2Enum,V3a: A_27a] : ( V0P @ V2k @ ( c_2Esptree_2ELS @ A_27a @ V3a ) )
& ! [V4k: tyop_2Enum_2Enum,V5t1: tyop_2Esptree_2Espt @ A_27a,V6t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( (~) @ ( V4k = c_2Enum_2E0 ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V4k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V4k ) @ V5t1 @ V6t2 ) ) )
=> ( V0P @ V4k @ ( c_2Esptree_2EBN @ A_27a @ V5t1 @ V6t2 ) ) )
& ! [V7k: tyop_2Enum_2Enum,V8t1: tyop_2Esptree_2Espt @ A_27a,V9a: A_27a,V10t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( (~) @ ( V7k = c_2Enum_2E0 ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V7k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V7k ) @ V8t1 @ V10t2 ) ) )
=> ( V0P @ V7k @ ( c_2Esptree_2EBS @ A_27a @ V8t1 @ V9a @ V10t2 ) ) ) )
=> ! [V11v: tyop_2Enum_2Enum,V12v1: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V11v @ V12v1 ) ) ).
thf(thm_2Esptree_2Elookup__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V0k @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1k: tyop_2Enum_2Enum,V2a: A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V1k @ ( c_2Esptree_2ELS @ A_27a @ V2a ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V2a ) @ ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V3t2: tyop_2Esptree_2Espt @ A_27a,V4t1: tyop_2Esptree_2Espt @ A_27a,V5k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V5k @ ( c_2Esptree_2EBN @ A_27a @ V4t1 @ V3t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V5k @ c_2Enum_2E0 ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V5k ) @ V4t1 @ V3t2 ) ) ) )
& ! [V6t2: tyop_2Esptree_2Espt @ A_27a,V7t1: tyop_2Esptree_2Espt @ A_27a,V8k: tyop_2Enum_2Enum,V9a: A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V8k @ ( c_2Esptree_2EBS @ A_27a @ V7t1 @ V9a @ V6t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V8k @ c_2Enum_2E0 ) @ ( c_2Eoption_2ESOME @ A_27a @ V9a ) @ ( c_2Esptree_2Elookup @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V8k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V8k ) @ V7t1 @ V6t2 ) ) ) ) ) ).
thf(thm_2Esptree_2Einsert__ind,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ! [V1k: tyop_2Enum_2Enum,V2a: A_27a] :
( ( ( ( ( (~) @ ( V1k = c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ V1k ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V1k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2a @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ( (~) @ ( V1k = c_2Enum_2E0 ) )
& ( (~) @ ( c_2Earithmetic_2EEVEN @ V1k ) ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V1k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2a @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
=> ( V0P @ V1k @ V2a @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V3k: tyop_2Enum_2Enum,V4a: A_27a,V5a_27: A_27a] :
( ( ( ( ( (~) @ ( V3k = c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ V3k ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V4a @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ( (~) @ ( V3k = c_2Enum_2E0 ) )
& ( (~) @ ( c_2Earithmetic_2EEVEN @ V3k ) ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V4a @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
=> ( V0P @ V3k @ V4a @ ( c_2Esptree_2ELS @ A_27a @ V5a_27 ) ) )
& ! [V6k: tyop_2Enum_2Enum,V7a: A_27a,V8t1: tyop_2Esptree_2Espt @ A_27a,V9t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( ( ( (~) @ ( V6k = c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ V6k ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V7a @ V8t1 ) )
& ( ( ( (~) @ ( V6k = c_2Enum_2E0 ) )
& ( (~) @ ( c_2Earithmetic_2EEVEN @ V6k ) ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V7a @ V9t2 ) ) )
=> ( V0P @ V6k @ V7a @ ( c_2Esptree_2EBN @ A_27a @ V8t1 @ V9t2 ) ) )
& ! [V10k: tyop_2Enum_2Enum,V11a: A_27a,V12t1: tyop_2Esptree_2Espt @ A_27a,V13a_27: A_27a,V14t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( ( ( (~) @ ( V10k = c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ V10k ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V10k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V11a @ V12t1 ) )
& ( ( ( (~) @ ( V10k = c_2Enum_2E0 ) )
& ( (~) @ ( c_2Earithmetic_2EEVEN @ V10k ) ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V10k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V11a @ V14t2 ) ) )
=> ( V0P @ V10k @ V11a @ ( c_2Esptree_2EBS @ A_27a @ V12t1 @ V13a_27 @ V14t2 ) ) ) )
=> ! [V15v: tyop_2Enum_2Enum,V16v1: A_27a,V17v2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V15v @ V16v1 @ V17v2 ) ) ).
thf(thm_2Esptree_2Einsert__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum,V1a: A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1a @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0k @ c_2Enum_2E0 ) @ ( c_2Esptree_2ELS @ A_27a @ V1a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V0k ) @ ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1a @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1a @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) )
& ! [V2k: tyop_2Enum_2Enum,V3a_27: A_27a,V4a: A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V2k @ V4a @ ( c_2Esptree_2ELS @ A_27a @ V3a_27 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2k @ c_2Enum_2E0 ) @ ( c_2Esptree_2ELS @ A_27a @ V4a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V2k ) @ ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V4a @ ( c_2Esptree_2ELN @ A_27a ) ) @ V3a_27 @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V3a_27 @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V4a @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) )
& ! [V5t2: tyop_2Esptree_2Espt @ A_27a,V6t1: tyop_2Esptree_2Espt @ A_27a,V7k: tyop_2Enum_2Enum,V8a: A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V7k @ V8a @ ( c_2Esptree_2EBN @ A_27a @ V6t1 @ V5t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V7k @ c_2Enum_2E0 ) @ ( c_2Esptree_2EBS @ A_27a @ V6t1 @ V8a @ V5t2 ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V7k ) @ ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V7k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V8a @ V6t1 ) @ V5t2 ) @ ( c_2Esptree_2EBN @ A_27a @ V6t1 @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V7k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V8a @ V5t2 ) ) ) ) )
& ! [V9t2: tyop_2Esptree_2Espt @ A_27a,V10t1: tyop_2Esptree_2Espt @ A_27a,V11k: tyop_2Enum_2Enum,V12a_27: A_27a,V13a: A_27a] :
( ( c_2Esptree_2Einsert @ A_27a @ V11k @ V13a @ ( c_2Esptree_2EBS @ A_27a @ V10t1 @ V12a_27 @ V9t2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V11k @ c_2Enum_2E0 ) @ ( c_2Esptree_2EBS @ A_27a @ V10t1 @ V13a @ V9t2 ) @ ( c_2Ebool_2ECOND @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Earithmetic_2EEVEN @ V11k ) @ ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V11k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V13a @ V10t1 ) @ V12a_27 @ V9t2 ) @ ( c_2Esptree_2EBS @ A_27a @ V10t1 @ V12a_27 @ ( c_2Esptree_2Einsert @ A_27a @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V11k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V13a @ V9t2 ) ) ) ) ) ) ).
thf(thm_2Esptree_2Emk__BN__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esptree_2Espt @ A_27a ) > ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1v14: A_27a] : ( V0P @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELS @ A_27a @ V1v14 ) )
& ! [V2v15: tyop_2Esptree_2Espt @ A_27a,V3v16: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBN @ A_27a @ V2v15 @ V3v16 ) )
& ! [V4v17: tyop_2Esptree_2Espt @ A_27a,V5v18: A_27a,V6v19: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBS @ A_27a @ V4v17 @ V5v18 @ V6v19 ) )
& ! [V7v2: A_27a,V8t2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2ELS @ A_27a @ V7v2 ) @ V8t2 )
& ! [V9v3: tyop_2Esptree_2Espt @ A_27a,V10v4: tyop_2Esptree_2Espt @ A_27a,V11t2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2EBN @ A_27a @ V9v3 @ V10v4 ) @ V11t2 )
& ! [V12v5: tyop_2Esptree_2Espt @ A_27a,V13v6: A_27a,V14v7: tyop_2Esptree_2Espt @ A_27a,V15t2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ ( c_2Esptree_2EBS @ A_27a @ V12v5 @ V13v6 @ V14v7 ) @ V15t2 ) )
=> ! [V16v: tyop_2Esptree_2Espt @ A_27a,V17v1: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V16v @ V17v1 ) ) ).
thf(thm_2Esptree_2Emk__BN__def,axiom,
! [A_27a: $tType,V0v7: tyop_2Esptree_2Espt @ A_27a,V1v6: A_27a,V2v5: tyop_2Esptree_2Espt @ A_27a,V3v4: tyop_2Esptree_2Espt @ A_27a,V4v3: tyop_2Esptree_2Espt @ A_27a,V5v2: A_27a,V6v19: tyop_2Esptree_2Espt @ A_27a,V7v18: A_27a,V8v17: tyop_2Esptree_2Espt @ A_27a,V9v16: tyop_2Esptree_2Espt @ A_27a,V10v15: tyop_2Esptree_2Espt @ A_27a,V11v14: A_27a,V12t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELS @ A_27a @ V11v14 ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2ELS @ A_27a @ V11v14 ) ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBN @ A_27a @ V10v15 @ V9v16 ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBN @ A_27a @ V10v15 @ V9v16 ) ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBS @ A_27a @ V8v17 @ V7v18 @ V6v19 ) )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ ( c_2Esptree_2EBS @ A_27a @ V8v17 @ V7v18 @ V6v19 ) ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V5v2 ) @ V12t2 )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V5v2 ) @ V12t2 ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V4v3 @ V3v4 ) @ V12t2 )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V4v3 @ V3v4 ) @ V12t2 ) )
& ( ( c_2Esptree_2Emk__BN @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V2v5 @ V1v6 @ V0v7 ) @ V12t2 )
= ( c_2Esptree_2EBN @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V2v5 @ V1v6 @ V0v7 ) @ V12t2 ) ) ) ).
thf(thm_2Esptree_2Emk__BS__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esptree_2Espt @ A_27a ) > A_27a > ( tyop_2Esptree_2Espt @ A_27a ) > $o] :
( ( ! [V1x: A_27a] : ( V0P @ ( c_2Esptree_2ELN @ A_27a ) @ V1x @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V2v16: A_27a,V3x: A_27a] : ( V0P @ ( c_2Esptree_2ELS @ A_27a @ V2v16 ) @ V3x @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V4v17: tyop_2Esptree_2Espt @ A_27a,V5v18: tyop_2Esptree_2Espt @ A_27a,V6x: A_27a] : ( V0P @ ( c_2Esptree_2EBN @ A_27a @ V4v17 @ V5v18 ) @ V6x @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V7v19: tyop_2Esptree_2Espt @ A_27a,V8v20: A_27a,V9v21: tyop_2Esptree_2Espt @ A_27a,V10x: A_27a] : ( V0P @ ( c_2Esptree_2EBS @ A_27a @ V7v19 @ V8v20 @ V9v21 ) @ V10x @ ( c_2Esptree_2ELN @ A_27a ) )
& ! [V11t1: tyop_2Esptree_2Espt @ A_27a,V12x: A_27a,V13v4: A_27a] : ( V0P @ V11t1 @ V12x @ ( c_2Esptree_2ELS @ A_27a @ V13v4 ) )
& ! [V14t1: tyop_2Esptree_2Espt @ A_27a,V15x: A_27a,V16v5: tyop_2Esptree_2Espt @ A_27a,V17v6: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V14t1 @ V15x @ ( c_2Esptree_2EBN @ A_27a @ V16v5 @ V17v6 ) )
& ! [V18t1: tyop_2Esptree_2Espt @ A_27a,V19x: A_27a,V20v7: tyop_2Esptree_2Espt @ A_27a,V21v8: A_27a,V22v9: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V18t1 @ V19x @ ( c_2Esptree_2EBS @ A_27a @ V20v7 @ V21v8 @ V22v9 ) ) )
=> ! [V23v: tyop_2Esptree_2Espt @ A_27a,V24v1: A_27a,V25v2: tyop_2Esptree_2Espt @ A_27a] : ( V0P @ V23v @ V24v1 @ V25v2 ) ) ).
thf(thm_2Esptree_2Emk__BS__def,axiom,
! [A_27a: $tType,V0x: A_27a,V1v9: tyop_2Esptree_2Espt @ A_27a,V2v8: A_27a,V3v7: tyop_2Esptree_2Espt @ A_27a,V4v6: tyop_2Esptree_2Espt @ A_27a,V5v5: tyop_2Esptree_2Espt @ A_27a,V6v4: A_27a,V7v21: tyop_2Esptree_2Espt @ A_27a,V8v20: A_27a,V9v19: tyop_2Esptree_2Espt @ A_27a,V10v18: tyop_2Esptree_2Espt @ A_27a,V11v17: tyop_2Esptree_2Espt @ A_27a,V12v16: A_27a,V13t1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2ELN @ A_27a ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELS @ A_27a @ V0x ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V12v16 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V12v16 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V11v17 @ V10v18 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V11v17 @ V10v18 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V9v19 @ V8v20 @ V7v21 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2EBS @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V9v19 @ V8v20 @ V7v21 ) @ V0x @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2ELS @ A_27a @ V6v4 ) )
= ( c_2Esptree_2EBS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2ELS @ A_27a @ V6v4 ) ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2EBN @ A_27a @ V5v5 @ V4v6 ) )
= ( c_2Esptree_2EBS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2EBN @ A_27a @ V5v5 @ V4v6 ) ) )
& ( ( c_2Esptree_2Emk__BS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2EBS @ A_27a @ V3v7 @ V2v8 @ V1v9 ) )
= ( c_2Esptree_2EBS @ A_27a @ V13t1 @ V0x @ ( c_2Esptree_2EBS @ A_27a @ V3v7 @ V2v8 @ V1v9 ) ) ) ) ).
thf(thm_2Esptree_2Einsert__notEmpty,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum,V2a: A_27a] :
( (~)
@ ( ( c_2Esptree_2Einsert @ A_27a @ V1k @ V2a @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) ) ) ).
thf(thm_2Esptree_2Ewf__insert,axiom,
! [A_27a: $tType,V0k: tyop_2Enum_2Enum,V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Ewf @ A_27a @ V2t )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1a @ V2t ) ) ) ).
thf(thm_2Esptree_2Ewf__delete,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Ewf @ A_27a @ V0t )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Edelete @ A_27a @ V1k @ V0t ) ) ) ).
thf(thm_2Esptree_2Elookup__insert1,axiom,
! [A_27a: $tType,V0k: tyop_2Enum_2Enum,V1a: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V0k @ ( c_2Esptree_2Einsert @ A_27a @ V0k @ V1a @ V2t ) )
= ( c_2Eoption_2ESOME @ A_27a @ V1a ) ) ).
thf(thm_2Esptree_2Elookup__insert,axiom,
! [A_27a: $tType,V0k2: tyop_2Enum_2Enum,V1v: A_27a,V2t: tyop_2Esptree_2Espt @ A_27a,V3k1: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V3k1 @ ( c_2Esptree_2Einsert @ A_27a @ V0k2 @ V1v @ V2t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3k1 @ V0k2 ) @ ( c_2Eoption_2ESOME @ A_27a @ V1v ) @ ( c_2Esptree_2Elookup @ A_27a @ V3k1 @ V2t ) ) ) ).
thf(thm_2Esptree_2EisEmpty__union,axiom,
! [A_27a: $tType,V0m2: tyop_2Esptree_2Espt @ A_27a,V1m1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Eunion @ A_27a @ V1m1 @ V0m2 )
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( V1m1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V0m2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ).
thf(thm_2Esptree_2Ewf__union,axiom,
! [A_27a: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Ewf @ A_27a @ V0m1 )
& ( c_2Esptree_2Ewf @ A_27a @ V1m2 ) )
=> ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Eunion @ A_27a @ V0m1 @ V1m2 ) ) ) ).
thf(thm_2Esptree_2Elookup__union,axiom,
! [A_27a: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2k @ ( c_2Esptree_2Eunion @ A_27a @ V0m1 @ V1m2 ) )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V0m1 ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V1m2 )
@ ^ [V3v: A_27a] : ( c_2Eoption_2ESOME @ A_27a @ V3v ) ) ) ).
thf(thm_2Esptree_2Ewf__inter,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27b] : ( c_2Esptree_2Ewf @ A_27a @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0m1 @ V1m2 ) ) ).
thf(thm_2Esptree_2Elookup__inter,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27b,V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2k @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0m1 @ V1m2 ) )
= ( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Epair_2E_2C @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V0m1 ) @ ( c_2Esptree_2Elookup @ A_27b @ V2k @ V1m2 ) )
@ ^ [V3v3: tyop_2Eoption_2Eoption @ A_27a,V4v4: tyop_2Eoption_2Eoption @ A_27b] :
( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V3v3 @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V5v: A_27a] :
( c_2Eoption_2Eoption__CASE @ A_27b @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V4v4 @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V6w: A_27b] : ( c_2Eoption_2ESOME @ A_27a @ V5v ) ) ) ) ) ).
thf(thm_2Esptree_2Elookup__inter__eq,axiom,
! [A_27a: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2k @ ( c_2Esptree_2Einter__eq @ A_27a @ V0m1 @ V1m2 ) )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V0m1 ) @ ( c_2Eoption_2ENONE @ A_27a )
@ ^ [V3v: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V1m2 ) @ ( c_2Eoption_2ESOME @ A_27a @ V3v ) ) @ ( c_2Eoption_2ESOME @ A_27a @ V3v ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(thm_2Esptree_2Elookup__inter__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a,V1x: tyop_2Enum_2Enum,V2t2: tyop_2Esptree_2Espt @ A_27b,V3t1: tyop_2Esptree_2Espt @ A_27a] :
( ( ( ( c_2Esptree_2Elookup @ A_27a @ V1x @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V3t1 @ V2t2 ) )
= ( c_2Eoption_2ESOME @ A_27a @ V0y ) )
<=> ( ( ( c_2Esptree_2Elookup @ A_27a @ V1x @ V3t1 )
= ( c_2Eoption_2ESOME @ A_27a @ V0y ) )
& ( (~)
@ ( ( c_2Esptree_2Elookup @ A_27b @ V1x @ V2t2 )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) )
& ( ( ( c_2Esptree_2Elookup @ A_27a @ V1x @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V3t1 @ V2t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( ( ( c_2Esptree_2Elookup @ A_27a @ V1x @ V3t1 )
= ( c_2Eoption_2ENONE @ A_27a ) )
| ( ( c_2Esptree_2Elookup @ A_27b @ V1x @ V2t2 )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ).
thf(thm_2Esptree_2Elookup__inter__assoc,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: tyop_2Enum_2Enum,V1t3: tyop_2Esptree_2Espt @ A_27c,V2t2: tyop_2Esptree_2Espt @ A_27b,V3t1: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Elookup @ A_27a @ V0x @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V3t1 @ ( c_2Esptree_2Einter @ A_27b @ A_27c @ V2t2 @ V1t3 ) ) )
= ( c_2Esptree_2Elookup @ A_27a @ V0x @ ( c_2Esptree_2Einter @ A_27a @ A_27c @ ( c_2Esptree_2Einter @ A_27a @ A_27b @ V3t1 @ V2t2 ) @ V1t3 ) ) ) ).
thf(thm_2Esptree_2Elookup__difference,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Esptree_2Espt @ A_27a,V1m2: tyop_2Esptree_2Espt @ A_27b,V2k: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27a @ V2k @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V0m1 @ V1m2 ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Esptree_2Elookup @ A_27b @ V2k @ V1m2 ) @ ( c_2Eoption_2ENONE @ A_27b ) ) @ ( c_2Esptree_2Elookup @ A_27a @ V2k @ V0m1 ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Esptree_2Elrnext__ind,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] :
( ( ( (~) @ ( V1n = c_2Enum_2E0 ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( V0P @ V1n ) )
=> ! [V2v: tyop_2Enum_2Enum] : ( V0P @ V2v ) ) ).
thf(thm_2Esptree_2Elrnext__def,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elrnext @ V0n )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Esptree_2Elrnext__thm,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: A_27a] :
( ( c_2Esptree_2Elrnext @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V1n: tyop_2Enum_2Enum,V2a: A_27b] :
( ( c_2Esptree_2Elrnext @ ( c_2Earithmetic_2ENUMERAL @ V1n ) )
= ( c_2Esptree_2Elrnext @ V1n ) )
& ( ( c_2Esptree_2Elrnext @ c_2Earithmetic_2EZERO )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elrnext @ ( c_2Earithmetic_2EBIT1 @ V3n ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V3n ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elrnext @ ( c_2Earithmetic_2EBIT2 @ V4n ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2Elrnext @ V4n ) ) ) ) ).
thf(thm_2Esptree_2EFINITE__domain,axiom,
! [A_27a: $tType,V0t: tyop_2Esptree_2Espt @ A_27a] : ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V0t ) ) ).
thf(thm_2Esptree_2EIN__domain,axiom,
! [A_27a: $tType,A_27b: $tType,V0n: tyop_2Enum_2Enum,V1x: A_27a,V2t1: tyop_2Esptree_2Espt @ A_27a,V3t2: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0n @ ( c_2Esptree_2Edomain @ A_27b @ ( c_2Esptree_2ELN @ A_27b ) ) )
= c_2Ebool_2EF )
& ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0n @ ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2ELS @ A_27a @ V1x ) ) )
<=> ( V0n = c_2Enum_2E0 ) )
& ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0n @ ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EBN @ A_27a @ V2t1 @ V3t2 ) ) )
<=> ( ( (~) @ ( V0n = c_2Enum_2E0 ) )
& ( c_2Ebool_2ECOND @ $o @ ( c_2Earithmetic_2EEVEN @ V0n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Esptree_2Edomain @ A_27a @ V2t1 ) ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Esptree_2Edomain @ A_27a @ V3t2 ) ) ) ) )
& ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0n @ ( c_2Esptree_2Edomain @ A_27a @ ( c_2Esptree_2EBS @ A_27a @ V2t1 @ V1x @ V3t2 ) ) )
<=> ( ( V0n = c_2Enum_2E0 )
| ( c_2Ebool_2ECOND @ $o @ ( c_2Earithmetic_2EEVEN @ V0n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Esptree_2Edomain @ A_27a @ V2t1 ) ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Esptree_2Edomain @ A_27a @ V3t2 ) ) ) ) ) ) ).
thf(thm_2Esptree_2Emap__map__K,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1a: A_27b,V2t: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Emap @ A_27b @ A_27c @ ( c_2Ecombin_2EK @ A_27b @ A_27c @ V1a ) @ ( c_2Esptree_2Emap @ A_27c @ A_27a @ V0f @ V2t ) )
= ( c_2Esptree_2Emap @ A_27b @ A_27a @ ( c_2Ecombin_2EK @ A_27b @ A_27a @ V1a ) @ V2t ) ) ).
thf(thm_2Esptree_2Elookup__map__K,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1t: tyop_2Esptree_2Espt @ A_27a,V2n: tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ A_27b @ V2n @ ( c_2Esptree_2Emap @ A_27b @ A_27a @ ( c_2Ecombin_2EK @ A_27b @ A_27a @ V0x ) @ V1t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ ( c_2Esptree_2Edomain @ A_27a @ V1t ) ) @ ( c_2Eoption_2ESOME @ A_27b @ V0x ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Esptree_2EIMP__size__LESS__size,axiom,
! [A_27a: $tType,V0x: tyop_2Esptree_2Espt @ A_27a,V1y: tyop_2Esptree_2Espt @ A_27a] :
( ( ( c_2Esptree_2Esubspt @ A_27a @ V0x @ V1y )
& ( (~)
@ ( ( c_2Esptree_2Edomain @ A_27a @ V0x )
= ( c_2Esptree_2Edomain @ A_27a @ V1y ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Esptree_2Esize @ A_27a @ V0x ) @ ( c_2Esptree_2Esize @ A_27a @ V1y ) ) ) ).
thf(thm_2Esptree_2Esize__diff__less,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: tyop_2Esptree_2Espt @ A_27a,V1y: tyop_2Esptree_2Espt @ A_27b,V2z: tyop_2Esptree_2Espt @ A_27c,V3t: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27c @ V2z ) @ ( c_2Esptree_2Edomain @ A_27b @ V1y ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3t @ ( c_2Esptree_2Edomain @ A_27b @ V1y ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3t @ ( c_2Esptree_2Edomain @ A_27c @ V2z ) ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3t @ ( c_2Esptree_2Edomain @ A_27a @ V0x ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27b @ V0x @ V1y ) ) @ ( c_2Esptree_2Esize @ A_27a @ ( c_2Esptree_2Edifference @ A_27a @ A_27c @ V0x @ V2z ) ) ) ) ).
thf(thm_2Esptree_2Einter__eq__LN,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esptree_2Espt @ A_27a,V1y: tyop_2Esptree_2Espt @ A_27b] :
( ( ( c_2Esptree_2Einter @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( c_2Epred__set_2EDISJOINT @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V0x ) @ ( c_2Esptree_2Edomain @ A_27b @ V1y ) ) ) ).
thf(thm_2Esptree_2Edomain__list__to__num__set,axiom,
! [V0x: tyop_2Enum_2Enum,V1xs: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ ( c_2Esptree_2Elist__to__num__set @ V1xs ) ) )
= ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ V1xs ) ) ) ).
thf(thm_2Esptree_2Edomain__list__insert,axiom,
! [V0xs: tyop_2Elist_2Elist @ tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2t: tyop_2Esptree_2Espt @ tyop_2Eone_2Eone] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ ( c_2Esptree_2Elist__insert @ V0xs @ V2t ) ) )
<=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ V0xs ) )
| ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ ( c_2Esptree_2Edomain @ tyop_2Eone_2Eone @ V2t ) ) ) ) ).
thf(thm_2Esptree_2Edomain__FOLDR__delete,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ tyop_2Enum_2Enum,V1live: tyop_2Esptree_2Espt @ A_27a] :
( ( c_2Esptree_2Edomain @ A_27a @ ( c_2Elist_2EFOLDR @ tyop_2Enum_2Enum @ ( tyop_2Esptree_2Espt @ A_27a ) @ ( c_2Esptree_2Edelete @ A_27a ) @ V1live @ V0ls ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ ( c_2Esptree_2Edomain @ A_27a @ V1live ) @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ V0ls ) ) ) ).
thf(thm_2Esptree_2Elookup__list__to__num__set,axiom,
! [V0x: tyop_2Enum_2Enum,V1xs: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elookup @ tyop_2Eone_2Eone @ V0x @ ( c_2Esptree_2Elist__to__num__set @ V1xs ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Eone_2Eone ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ V1xs ) ) @ ( c_2Eoption_2ESOME @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) @ ( c_2Eoption_2ENONE @ tyop_2Eone_2Eone ) ) ) ).
thf(thm_2Esptree_2Elist__to__num__set__append,axiom,
! [V0l1: tyop_2Elist_2Elist @ tyop_2Enum_2Enum,V1l2: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Esptree_2Elist__to__num__set @ ( c_2Elist_2EAPPEND @ tyop_2Enum_2Enum @ V0l1 @ V1l2 ) )
= ( c_2Esptree_2Eunion @ tyop_2Eone_2Eone @ ( c_2Esptree_2Elist__to__num__set @ V0l1 ) @ ( c_2Esptree_2Elist__to__num__set @ V1l2 ) ) ) ).
%------------------------------------------------------------------------------