ITP001 Axioms: ITP101^7.ax
%------------------------------------------------------------------------------
% File : ITP101^7 : 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 : patricia.ax [Gau19]
% : HL4101^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 239 ( 37 unt; 103 typ; 0 def)
% Number of atoms : 517 ( 172 equ; 27 cnn)
% Maximal formula atoms : 19 ( 2 avg)
% Number of connectives : 3991 ( 27 ~; 13 |; 124 &;3712 @)
% ( 27 <=>; 88 =>; 0 <=; 0 <~>)
% Maximal formula depth : 44 ( 9 avg;3712 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 350 ( 350 >; 0 *; 0 +; 0 <<)
% Number of symbols : 102 ( 100 usr; 6 con; 0-6 aty)
% Number of variables : 825 ( 66 ^ 666 !; 17 ?; 825 :)
% ( 76 !>; 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_2Epatricia_2Eptree,type,
tyop_2Epatricia_2Eptree: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
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_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_2Epatricia_2EADD,type,
c_2Epatricia_2EADD:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Epatricia_2EADD__LIST,type,
c_2Epatricia_2EADD__LIST:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
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_2Ebit_2EBIT,type,
c_2Ebit_2EBIT: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
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_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Epatricia_2EBRANCH,type,
c_2Epatricia_2EBRANCH:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Epatricia_2EBRANCHING__BIT,type,
c_2Epatricia_2EBRANCHING__BIT: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epatricia_2EBranch,type,
c_2Epatricia_2EBranch:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Ecombin_2EC,type,
c_2Ecombin_2EC:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > A_27b > A_27a > A_27c ) ).
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_2Epatricia_2EDEPTH,type,
c_2Epatricia_2EDEPTH:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Earithmetic_2EDIV2,type,
c_2Earithmetic_2EDIV2: 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_2Epatricia_2EEVERY__LEAF,type,
c_2Epatricia_2EEVERY__LEAF:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a > $o ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > $o ) ).
thf(c_2Epatricia_2EEXISTS__LEAF,type,
c_2Epatricia_2EEXISTS__LEAF:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a > $o ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > $o ) ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epatricia_2EEmpty,type,
c_2Epatricia_2EEmpty:
!>[A_27a: $tType] : ( tyop_2Epatricia_2Eptree @ A_27a ) ).
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_2Elist_2EFILTER,type,
c_2Elist_2EFILTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epatricia_2EFIND,type,
c_2Epatricia_2EFIND:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ 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_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
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_2Epatricia_2EINSERT__PTREE,type,
c_2Epatricia_2EINSERT__PTREE: tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) ).
thf(c_2Epatricia_2EIN__PTREE,type,
c_2Epatricia_2EIN__PTREE: tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > $o ).
thf(c_2Epatricia_2EIS__EMPTY,type,
c_2Epatricia_2EIS__EMPTY:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > $o ) ).
thf(c_2Epatricia_2EIS__PTREE,type,
c_2Epatricia_2EIS__PTREE:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > $o ) ).
thf(c_2Eoption_2EIS__SOME,type,
c_2Eoption_2EIS__SOME:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > $o ) ).
thf(c_2Epatricia_2EJOIN,type,
c_2Epatricia_2EJOIN:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Epatricia_2EKEYS,type,
c_2Epatricia_2EKEYS:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ) ).
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_2Epatricia_2ELeaf,type,
c_2Epatricia_2ELeaf:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Earithmetic_2EMAX,type,
c_2Earithmetic_2EMAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EMOD__2EXP,type,
c_2Ebit_2EMOD__2EXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebit_2EMOD__2EXP__EQ,type,
c_2Ebit_2EMOD__2EXP__EQ: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
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_2Epatricia_2ENUMSET__OF__PTREE,type,
c_2Epatricia_2ENUMSET__OF__PTREE: ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).
thf(c_2Epatricia_2EPEEK,type,
c_2Epatricia_2EPEEK:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Esorting_2EPERM,type,
c_2Esorting_2EPERM:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Epatricia_2EPTREE__OF__NUMSET,type,
c_2Epatricia_2EPTREE__OF__NUMSET: ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > ( tyop_2Enum_2Enum > $o ) > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) ).
thf(c_2Esorting_2EQSORT,type,
c_2Esorting_2EQSORT:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epatricia_2EREMOVE,type,
c_2Epatricia_2EREMOVE:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Elist_2ESET__TO__LIST,type,
c_2Elist_2ESET__TO__LIST:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epatricia_2ESIZE,type,
c_2Epatricia_2ESIZE:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
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_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
thf(c_2Epatricia_2ETRANSFORM,type,
c_2Epatricia_2ETRANSFORM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27b ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) ).
thf(c_2Epatricia_2ETRAVERSE,type,
c_2Epatricia_2ETRAVERSE:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ) ).
thf(c_2Epatricia_2ETRAVERSE__AUX,type,
c_2Epatricia_2ETRAVERSE__AUX:
!>[A_27a: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ) ).
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_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epatricia_2EUNION__PTREE,type,
c_2Epatricia_2EUNION__PTREE: ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) > ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) ).
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_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_2Epatricia_2Eptree__CASE,type,
c_2Epatricia_2Eptree__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b > ( tyop_2Enum_2Enum > A_27a > A_27b ) > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Epatricia_2Eptree__size,type,
c_2Epatricia_2Eptree__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum ) ).
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_2Epatricia_2Eptree__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ ( tyop_2Epatricia_2Eptree @ A_27a )
@ ^ [V1a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) )] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) > $o )
@ ^ [V2_27ptree_27: ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) )
@ ^ [V3a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) )] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ V3a0_27
@ ( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) @ c_2Enum_2E0 @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) @ ( c_2Ebool_2EARB @ tyop_2Enum_2Enum ) @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Enum_2Enum @ ( c_2Ebool_2EARB @ A_27a ) @ ( c_2Ebool_2EARB @ tyop_2Enum_2Enum ) ) )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V5a0: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V6a1: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ V3a0_27
@ ( ^ [V7a0: tyop_2Enum_2Enum,V8a1: A_27a] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) @ V7a0 @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Enum_2Enum @ V8a1 @ ( c_2Ebool_2EARB @ tyop_2Enum_2Enum ) ) )
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) )
@ V5a0
@ V6a1 ) ) ) )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V10a0: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V11a1: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) )
@ ^ [V12a2: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) )] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) )
@ ^ [V13a3: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ V3a0_27
@ ( ^ [V14a0: tyop_2Enum_2Enum,V15a1: tyop_2Enum_2Enum,V16a2: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ),V17a3: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) )] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) @ V14a0 @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Enum_2Enum @ ( c_2Ebool_2EARB @ A_27a ) @ V15a1 ) )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ V16a2
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) @ V17a3
@ ^ [V18n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Enum_2Enum ) ) ) ) ) )
@ V10a0
@ V11a1
@ V12a2
@ V13a3 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27ptree_27 @ V12a2 ) @ ( V2_27ptree_27 @ V13a3 ) ) ) ) ) ) ) ) )
@ ( V2_27ptree_27 @ V3a0_27 ) ) )
@ ( V2_27ptree_27 @ V1a0_27 ) ) )
@ V0rep ) ).
thf(thm_2Epatricia_2Eptree__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v: A_27b,V1f: tyop_2Enum_2Enum > A_27a > A_27b,V2f1: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b] :
( ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0v @ V1f @ V2f1 )
= V0v )
& ! [V3a0: tyop_2Enum_2Enum,V4a1: A_27a,V5v: A_27b,V6f: tyop_2Enum_2Enum > A_27a > A_27b,V7f1: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b] :
( ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ ( c_2Epatricia_2ELeaf @ A_27a @ V3a0 @ V4a1 ) @ V5v @ V6f @ V7f1 )
= ( V6f @ V3a0 @ V4a1 ) )
& ! [V8a0: tyop_2Enum_2Enum,V9a1: tyop_2Enum_2Enum,V10a2: tyop_2Epatricia_2Eptree @ A_27a,V11a3: tyop_2Epatricia_2Eptree @ A_27a,V12v: A_27b,V13f: tyop_2Enum_2Enum > A_27a > A_27b,V14f1: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b] :
( ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ ( c_2Epatricia_2EBranch @ A_27a @ V8a0 @ V9a1 @ V10a2 @ V11a3 ) @ V12v @ V13f @ V14f1 )
= ( V14f1 @ V8a0 @ V9a1 @ V10a2 @ V11a3 ) ) ) ).
thf(thm_2Epatricia_2Eptree__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Epatricia_2Eptree__size @ A_27a @ V0f @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Enum_2E0 )
& ! [V1f: A_27a > tyop_2Enum_2Enum,V2a0: tyop_2Enum_2Enum,V3a1: A_27a] :
( ( c_2Epatricia_2Eptree__size @ A_27a @ V1f @ ( c_2Epatricia_2ELeaf @ A_27a @ V2a0 @ V3a1 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ V2a0 @ ( V1f @ V3a1 ) ) ) )
& ! [V4f: A_27a > tyop_2Enum_2Enum,V5a0: tyop_2Enum_2Enum,V6a1: tyop_2Enum_2Enum,V7a2: tyop_2Epatricia_2Eptree @ A_27a,V8a3: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2Eptree__size @ A_27a @ V4f @ ( c_2Epatricia_2EBranch @ A_27a @ V5a0 @ V6a1 @ V7a2 @ V8a3 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ V5a0 @ ( c_2Earithmetic_2E_2B @ V6a1 @ ( c_2Earithmetic_2E_2B @ ( c_2Epatricia_2Eptree__size @ A_27a @ V4f @ V7a2 ) @ ( c_2Epatricia_2Eptree__size @ A_27a @ V4f @ V8a3 ) ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EJOIN__def,axiom,
! [A_27a: $tType,V0p0: tyop_2Enum_2Enum,V1t0: tyop_2Epatricia_2Eptree @ A_27a,V2p1: tyop_2Enum_2Enum,V3t1: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EJOIN @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V0p0 @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V1t0 @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V2p1 @ V3t1 ) ) ) )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a )
@ ^ [V4m: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EBIT @ V4m @ V0p0 ) @ ( c_2Epatricia_2EBranch @ A_27a @ ( c_2Ebit_2EMOD__2EXP @ V4m @ V0p0 ) @ V4m @ V1t0 @ V3t1 ) @ ( c_2Epatricia_2EBranch @ A_27a @ ( c_2Ebit_2EMOD__2EXP @ V4m @ V0p0 ) @ V4m @ V3t1 @ V1t0 ) )
@ ( c_2Epatricia_2EBRANCHING__BIT @ V0p0 @ V2p1 ) ) ) ).
thf(thm_2Epatricia_2EBRANCH__primitive__def,axiom,
! [A_27a: $tType] :
( ( c_2Epatricia_2EBRANCH @ A_27a )
= ( c_2Erelation_2EWFREC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( tyop_2Epatricia_2Eptree @ A_27a )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > $o )
@ ^ [V0R: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > $o] : ( c_2Erelation_2EWF @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) @ V0R ) )
@ ^ [V1BRANCH: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > ( tyop_2Epatricia_2Eptree @ A_27a ),V2a: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) )] :
( c_2Epair_2Epair__CASE @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V2a
@ ^ [V3p: tyop_2Enum_2Enum,V4v1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) )] :
( c_2Epair_2Epair__CASE @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V4v1
@ ^ [V5m: tyop_2Enum_2Enum,V6v3: tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a )] :
( c_2Epair_2Epair__CASE @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V6v3
@ ^ [V7v4: tyop_2Epatricia_2Eptree @ A_27a,V8t: tyop_2Epatricia_2Eptree @ A_27a] :
( c_2Epatricia_2Eptree__CASE @ A_27a @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V7v4 @ ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V8t )
@ ^ [V9v18: tyop_2Enum_2Enum,V10v19: A_27a] :
( c_2Epatricia_2Eptree__CASE @ A_27a @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V8t @ ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V9v18 @ V10v19 ) )
@ ^ [V11v30: tyop_2Enum_2Enum,V12v31: A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V3p @ V5m @ ( c_2Epatricia_2ELeaf @ A_27a @ V9v18 @ V10v19 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V11v30 @ V12v31 ) ) )
@ ^ [V13v32: tyop_2Enum_2Enum,V14v33: tyop_2Enum_2Enum,V15v34: tyop_2Epatricia_2Eptree @ A_27a,V16v35: tyop_2Epatricia_2Eptree @ A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V3p @ V5m @ ( c_2Epatricia_2ELeaf @ A_27a @ V9v18 @ V10v19 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V13v32 @ V14v33 @ V15v34 @ V16v35 ) ) ) )
@ ^ [V17v20: tyop_2Enum_2Enum,V18v21: tyop_2Enum_2Enum,V19v22: tyop_2Epatricia_2Eptree @ A_27a,V20v23: tyop_2Epatricia_2Eptree @ A_27a] :
( c_2Epatricia_2Eptree__CASE @ A_27a @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V8t @ ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V17v20 @ V18v21 @ V19v22 @ V20v23 ) )
@ ^ [V21v42: tyop_2Enum_2Enum,V22v43: A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V3p @ V5m @ ( c_2Epatricia_2EBranch @ A_27a @ V17v20 @ V18v21 @ V19v22 @ V20v23 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V21v42 @ V22v43 ) ) )
@ ^ [V23v44: tyop_2Enum_2Enum,V24v45: tyop_2Enum_2Enum,V25v46: tyop_2Epatricia_2Eptree @ A_27a,V26v47: tyop_2Epatricia_2Eptree @ A_27a] : ( c_2Ecombin_2EI @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V3p @ V5m @ ( c_2Epatricia_2EBranch @ A_27a @ V17v20 @ V18v21 @ V19v22 @ V20v23 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V23v44 @ V24v45 @ V25v46 @ V26v47 ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EREMOVE__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0k )
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ! [V1j: tyop_2Enum_2Enum,V2d: A_27a,V3k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V1j @ V2d ) @ V3k )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1j @ V3k ) @ ( c_2Epatricia_2EEmpty @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V1j @ V2d ) ) )
& ! [V4p: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum,V6l: tyop_2Epatricia_2Eptree @ A_27a,V7r: tyop_2Epatricia_2Eptree @ A_27a,V8k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V4p @ V5m @ V6l @ V7r ) @ V8k )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EMOD__2EXP__EQ @ V5m @ V8k @ V4p ) @ ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EBIT @ V5m @ V8k ) @ ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V4p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V5m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EREMOVE @ A_27a @ V6l @ V8k ) @ V7r ) ) ) ) @ ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V4p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V5m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V6l @ ( c_2Epatricia_2EREMOVE @ A_27a @ V7r @ V8k ) ) ) ) ) ) @ ( c_2Epatricia_2EBranch @ A_27a @ V4p @ V5m @ V6l @ V7r ) ) ) ) ).
thf(thm_2Epatricia_2ETRAVERSE__AUX__def,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0a )
= V0a )
& ! [V1k: tyop_2Enum_2Enum,V2d: A_27a,V3a: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V1k @ V2d ) @ V3a )
= ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V1k @ V3a ) )
& ! [V4p: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum,V6l: tyop_2Epatricia_2Eptree @ A_27a,V7r: tyop_2Epatricia_2Eptree @ A_27a,V8a: tyop_2Elist_2Elist @ tyop_2Enum_2Enum] :
( ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V4p @ V5m @ V6l @ V7r ) @ V8a )
= ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ V6l @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ V7r @ V8a ) ) ) ) ).
thf(thm_2Epatricia_2ETRAVERSE__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) )
& ! [V0j: tyop_2Enum_2Enum,V1d: A_27a] :
( ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V0j @ V1d ) )
= ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V0j @ ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) )
& ! [V2p: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4l: tyop_2Epatricia_2Eptree @ A_27a,V5r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V2p @ V3m @ V4l @ V5r ) )
= ( c_2Elist_2EAPPEND @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V4l ) @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V5r ) ) ) ) ).
thf(thm_2Epatricia_2EKEYS__def,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EKEYS @ A_27a @ V0t )
= ( c_2Esorting_2EQSORT @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ).
thf(thm_2Epatricia_2ETRANSFORM__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27b > A_27a] :
( ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b @ V0f @ ( c_2Epatricia_2EEmpty @ A_27b ) )
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ! [V1f: A_27b > A_27a,V2j: tyop_2Enum_2Enum,V3d: A_27b] :
( ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b @ V1f @ ( c_2Epatricia_2ELeaf @ A_27b @ V2j @ V3d ) )
= ( c_2Epatricia_2ELeaf @ A_27a @ V2j @ ( V1f @ V3d ) ) )
& ! [V4f: A_27b > A_27a,V5p: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7l: tyop_2Epatricia_2Eptree @ A_27b,V8r: tyop_2Epatricia_2Eptree @ A_27b] :
( ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b @ V4f @ ( c_2Epatricia_2EBranch @ A_27b @ V5p @ V6m @ V7l @ V8r ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V5p @ V6m @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b @ V4f @ V7l ) @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b @ V4f @ V8r ) ) ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__def,axiom,
! [A_27a: $tType] :
( ! [V0P: tyop_2Enum_2Enum > A_27a > $o] :
( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Ebool_2ET )
& ! [V1P: tyop_2Enum_2Enum > A_27a > $o,V2j: tyop_2Enum_2Enum,V3d: A_27a] :
( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V1P @ ( c_2Epatricia_2ELeaf @ A_27a @ V2j @ V3d ) )
= ( V1P @ V2j @ V3d ) )
& ! [V4P: tyop_2Enum_2Enum > A_27a > $o,V5p: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7l: tyop_2Epatricia_2Eptree @ A_27a,V8r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V4P @ ( c_2Epatricia_2EBranch @ A_27a @ V5p @ V6m @ V7l @ V8r ) )
<=> ( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V4P @ V7l )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V4P @ V8r ) ) ) ) ).
thf(thm_2Epatricia_2EEXISTS__LEAF__def,axiom,
! [A_27a: $tType] :
( ! [V0P: tyop_2Enum_2Enum > A_27a > $o] :
( ( c_2Epatricia_2EEXISTS__LEAF @ A_27a @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Ebool_2EF )
& ! [V1P: tyop_2Enum_2Enum > A_27a > $o,V2j: tyop_2Enum_2Enum,V3d: A_27a] :
( ( c_2Epatricia_2EEXISTS__LEAF @ A_27a @ V1P @ ( c_2Epatricia_2ELeaf @ A_27a @ V2j @ V3d ) )
= ( V1P @ V2j @ V3d ) )
& ! [V4P: tyop_2Enum_2Enum > A_27a > $o,V5p: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7l: tyop_2Epatricia_2Eptree @ A_27a,V8r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EEXISTS__LEAF @ A_27a @ V4P @ ( c_2Epatricia_2EBranch @ A_27a @ V5p @ V6m @ V7l @ V8r ) )
<=> ( ( c_2Epatricia_2EEXISTS__LEAF @ A_27a @ V4P @ V7l )
| ( c_2Epatricia_2EEXISTS__LEAF @ A_27a @ V4P @ V8r ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE__def,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ESIZE @ A_27a @ V0t )
= ( c_2Elist_2ELENGTH @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ).
thf(thm_2Epatricia_2EDEPTH__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Epatricia_2EDEPTH @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Enum_2E0 )
& ! [V0j: tyop_2Enum_2Enum,V1d: A_27a] :
( ( c_2Epatricia_2EDEPTH @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V0j @ V1d ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V2p: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4l: tyop_2Epatricia_2Eptree @ A_27a,V5r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EDEPTH @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V2p @ V3m @ V4l @ V5r ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2EMAX @ ( c_2Epatricia_2EDEPTH @ A_27a @ V4l ) @ ( c_2Epatricia_2EDEPTH @ A_27a @ V5r ) ) ) ) ) ).
thf(thm_2Epatricia_2EIS__PTREE__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Ebool_2ET )
& ! [V0k: tyop_2Enum_2Enum,V1d: A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V0k @ V1d ) )
= c_2Ebool_2ET )
& ! [V2p: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4l: tyop_2Epatricia_2Eptree @ A_27a,V5r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V2p @ V3m @ V4l @ V5r ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V2p @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3m ) )
& ( c_2Epatricia_2EIS__PTREE @ A_27a @ V4l )
& ( c_2Epatricia_2EIS__PTREE @ A_27a @ V5r )
& ( (~)
@ ( V4l
= ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ( (~)
@ ( V5r
= ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a
@ ^ [V6k: tyop_2Enum_2Enum,V7d: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebit_2EMOD__2EXP__EQ @ V3m @ V6k @ V2p ) @ ( c_2Ebit_2EBIT @ V3m @ V6k ) )
@ V4l )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a
@ ^ [V8k: tyop_2Enum_2Enum,V9d: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebit_2EMOD__2EXP__EQ @ V3m @ V8k @ V2p ) @ ( c_2Ebool_2E_7E @ ( c_2Ebit_2EBIT @ V3m @ V8k ) ) )
@ V5r ) ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EIN__PTREE @ V0n @ V1t )
= ( c_2Eoption_2EIS__SOME @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPEEK @ tyop_2Eone_2Eone @ V1t @ V0n ) ) ) ).
thf(thm_2Epatricia_2EINSERT__PTREE__def,axiom,
! [V0n: tyop_2Enum_2Enum,V1t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EINSERT__PTREE @ V0n @ V1t )
= ( c_2Epatricia_2EADD @ tyop_2Eone_2Eone @ V1t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Eone_2Eone @ V0n @ c_2Eone_2Eone ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__def,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s )
= ( c_2Elist_2EFOLDL @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ ( c_2Ecombin_2EC @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ c_2Epatricia_2EINSERT__PTREE ) @ V0t @ ( c_2Elist_2ESET__TO__LIST @ tyop_2Enum_2Enum @ V1s ) ) ) ).
thf(thm_2Epatricia_2ENUMSET__OF__PTREE__def,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t )
= ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__def,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ V1t2 )
= ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t1 @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V1t2 ) ) ) ).
thf(thm_2Epatricia_2EIS__EMPTY__primitive__def,axiom,
! [A_27a: $tType] :
( ( c_2Epatricia_2EIS__EMPTY @ A_27a )
= ( c_2Erelation_2EWFREC @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ $o
@ ( c_2Emin_2E_40 @ ( ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > $o )
@ ^ [V0R: ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > $o] : ( c_2Erelation_2EWF @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V0R ) )
@ ^ [V1IS__EMPTY: ( tyop_2Epatricia_2Eptree @ A_27a ) > $o,V2a: tyop_2Epatricia_2Eptree @ A_27a] :
( c_2Epatricia_2Eptree__CASE @ A_27a @ $o @ V2a @ ( c_2Ecombin_2EI @ $o @ c_2Ebool_2ET )
@ ^ [V3v6: tyop_2Enum_2Enum,V4v7: A_27a] : ( c_2Ecombin_2EI @ $o @ c_2Ebool_2EF )
@ ^ [V5v8: tyop_2Enum_2Enum,V6v9: tyop_2Enum_2Enum,V7v10: tyop_2Epatricia_2Eptree @ A_27a,V8v11: tyop_2Epatricia_2Eptree @ A_27a] : ( c_2Ecombin_2EI @ $o @ c_2Ebool_2EF ) ) ) ) ).
thf(thm_2Epatricia_2EFIND__def,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EFIND @ A_27a @ V0t @ V1k )
= ( c_2Eoption_2ETHE @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V0t @ V1k ) ) ) ).
thf(thm_2Epatricia_2EADD__LIST__def,axiom,
! [A_27a: $tType] :
( ( c_2Epatricia_2EADD__LIST @ A_27a )
= ( c_2Elist_2EFOLDL @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EADD @ A_27a ) ) ) ).
thf(thm_2Epatricia_2Edatatype__ptree,axiom,
! [A_27a: $tType,V0ptree: ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Enum_2Enum > A_27a > ( tyop_2Epatricia_2Eptree @ A_27a ) ) > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0ptree @ ( c_2Epatricia_2EEmpty @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a ) ) ) ).
thf(thm_2Epatricia_2Eptree__11,axiom,
! [A_27a: $tType] :
( ! [V0a0: tyop_2Enum_2Enum,V1a1: A_27a,V2a0_27: tyop_2Enum_2Enum,V3a1_27: A_27a] :
( ( ( c_2Epatricia_2ELeaf @ A_27a @ V0a0 @ V1a1 )
= ( c_2Epatricia_2ELeaf @ A_27a @ V2a0_27 @ V3a1_27 ) )
<=> ( ( V0a0 = V2a0_27 )
& ( V1a1 = V3a1_27 ) ) )
& ! [V4a0: tyop_2Enum_2Enum,V5a1: tyop_2Enum_2Enum,V6a2: tyop_2Epatricia_2Eptree @ A_27a,V7a3: tyop_2Epatricia_2Eptree @ A_27a,V8a0_27: tyop_2Enum_2Enum,V9a1_27: tyop_2Enum_2Enum,V10a2_27: tyop_2Epatricia_2Eptree @ A_27a,V11a3_27: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ( c_2Epatricia_2EBranch @ A_27a @ V4a0 @ V5a1 @ V6a2 @ V7a3 )
= ( c_2Epatricia_2EBranch @ A_27a @ V8a0_27 @ V9a1_27 @ V10a2_27 @ V11a3_27 ) )
<=> ( ( V4a0 = V8a0_27 )
& ( V5a1 = V9a1_27 )
& ( V6a2 = V10a2_27 )
& ( V7a3 = V11a3_27 ) ) ) ) ).
thf(thm_2Epatricia_2Eptree__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a1: A_27a,V1a0: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Epatricia_2EEmpty @ A_27a )
= ( c_2Epatricia_2ELeaf @ A_27a @ V1a0 @ V0a1 ) ) )
& ! [V2a3: tyop_2Epatricia_2Eptree @ A_27a,V3a2: tyop_2Epatricia_2Eptree @ A_27a,V4a1: tyop_2Enum_2Enum,V5a0: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Epatricia_2EEmpty @ A_27a )
= ( c_2Epatricia_2EBranch @ A_27a @ V5a0 @ V4a1 @ V3a2 @ V2a3 ) ) )
& ! [V6a3: tyop_2Epatricia_2Eptree @ A_27a,V7a2: tyop_2Epatricia_2Eptree @ A_27a,V8a1_27: tyop_2Enum_2Enum,V9a1: A_27a,V10a0_27: tyop_2Enum_2Enum,V11a0: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Epatricia_2ELeaf @ A_27a @ V11a0 @ V9a1 )
= ( c_2Epatricia_2EBranch @ A_27a @ V10a0_27 @ V8a1_27 @ V7a2 @ V6a3 ) ) ) ) ).
thf(thm_2Epatricia_2Eptree__nchotomy,axiom,
! [A_27a: $tType,V0pp: tyop_2Epatricia_2Eptree @ A_27a] :
( ( V0pp
= ( c_2Epatricia_2EEmpty @ A_27a ) )
| ? [V1n: tyop_2Enum_2Enum,V2a: A_27a] :
( V0pp
= ( c_2Epatricia_2ELeaf @ A_27a @ V1n @ V2a ) )
| ? [V3n0: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum,V5p: tyop_2Epatricia_2Eptree @ A_27a,V6p0: tyop_2Epatricia_2Eptree @ A_27a] :
( V0pp
= ( c_2Epatricia_2EBranch @ A_27a @ V3n0 @ V4n @ V5p @ V6p0 ) ) ) ).
thf(thm_2Epatricia_2Eptree__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27b,V1f1: tyop_2Enum_2Enum > A_27a > A_27b,V2f2: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b > A_27b > A_27b] :
? [V3fn: ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b] :
( ( ( V3fn @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= V0f0 )
& ! [V4a0: tyop_2Enum_2Enum,V5a1: A_27a] :
( ( V3fn @ ( c_2Epatricia_2ELeaf @ A_27a @ V4a0 @ V5a1 ) )
= ( V1f1 @ V4a0 @ V5a1 ) )
& ! [V6a0: tyop_2Enum_2Enum,V7a1: tyop_2Enum_2Enum,V8a2: tyop_2Epatricia_2Eptree @ A_27a,V9a3: tyop_2Epatricia_2Eptree @ A_27a] :
( ( V3fn @ ( c_2Epatricia_2EBranch @ A_27a @ V6a0 @ V7a1 @ V8a2 @ V9a3 ) )
= ( V2f2 @ V6a0 @ V7a1 @ V8a2 @ V9a3 @ ( V3fn @ V8a2 ) @ ( V3fn @ V9a3 ) ) ) ) ).
thf(thm_2Epatricia_2Eptree__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Epatricia_2Eptree @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) )
& ! [V1n: tyop_2Enum_2Enum,V2a: A_27a] : ( V0P @ ( c_2Epatricia_2ELeaf @ A_27a @ V1n @ V2a ) )
& ! [V3p: tyop_2Epatricia_2Eptree @ A_27a,V4p0: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ( V0P @ V3p )
& ( V0P @ V4p0 ) )
=> ! [V5n: tyop_2Enum_2Enum,V6n0: tyop_2Enum_2Enum] : ( V0P @ ( c_2Epatricia_2EBranch @ A_27a @ V6n0 @ V5n @ V3p @ V4p0 ) ) ) )
=> ! [V7p: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ V7p ) ) ).
thf(thm_2Epatricia_2Eptree__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27: A_27b,V1f1_27: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b,V2f_27: tyop_2Enum_2Enum > A_27a > A_27b,V3M: tyop_2Epatricia_2Eptree @ A_27a,V4M_27: tyop_2Epatricia_2Eptree @ A_27a,V5v: A_27b,V6f: tyop_2Enum_2Enum > A_27a > A_27b,V7f1: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b] :
( ( ( V3M = V4M_27 )
& ( ( V4M_27
= ( c_2Epatricia_2EEmpty @ A_27a ) )
=> ( V5v = V0v_27 ) )
& ! [V8a0: tyop_2Enum_2Enum,V9a1: A_27a] :
( ( V4M_27
= ( c_2Epatricia_2ELeaf @ A_27a @ V8a0 @ V9a1 ) )
=> ( ( V6f @ V8a0 @ V9a1 )
= ( V2f_27 @ V8a0 @ V9a1 ) ) )
& ! [V10a0: tyop_2Enum_2Enum,V11a1: tyop_2Enum_2Enum,V12a2: tyop_2Epatricia_2Eptree @ A_27a,V13a3: tyop_2Epatricia_2Eptree @ A_27a] :
( ( V4M_27
= ( c_2Epatricia_2EBranch @ A_27a @ V10a0 @ V11a1 @ V12a2 @ V13a3 ) )
=> ( ( V7f1 @ V10a0 @ V11a1 @ V12a2 @ V13a3 )
= ( V1f1_27 @ V10a0 @ V11a1 @ V12a2 @ V13a3 ) ) ) )
=> ( ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ V3M @ V5v @ V6f @ V7f1 )
= ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ V4M_27 @ V0v_27 @ V2f_27 @ V1f1_27 ) ) ) ).
thf(thm_2Epatricia_2Eptree__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epatricia_2Eptree @ A_27a,V1v_27: A_27b,V2v: A_27b,V3f1: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epatricia_2Eptree @ A_27a ) > A_27b,V4f: tyop_2Enum_2Enum > A_27a > A_27b] :
( ( ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b @ V0x @ V2v @ V4f @ V3f1 )
= V1v_27 )
<=> ( ( ( V0x
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( V2v = V1v_27 ) )
| ? [V5n: tyop_2Enum_2Enum,V6a: A_27a] :
( ( V0x
= ( c_2Epatricia_2ELeaf @ A_27a @ V5n @ V6a ) )
& ( ( V4f @ V5n @ V6a )
= V1v_27 ) )
| ? [V7n0: tyop_2Enum_2Enum,V8n: tyop_2Enum_2Enum,V9p: tyop_2Epatricia_2Eptree @ A_27a,V10p0: tyop_2Epatricia_2Eptree @ A_27a] :
( ( V0x
= ( c_2Epatricia_2EBranch @ A_27a @ V7n0 @ V8n @ V9p @ V10p0 ) )
& ( ( V3f1 @ V7n0 @ V8n @ V9p @ V10p0 )
= V1v_27 ) ) ) ) ).
thf(thm_2Epatricia_2EBRANCHING__BIT__ind,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ! [V1p0: tyop_2Enum_2Enum,V2p1: tyop_2Enum_2Enum] :
( ( ( (~)
@ ( ( ( c_2Earithmetic_2EODD @ V1p0 )
= ( c_2Earithmetic_2EEVEN @ V2p1 ) )
| ( V1p0 = V2p1 ) ) )
=> ( V0P @ ( c_2Earithmetic_2EDIV2 @ V1p0 ) @ ( c_2Earithmetic_2EDIV2 @ V2p1 ) ) )
=> ( V0P @ V1p0 @ V2p1 ) )
=> ! [V3v: tyop_2Enum_2Enum,V4v1: tyop_2Enum_2Enum] : ( V0P @ V3v @ V4v1 ) ) ).
thf(thm_2Epatricia_2EBRANCHING__BIT__def,axiom,
! [V0p1: tyop_2Enum_2Enum,V1p0: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EBRANCHING__BIT @ V1p0 @ V0p1 )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ $o @ ( c_2Earithmetic_2EODD @ V1p0 ) @ ( c_2Earithmetic_2EEVEN @ V0p1 ) ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1p0 @ V0p1 ) ) @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ ( c_2Epatricia_2EBRANCHING__BIT @ ( c_2Earithmetic_2EDIV2 @ V1p0 ) @ ( c_2Earithmetic_2EDIV2 @ V0p1 ) ) ) ) ) ).
thf(thm_2Epatricia_2EPEEK__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Epatricia_2Eptree @ A_27a ) > tyop_2Enum_2Enum > $o] :
( ( ! [V1k: tyop_2Enum_2Enum] : ( V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V1k )
& ! [V2j: tyop_2Enum_2Enum,V3d: A_27a,V4k: tyop_2Enum_2Enum] : ( V0P @ ( c_2Epatricia_2ELeaf @ A_27a @ V2j @ V3d ) @ V4k )
& ! [V5p: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7l: tyop_2Epatricia_2Eptree @ A_27a,V8r: tyop_2Epatricia_2Eptree @ A_27a,V9k: tyop_2Enum_2Enum] :
( ( V0P @ ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EBIT @ V6m @ V9k ) @ V7l @ V8r ) @ V9k )
=> ( V0P @ ( c_2Epatricia_2EBranch @ A_27a @ V5p @ V6m @ V7l @ V8r ) @ V9k ) ) )
=> ! [V10v: tyop_2Epatricia_2Eptree @ A_27a,V11v1: tyop_2Enum_2Enum] : ( V0P @ V10v @ V11v1 ) ) ).
thf(thm_2Epatricia_2EPEEK__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0k )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1k: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum,V3d: A_27a] :
( ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V2j @ V3d ) @ V1k )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ V2j ) @ ( c_2Eoption_2ESOME @ A_27a @ V3d ) @ ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V4r: tyop_2Epatricia_2Eptree @ A_27a,V5p: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7l: tyop_2Epatricia_2Eptree @ A_27a,V8k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V5p @ V6m @ V7l @ V4r ) @ V8k )
= ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EBIT @ V6m @ V8k ) @ V7l @ V4r ) @ V8k ) ) ) ).
thf(thm_2Epatricia_2EADD__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Epatricia_2Eptree @ A_27a ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a ) > $o] :
( ( ! [V1k: tyop_2Enum_2Enum,V2e: A_27a] : ( V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2e ) )
& ! [V3j: tyop_2Enum_2Enum,V4d: A_27a,V5k: tyop_2Enum_2Enum,V6e: A_27a] : ( V0P @ ( c_2Epatricia_2ELeaf @ A_27a @ V3j @ V4d ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V5k @ V6e ) )
& ! [V7p: tyop_2Enum_2Enum,V8m: tyop_2Enum_2Enum,V9l: tyop_2Epatricia_2Eptree @ A_27a,V10r: tyop_2Epatricia_2Eptree @ A_27a,V11k: tyop_2Enum_2Enum,V12e: A_27a] :
( ( ( ( ( c_2Ebit_2EMOD__2EXP__EQ @ V8m @ V11k @ V7p )
& ( (~) @ ( c_2Ebit_2EBIT @ V8m @ V11k ) ) )
=> ( V0P @ V10r @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V11k @ V12e ) ) )
& ( ( ( c_2Ebit_2EMOD__2EXP__EQ @ V8m @ V11k @ V7p )
& ( c_2Ebit_2EBIT @ V8m @ V11k ) )
=> ( V0P @ V9l @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V11k @ V12e ) ) ) )
=> ( V0P @ ( c_2Epatricia_2EBranch @ A_27a @ V7p @ V8m @ V9l @ V10r ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V11k @ V12e ) ) ) )
=> ! [V13v: tyop_2Epatricia_2Eptree @ A_27a,V14v1: tyop_2Enum_2Enum,V15v2: A_27a] : ( V0P @ V13v @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V14v1 @ V15v2 ) ) ) ).
thf(thm_2Epatricia_2EADD__def,axiom,
! [A_27a: $tType] :
( ! [V0k: tyop_2Enum_2Enum,V1e: A_27a] :
( ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V0k @ V1e ) )
= ( c_2Epatricia_2ELeaf @ A_27a @ V0k @ V1e ) )
& ! [V2k: tyop_2Enum_2Enum,V3j: tyop_2Enum_2Enum,V4e: A_27a,V5d: A_27a] :
( ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V3j @ V5d ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2k @ V4e ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3j @ V2k ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V2k @ V4e ) @ ( c_2Epatricia_2EJOIN @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V2k @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V2k @ V4e ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V3j @ ( c_2Epatricia_2ELeaf @ A_27a @ V3j @ V5d ) ) ) ) ) ) )
& ! [V6r: tyop_2Epatricia_2Eptree @ A_27a,V7p: tyop_2Enum_2Enum,V8m: tyop_2Enum_2Enum,V9l: tyop_2Epatricia_2Eptree @ A_27a,V10k: tyop_2Enum_2Enum,V11e: A_27a] :
( ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V7p @ V8m @ V9l @ V6r ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V10k @ V11e ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EMOD__2EXP__EQ @ V8m @ V10k @ V7p ) @ ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Ebit_2EBIT @ V8m @ V10k ) @ ( c_2Epatricia_2EBranch @ A_27a @ V7p @ V8m @ ( c_2Epatricia_2EADD @ A_27a @ V9l @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V10k @ V11e ) ) @ V6r ) @ ( c_2Epatricia_2EBranch @ A_27a @ V7p @ V8m @ V9l @ ( c_2Epatricia_2EADD @ A_27a @ V6r @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V10k @ V11e ) ) ) ) @ ( c_2Epatricia_2EJOIN @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V10k @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V10k @ V11e ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V7p @ ( c_2Epatricia_2EBranch @ A_27a @ V7p @ V8m @ V9l @ V6r ) ) ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EBRANCH__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) ) > $o] :
( ( ! [V1p: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3t: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V1p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V2m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V3t ) ) ) )
& ! [V4p: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum,V6v6: tyop_2Enum_2Enum,V7v7: A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V4p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V5m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V6v6 @ V7v7 ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
& ! [V8p: tyop_2Enum_2Enum,V9m: tyop_2Enum_2Enum,V10v8: tyop_2Enum_2Enum,V11v9: tyop_2Enum_2Enum,V12v10: tyop_2Epatricia_2Eptree @ A_27a,V13v11: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V8p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V9m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V10v8 @ V11v9 @ V12v10 @ V13v11 ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
& ! [V14p: tyop_2Enum_2Enum,V15m: tyop_2Enum_2Enum,V16v12: tyop_2Enum_2Enum,V17v13: A_27a,V18v24: tyop_2Enum_2Enum,V19v25: A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V14p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V15m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V16v12 @ V17v13 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V18v24 @ V19v25 ) ) ) ) )
& ! [V20p: tyop_2Enum_2Enum,V21m: tyop_2Enum_2Enum,V22v12: tyop_2Enum_2Enum,V23v13: A_27a,V24v26: tyop_2Enum_2Enum,V25v27: tyop_2Enum_2Enum,V26v28: tyop_2Epatricia_2Eptree @ A_27a,V27v29: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V20p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V21m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V22v12 @ V23v13 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V24v26 @ V25v27 @ V26v28 @ V27v29 ) ) ) ) )
& ! [V28p: tyop_2Enum_2Enum,V29m: tyop_2Enum_2Enum,V30v14: tyop_2Enum_2Enum,V31v15: tyop_2Enum_2Enum,V32v16: tyop_2Epatricia_2Eptree @ A_27a,V33v17: tyop_2Epatricia_2Eptree @ A_27a,V34v36: tyop_2Enum_2Enum,V35v37: A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V28p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V29m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V30v14 @ V31v15 @ V32v16 @ V33v17 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V34v36 @ V35v37 ) ) ) ) )
& ! [V36p: tyop_2Enum_2Enum,V37m: tyop_2Enum_2Enum,V38v14: tyop_2Enum_2Enum,V39v15: tyop_2Enum_2Enum,V40v16: tyop_2Epatricia_2Eptree @ A_27a,V41v17: tyop_2Epatricia_2Eptree @ A_27a,V42v38: tyop_2Enum_2Enum,V43v39: tyop_2Enum_2Enum,V44v40: tyop_2Epatricia_2Eptree @ A_27a,V45v41: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V36p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V37m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V38v14 @ V39v15 @ V40v16 @ V41v17 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V42v38 @ V43v39 @ V44v40 @ V45v41 ) ) ) ) ) )
=> ! [V46v: tyop_2Enum_2Enum,V47v1: tyop_2Enum_2Enum,V48v2: tyop_2Epatricia_2Eptree @ A_27a,V49v3: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V46v @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V47v1 @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V48v2 @ V49v3 ) ) ) ) ) ).
thf(thm_2Epatricia_2EBRANCH__def,axiom,
! [A_27a: $tType,V0v9: tyop_2Enum_2Enum,V1v8: tyop_2Enum_2Enum,V2v7: A_27a,V3v6: tyop_2Enum_2Enum,V4v41: tyop_2Epatricia_2Eptree @ A_27a,V5v40: tyop_2Epatricia_2Eptree @ A_27a,V6v39: tyop_2Enum_2Enum,V7v38: tyop_2Enum_2Enum,V8v37: A_27a,V9v36: tyop_2Enum_2Enum,V10v29: tyop_2Epatricia_2Eptree @ A_27a,V11v28: tyop_2Epatricia_2Eptree @ A_27a,V12v27: tyop_2Enum_2Enum,V13v26: tyop_2Enum_2Enum,V14v25: A_27a,V15v24: tyop_2Enum_2Enum,V16v17: tyop_2Epatricia_2Eptree @ A_27a,V17v16: tyop_2Epatricia_2Eptree @ A_27a,V18v15: tyop_2Enum_2Enum,V19v14: tyop_2Enum_2Enum,V20v13: A_27a,V21v12: tyop_2Enum_2Enum,V22v11: tyop_2Epatricia_2Eptree @ A_27a,V23v10: tyop_2Epatricia_2Eptree @ A_27a,V24t: tyop_2Epatricia_2Eptree @ A_27a,V25p: tyop_2Enum_2Enum,V26m: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V24t ) ) ) )
= V24t )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V3v6 @ V2v7 ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
= ( c_2Epatricia_2ELeaf @ A_27a @ V3v6 @ V2v7 ) )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V1v8 @ V0v9 @ V23v10 @ V22v11 ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V1v8 @ V0v9 @ V23v10 @ V22v11 ) )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V21v12 @ V20v13 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V15v24 @ V14v25 ) ) ) ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V25p @ V26m @ ( c_2Epatricia_2ELeaf @ A_27a @ V21v12 @ V20v13 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V15v24 @ V14v25 ) ) )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V21v12 @ V20v13 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V13v26 @ V12v27 @ V11v28 @ V10v29 ) ) ) ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V25p @ V26m @ ( c_2Epatricia_2ELeaf @ A_27a @ V21v12 @ V20v13 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V13v26 @ V12v27 @ V11v28 @ V10v29 ) ) )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V19v14 @ V18v15 @ V17v16 @ V16v17 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V9v36 @ V8v37 ) ) ) ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V25p @ V26m @ ( c_2Epatricia_2EBranch @ A_27a @ V19v14 @ V18v15 @ V17v16 @ V16v17 ) @ ( c_2Epatricia_2ELeaf @ A_27a @ V9v36 @ V8v37 ) ) )
& ( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V25p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V26m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Epatricia_2EBranch @ A_27a @ V19v14 @ V18v15 @ V17v16 @ V16v17 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V7v38 @ V6v39 @ V5v40 @ V4v41 ) ) ) ) )
= ( c_2Epatricia_2EBranch @ A_27a @ V25p @ V26m @ ( c_2Epatricia_2EBranch @ A_27a @ V19v14 @ V18v15 @ V17v16 @ V16v17 ) @ ( c_2Epatricia_2EBranch @ A_27a @ V7v38 @ V6v39 @ V5v40 @ V4v41 ) ) ) ) ).
thf(thm_2Epatricia_2EIS__EMPTY__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Epatricia_2Eptree @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) )
& ! [V1v: tyop_2Enum_2Enum,V2v1: A_27a] : ( V0P @ ( c_2Epatricia_2ELeaf @ A_27a @ V1v @ V2v1 ) )
& ! [V3v2: tyop_2Enum_2Enum,V4v3: tyop_2Enum_2Enum,V5v4: tyop_2Epatricia_2Eptree @ A_27a,V6v5: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ ( c_2Epatricia_2EBranch @ A_27a @ V3v2 @ V4v3 @ V5v4 @ V6v5 ) ) )
=> ! [V7v: tyop_2Epatricia_2Eptree @ A_27a] : ( V0P @ V7v ) ) ).
thf(thm_2Epatricia_2EIS__EMPTY__def,axiom,
! [A_27a: $tType,V0v5: tyop_2Epatricia_2Eptree @ A_27a,V1v4: tyop_2Epatricia_2Eptree @ A_27a,V2v3: tyop_2Enum_2Enum,V3v2: tyop_2Enum_2Enum,V4v1: A_27a,V5v: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__EMPTY @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Ebool_2ET )
& ( ( c_2Epatricia_2EIS__EMPTY @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V5v @ V4v1 ) )
= c_2Ebool_2EF )
& ( ( c_2Epatricia_2EIS__EMPTY @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V3v2 @ V2v3 @ V1v4 @ V0v5 ) )
= c_2Ebool_2EF ) ) ).
thf(thm_2Epatricia_2EBRANCHING__BIT,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( (~) @ ( V0a = V1b ) )
=> ( (~)
@ ( ( c_2Ebit_2EBIT @ ( c_2Epatricia_2EBRANCHING__BIT @ V0a @ V1b ) @ V0a )
= ( c_2Ebit_2EBIT @ ( c_2Epatricia_2EBRANCHING__BIT @ V0a @ V1b ) @ V1b ) ) ) ) ).
thf(thm_2Epatricia_2EBRANCHING__BIT__ZERO,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EBRANCHING__BIT @ V0a @ V1b )
= c_2Enum_2E0 )
<=> ( ( ( c_2Earithmetic_2EODD @ V0a )
= ( c_2Earithmetic_2EEVEN @ V1b ) )
| ( V0a = V1b ) ) ) ).
thf(thm_2Epatricia_2EBRANCHING__BIT__SYM,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EBRANCHING__BIT @ V0a @ V1b )
= ( c_2Epatricia_2EBRANCHING__BIT @ V1b @ V0a ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__ADD,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum,V3d: A_27a] :
( ( ( V0P @ V2k @ V3d )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V1t ) )
=> ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ ( c_2Epatricia_2EADD @ A_27a @ V1t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2k @ V3d ) ) ) ) ).
thf(thm_2Epatricia_2EMONO__EVERY__LEAF,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1Q: tyop_2Enum_2Enum > A_27a > $o,V2t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ! [V3k: tyop_2Enum_2Enum,V4d: A_27a] :
( ( V0P @ V3k @ V4d )
=> ( V1Q @ V3k @ V4d ) )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V2t ) )
=> ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V1Q @ V2t ) ) ).
thf(thm_2Epatricia_2ENOT__ADD__EMPTY,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a] :
( (~)
@ ( ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) )
= ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ).
thf(thm_2Epatricia_2EEMPTY__IS__PTREE,axiom,
! [A_27a: $tType] : ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ).
thf(thm_2Epatricia_2EADD__IS__PTREE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1x: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ V1x ) ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__BRANCH,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1p: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3l: tyop_2Epatricia_2Eptree @ A_27a,V4r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V1p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V2m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V3l @ V4r ) ) ) ) )
<=> ( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V3l )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V4r ) ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__REMOVE,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V1t )
=> ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ ( c_2Epatricia_2EREMOVE @ A_27a @ V1t @ V2k ) ) ) ).
thf(thm_2Epatricia_2EIS__PTREE__BRANCH,axiom,
! [A_27a: $tType,V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2l: tyop_2Epatricia_2Eptree @ A_27a,V3r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ( c_2Eprim__rec_2E_3C @ V0p @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) )
& ( (~)
@ ( ( V2l
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( V3r
= ( c_2Epatricia_2EEmpty @ A_27a ) ) ) )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a
@ ^ [V4k: tyop_2Enum_2Enum,V5d: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebit_2EMOD__2EXP__EQ @ V1m @ V4k @ V0p ) @ ( c_2Ebit_2EBIT @ V1m @ V4k ) )
@ V2l )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a
@ ^ [V6k: tyop_2Enum_2Enum,V7d: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebit_2EMOD__2EXP__EQ @ V1m @ V6k @ V0p ) @ ( c_2Ebool_2E_7E @ ( c_2Ebit_2EBIT @ V1m @ V6k ) ) )
@ V3r )
& ( c_2Epatricia_2EIS__PTREE @ A_27a @ V2l )
& ( c_2Epatricia_2EIS__PTREE @ A_27a @ V3r ) )
=> ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V0p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V1m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V2l @ V3r ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EREMOVE__IS__PTREE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) ) ) ).
thf(thm_2Epatricia_2EPEEK__NONE,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( ! [V3d: A_27a] : ( (~) @ ( V0P @ V2k @ V3d ) )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V1t ) )
=> ( ( c_2Epatricia_2EPEEK @ A_27a @ V1t @ V2k )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Epatricia_2EPEEK__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a,V3j: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) @ V3j )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ V3j ) @ ( c_2Eoption_2ESOME @ A_27a @ V2d ) @ ( c_2Epatricia_2EPEEK @ A_27a @ V0t @ V3j ) ) ) ) ).
thf(thm_2Epatricia_2EBRANCH,axiom,
! [A_27a: $tType,V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2l: tyop_2Epatricia_2Eptree @ A_27a,V3r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V0p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V1m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V2l @ V3r ) ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V2l @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V3r @ ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V3r @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V2l @ ( c_2Epatricia_2EBranch @ A_27a @ V0p @ V1m @ V2l @ V3r ) ) ) ) ).
thf(thm_2Epatricia_2EPEEK__REMOVE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) @ V2j )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ V2j ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Epatricia_2EPEEK @ A_27a @ V0t @ V2j ) ) ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__TRANSFORM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1Q: tyop_2Enum_2Enum > A_27b > $o,V2f: A_27a > A_27b,V3t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ! [V4k: tyop_2Enum_2Enum,V5d: A_27a] :
( ( V0P @ V4k @ V5d )
=> ( V1Q @ V4k @ ( V2f @ V5d ) ) )
& ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V3t ) )
=> ( c_2Epatricia_2EEVERY__LEAF @ A_27b @ V1Q @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V2f @ V3t ) ) ) ).
thf(thm_2Epatricia_2ETRANSFORM__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t )
= ( c_2Epatricia_2EEmpty @ A_27b ) )
<=> ( V1t
= ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ).
thf(thm_2Epatricia_2ETRANSFORM__IS__PTREE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V1t )
=> ( c_2Epatricia_2EIS__PTREE @ A_27b @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t ) ) ) ).
thf(thm_2Epatricia_2EPEEK__TRANSFORM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EPEEK @ A_27b @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t ) @ V2k )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Epatricia_2EPEEK @ A_27a @ V1t @ V2k ) @ ( c_2Eoption_2ENONE @ A_27b )
@ ^ [V3x: A_27a] : ( c_2Eoption_2ESOME @ A_27b @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Epatricia_2EADD__TRANSFORM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum,V3d: A_27a] :
( ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ ( c_2Epatricia_2EADD @ A_27a @ V1t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2k @ V3d ) ) )
= ( c_2Epatricia_2EADD @ A_27b @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27b @ V2k @ ( V0f @ V3d ) ) ) ) ).
thf(thm_2Epatricia_2ETRANSFORM__BRANCH,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1p: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3l: tyop_2Epatricia_2Eptree @ A_27a,V4r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ ( c_2Epatricia_2EBRANCH @ A_27a @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) ) @ V1p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) ) @ V2m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ V3l @ V4r ) ) ) ) )
= ( c_2Epatricia_2EBRANCH @ A_27b @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27b ) @ ( tyop_2Epatricia_2Eptree @ A_27b ) ) ) @ V1p @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ ( tyop_2Epatricia_2Eptree @ A_27b ) @ ( tyop_2Epatricia_2Eptree @ A_27b ) ) @ V2m @ ( c_2Epair_2E_2C @ ( tyop_2Epatricia_2Eptree @ A_27b ) @ ( tyop_2Epatricia_2Eptree @ A_27b ) @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V3l ) @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V4r ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EREMOVE__TRANSFORM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ ( c_2Epatricia_2EREMOVE @ A_27a @ V1t @ V2k ) )
= ( c_2Epatricia_2EREMOVE @ A_27b @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t ) @ V2k ) ) ).
thf(thm_2Epatricia_2EREMOVE__ADD__EQ,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a] :
( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) @ V1k )
= ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) ) ).
thf(thm_2Epatricia_2EADD__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a,V3e: A_27a] :
( ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V3e ) )
= ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V3e ) ) ) ).
thf(thm_2Epatricia_2EEVERY__LEAF__PEEK,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1t: tyop_2Epatricia_2Eptree @ A_27a,V2k: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EEVERY__LEAF @ A_27a @ V0P @ V1t )
& ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V1t @ V2k ) ) )
=> ( V0P @ V2k @ ( c_2Eoption_2ETHE @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V1t @ V2k ) ) ) ) ).
thf(thm_2Epatricia_2EIS__PTREE__PEEK,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0k: tyop_2Enum_2Enum] : ( (~) @ ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0k ) ) )
& ! [V1k: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum,V3b: A_27b] :
( ( c_2Eoption_2EIS__SOME @ A_27b @ ( c_2Epatricia_2EPEEK @ A_27b @ ( c_2Epatricia_2ELeaf @ A_27b @ V2j @ V3b ) @ V1k ) )
<=> ( V2j = V1k ) )
& ! [V4p: tyop_2Enum_2Enum,V5m: tyop_2Enum_2Enum,V6l: tyop_2Epatricia_2Eptree @ A_27c,V7r: tyop_2Epatricia_2Eptree @ A_27c] :
( ( c_2Epatricia_2EIS__PTREE @ A_27c @ ( c_2Epatricia_2EBranch @ A_27c @ V4p @ V5m @ V6l @ V7r ) )
=> ( ? [V8k: tyop_2Enum_2Enum] :
( ( c_2Ebit_2EBIT @ V5m @ V8k )
& ( c_2Eoption_2EIS__SOME @ A_27c @ ( c_2Epatricia_2EPEEK @ A_27c @ V6l @ V8k ) ) )
& ? [V9k: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Ebit_2EBIT @ V5m @ V9k ) )
& ( c_2Eoption_2EIS__SOME @ A_27c @ ( c_2Epatricia_2EPEEK @ A_27c @ V7r @ V9k ) ) )
& ! [V10k: tyop_2Enum_2Enum,V11n: tyop_2Enum_2Enum] :
( ( ( (~) @ ( c_2Ebit_2EMOD__2EXP__EQ @ V5m @ V10k @ V4p ) )
| ( ( c_2Eprim__rec_2E_3C @ V11n @ V5m )
& ( (~)
@ ( ( c_2Ebit_2EBIT @ V11n @ V4p )
= ( c_2Ebit_2EBIT @ V11n @ V10k ) ) ) ) )
=> ( ( (~) @ ( c_2Eoption_2EIS__SOME @ A_27c @ ( c_2Epatricia_2EPEEK @ A_27c @ V6l @ V10k ) ) )
& ( (~) @ ( c_2Eoption_2EIS__SOME @ A_27c @ ( c_2Epatricia_2EPEEK @ A_27c @ V7r @ V10k ) ) ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EPTREE__EQ,axiom,
! [A_27a: $tType,V0t1: tyop_2Epatricia_2Eptree @ A_27a,V1t2: tyop_2Epatricia_2Eptree @ A_27a] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ A_27a @ V1t2 ) )
=> ( ! [V2k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EPEEK @ A_27a @ V0t1 @ V2k )
= ( c_2Epatricia_2EPEEK @ A_27a @ V1t2 @ V2k ) )
<=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Epatricia_2EREMOVE__REMOVE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) @ V1k )
= ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) ) ) ).
thf(thm_2Epatricia_2EREMOVE__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a,V3j: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2EREMOVE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) @ V3j )
= ( c_2Ebool_2ECOND @ ( tyop_2Epatricia_2Eptree @ A_27a ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ V3j ) @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V3j ) @ ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V3j ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) ) ) ) ).
thf(thm_2Epatricia_2EADD__ADD__SYM,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum,V3d: A_27a,V4e: A_27a] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
& ( (~) @ ( V1k = V2j ) ) )
=> ( ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V3d ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2j @ V4e ) )
= ( c_2Epatricia_2EADD @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V2j @ V4e ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V3d ) ) ) ) ).
thf(thm_2Epatricia_2EFILTER__ALL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1l: tyop_2Elist_2Elist @ A_27a] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Elist_2ELENGTH @ A_27a @ V1l ) )
=> ( (~) @ ( V0P @ ( c_2Elist_2EEL @ A_27a @ V2n @ V1l ) ) ) )
<=> ( ( c_2Elist_2EFILTER @ A_27a @ V0P @ V1l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(thm_2Epatricia_2ETRAVERSE__TRANSFORM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ETRAVERSE @ A_27b @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a @ V0f @ V1t ) )
= ( c_2Epatricia_2ETRAVERSE @ A_27a @ V1t ) ) ).
thf(thm_2Epatricia_2EMEM__TRAVERSE__PEEK,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) )
= ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V0t @ V1k ) ) ) ) ).
thf(thm_2Epatricia_2EIN__NUMSET__OF__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1n: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1n @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) )
= ( c_2Epatricia_2EIN__PTREE @ V1n @ V0t ) ) ) ).
thf(thm_2Epatricia_2EADD__LIST__IS__PTREE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EADD__LIST @ A_27a @ V0t @ V1l ) ) ) ).
thf(thm_2Epatricia_2EADD__LIST__TO__EMPTY__IS__PTREE,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ A_27a )] : ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EADD__LIST @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) @ V0l ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE__EMPTY,axiom,
! [V0s: tyop_2Enum_2Enum > $o] : ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) ) ).
thf(thm_2Epatricia_2ENOT__KEY__LEFT__AND__RIGHT,axiom,
! [A_27a: $tType,V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2l: tyop_2Epatricia_2Eptree @ A_27a,V3r: tyop_2Epatricia_2Eptree @ A_27a,V4k: tyop_2Enum_2Enum,V5j: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V0p @ V1m @ V2l @ V3r ) )
& ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V2l @ V4k ) )
& ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Epatricia_2EPEEK @ A_27a @ V3r @ V5j ) ) )
=> ( (~) @ ( V4k = V5j ) ) ) ).
thf(thm_2Epatricia_2EALL__DISTINCT__TRAVERSE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( c_2Elist_2EALL__DISTINCT @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ).
thf(thm_2Epatricia_2EMEM__ALL__DISTINCT__IMP__PERM,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Elist_2EALL__DISTINCT @ A_27a @ V0l1 )
& ( c_2Elist_2EALL__DISTINCT @ A_27a @ V1l2 )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l1 ) )
= ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l2 ) ) ) )
=> ( c_2Esorting_2EPERM @ A_27a @ V0l1 @ V1l2 ) ) ).
thf(thm_2Epatricia_2EMEM__TRAVERSE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) ) )
= ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2EINSERT__PTREE__IS__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1x: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EINSERT__PTREE @ V1x @ V0t ) ) ) ).
thf(thm_2Epatricia_2EFINITE__NUMSET__OF__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] : ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) ) ).
thf(thm_2Epatricia_2EADD__INSERT,axiom,
! [V0v: tyop_2Eone_2Eone,V1t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V2n: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EADD @ tyop_2Eone_2Eone @ V1t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Eone_2Eone @ V2n @ V0v ) )
= ( c_2Epatricia_2EINSERT__PTREE @ V2n @ V1t ) ) ).
thf(thm_2Epatricia_2EPEEK__INSERT__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1k: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Epatricia_2EPEEK @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EINSERT__PTREE @ V1k @ V0t ) @ V2j )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Eone_2Eone ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1k @ V2j ) @ ( c_2Eoption_2ESOME @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) @ ( c_2Epatricia_2EPEEK @ tyop_2Eone_2Eone @ V0t @ V2j ) ) ) ) ).
thf(thm_2Epatricia_2EMEM__TRAVERSE__INSERT__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1x: tyop_2Enum_2Enum,V2h: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EINSERT__PTREE @ V2h @ V0t ) ) ) )
<=> ( ( V1x = V2h )
| ( ( (~) @ ( V1x = V2h ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) ) ) ) ) ) ) ).
thf(thm_2Epatricia_2EPERM__INSERT__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( c_2Esorting_2EPERM @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ ( c_2Elist_2EFOLDL @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ ( c_2Ecombin_2EC @ tyop_2Enum_2Enum @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ ( tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone ) @ c_2Epatricia_2EINSERT__PTREE ) @ V0t @ ( c_2Elist_2ESET__TO__LIST @ tyop_2Enum_2Enum @ V1s ) ) ) @ ( c_2Elist_2ESET__TO__LIST @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) @ V1s ) ) ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__OF__NUMSET,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s ) )
=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) )
<=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ V0t )
| ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V1s ) ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__EMPTY,axiom,
! [V0n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Epatricia_2EIN__PTREE @ V0n @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1n @ V0s )
= ( c_2Epatricia_2EIN__PTREE @ V1n @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) ) ) ) ).
thf(thm_2Epatricia_2EPTREE__EXTENSION,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t2 ) )
=> ( ( V0t1 = V1t2 )
<=> ! [V2x: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIN__PTREE @ V2x @ V0t1 )
= ( c_2Epatricia_2EIN__PTREE @ V2x @ V1t2 ) ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__NUMSET__OF__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s ) )
=> ( ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) @ V1s ) )
= ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) ) ) ).
thf(thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s ) )
=> ( ( c_2Epatricia_2ENUMSET__OF__PTREE @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) @ V1s ) ) ) ).
thf(thm_2Epatricia_2ENUMSET__OF__PTREE__EMPTY,axiom,
( ( c_2Epatricia_2ENUMSET__OF__PTREE @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__EMPTY,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= V0t ) ).
thf(thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( c_2Epatricia_2ENUMSET__OF__PTREE @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) )
= V0s ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__INSERT__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ ( c_2Epatricia_2EINSERT__PTREE @ V1m @ V0t ) )
<=> ( ( V1m = V2n )
| ( c_2Epatricia_2EIN__PTREE @ V2n @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__REMOVE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ ( c_2Epatricia_2EREMOVE @ tyop_2Eone_2Eone @ V0t @ V1m ) )
<=> ( ( (~) @ ( V2n = V1m ) )
& ( c_2Epatricia_2EIN__PTREE @ V2n @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2EIN__PTREE__UNION__PTREE,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t2 ) )
=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ V1t2 ) )
<=> ( ( c_2Epatricia_2EIN__PTREE @ V2n @ V0t1 )
| ( c_2Epatricia_2EIN__PTREE @ V2n @ V1t2 ) ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__IS__PTREE,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t2 ) )
=> ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ V1t2 ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__COMM,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t2 ) )
=> ( ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ V1t2 )
= ( c_2Epatricia_2EUNION__PTREE @ V1t2 @ V0t1 ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__COMM__EMPTY,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Epatricia_2EUNION__PTREE @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0t )
= ( c_2Epatricia_2EUNION__PTREE @ V0t @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__EMPTY,axiom,
( ! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EUNION__PTREE @ V0t @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) )
= V0t )
& ! [V1t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t )
=> ( ( c_2Epatricia_2EUNION__PTREE @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V1t )
= V1t ) ) ) ).
thf(thm_2Epatricia_2EUNION__PTREE__ASSOC,axiom,
! [V0t1: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1t2: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V2t3: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V1t2 )
& ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V2t3 ) )
=> ( ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ ( c_2Epatricia_2EUNION__PTREE @ V1t2 @ V2t3 ) )
= ( c_2Epatricia_2EUNION__PTREE @ ( c_2Epatricia_2EUNION__PTREE @ V0t1 @ V1t2 ) @ V2t3 ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__UNION,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s1: tyop_2Enum_2Enum > $o,V2s2: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s1 )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V2s2 ) )
=> ( ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V1s1 @ V2s2 ) )
= ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s1 ) @ V2s2 ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o,V2x: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s ) )
=> ( ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V2x @ V1s ) )
= ( c_2Epatricia_2EINSERT__PTREE @ V2x @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT__EMPTY,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V1x @ V0s ) )
= ( c_2Epatricia_2EINSERT__PTREE @ V1x @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) ) ) ) ).
thf(thm_2Epatricia_2EPTREE__OF__NUMSET__DELETE,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ V0s @ V1x ) )
= ( c_2Epatricia_2EREMOVE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) @ V1x ) ) ) ).
thf(thm_2Epatricia_2ETRAVERSE__AUX,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t )
= ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a @ V0t @ ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Epatricia_2EPTREE__TRAVERSE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: tyop_2Epatricia_2Eptree @ A_27a,V1t2: tyop_2Epatricia_2Eptree @ A_27b] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ A_27b @ V1t2 ) )
=> ( ! [V2k: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t1 ) ) )
= ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27b @ V1t2 ) ) ) )
<=> ( ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t1 )
= ( c_2Epatricia_2ETRAVERSE @ A_27b @ V1t2 ) ) ) ) ).
thf(thm_2Epatricia_2EQSORT__MEM__EQ,axiom,
! [A_27a: $tType,V0l2: tyop_2Elist_2Elist @ A_27a,V1l1: tyop_2Elist_2Elist @ A_27a,V2R: A_27a > A_27a > $o] :
( ( ( c_2Esorting_2EQSORT @ A_27a @ V2R @ V1l1 )
= ( c_2Esorting_2EQSORT @ A_27a @ V2R @ V0l2 ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l1 ) )
= ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l2 ) ) ) ) ).
thf(thm_2Epatricia_2EKEYS__PEEK,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: tyop_2Epatricia_2Eptree @ A_27a,V1t2: tyop_2Epatricia_2Eptree @ A_27b] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t1 )
& ( c_2Epatricia_2EIS__PTREE @ A_27b @ V1t2 ) )
=> ( ( ( c_2Epatricia_2EKEYS @ A_27a @ V0t1 )
= ( c_2Epatricia_2EKEYS @ A_27b @ V1t2 ) )
<=> ( ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t1 )
= ( c_2Epatricia_2ETRAVERSE @ A_27b @ V1t2 ) ) ) ) ).
thf(thm_2Epatricia_2EPERM__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ) )
=> ( c_2Esorting_2EPERM @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) ) @ ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V1k @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2EPERM__NOT__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) )
=> ( ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) )
= ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ).
thf(thm_2Epatricia_2EPERM__NOT__REMOVE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ) )
=> ( ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) )
= ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ).
thf(thm_2Epatricia_2EPERM__DELETE__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1k: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) ) ) )
=> ( c_2Esorting_2EPERM @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EREMOVE @ tyop_2Eone_2Eone @ V0t @ V1k ) )
@ ( c_2Elist_2EFILTER @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2x @ V1k ) )
@ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2EFILTER__NONE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1l: tyop_2Elist_2Elist @ A_27a] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Elist_2ELENGTH @ A_27a @ V1l ) )
=> ( V0P @ ( c_2Elist_2EEL @ A_27a @ V2n @ V1l ) ) )
=> ( ( c_2Elist_2EFILTER @ A_27a @ V0P @ V1l )
= V1l ) ) ).
thf(thm_2Epatricia_2EPERM__REMOVE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) )
=> ( c_2Esorting_2EPERM @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) )
@ ( c_2Elist_2EFILTER @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2x @ V1k ) )
@ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE__ADD,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum,V2d: A_27a] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2ESIZE @ A_27a @ ( c_2Epatricia_2EADD @ A_27a @ V0t @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ A_27a @ V1k @ V2d ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) @ ( c_2Epatricia_2ESIZE @ A_27a @ V0t ) @ ( c_2Earithmetic_2E_2B @ ( c_2Epatricia_2ESIZE @ A_27a @ V0t ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE__REMOVE,axiom,
! [A_27a: $tType,V0t: tyop_2Epatricia_2Eptree @ A_27a,V1k: tyop_2Enum_2Enum] :
( ( c_2Epatricia_2EIS__PTREE @ A_27a @ V0t )
=> ( ( c_2Epatricia_2ESIZE @ A_27a @ ( c_2Epatricia_2EREMOVE @ A_27a @ V0t @ V1k ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1k @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ A_27a @ V0t ) ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Epatricia_2ESIZE @ A_27a @ V0t ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Epatricia_2ESIZE @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE,axiom,
! [A_27a: $tType] :
( ( ( c_2Epatricia_2ESIZE @ A_27a @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= c_2Enum_2E0 )
& ! [V0k: tyop_2Enum_2Enum,V1d: A_27a] :
( ( c_2Epatricia_2ESIZE @ A_27a @ ( c_2Epatricia_2ELeaf @ A_27a @ V0k @ V1d ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V2p: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4l: tyop_2Epatricia_2Eptree @ A_27a,V5r: tyop_2Epatricia_2Eptree @ A_27a] :
( ( c_2Epatricia_2ESIZE @ A_27a @ ( c_2Epatricia_2EBranch @ A_27a @ V2p @ V3m @ V4l @ V5r ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Epatricia_2ESIZE @ A_27a @ V4l ) @ ( c_2Epatricia_2ESIZE @ A_27a @ V5r ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( ( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
& ( c_2Elist_2EALL__DISTINCT @ tyop_2Enum_2Enum @ ( c_2Elist_2EAPPEND @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ETRAVERSE @ tyop_2Eone_2Eone @ V0t ) @ ( c_2Elist_2ESET__TO__LIST @ tyop_2Enum_2Enum @ V1s ) ) ) )
=> ( ( c_2Epatricia_2ESIZE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ V0t @ V1s ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Epatricia_2ESIZE @ tyop_2Eone_2Eone @ V0t ) @ ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ V1s ) ) ) ) ) ).
thf(thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( c_2Epatricia_2ESIZE @ tyop_2Eone_2Eone @ ( c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ tyop_2Eone_2Eone ) @ V0s ) )
= ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ V0s ) ) ) ).
thf(thm_2Epatricia_2ECARD__LIST__TO__SET,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2EALL__DISTINCT @ A_27a @ V0l )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l ) )
= ( c_2Elist_2ELENGTH @ A_27a @ V0l ) ) ) ).
thf(thm_2Epatricia_2ECARD__NUMSET__OF__PTREE,axiom,
! [V0t: tyop_2Epatricia_2Eptree @ tyop_2Eone_2Eone] :
( ( c_2Epatricia_2EIS__PTREE @ tyop_2Eone_2Eone @ V0t )
=> ( ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Epatricia_2ENUMSET__OF__PTREE @ V0t ) )
= ( c_2Epatricia_2ESIZE @ tyop_2Eone_2Eone @ V0t ) ) ) ).
thf(thm_2Epatricia_2EDELETE__UNION,axiom,
! [A_27a: $tType,V0x: A_27a,V1s1: A_27a > $o,V2s2: A_27a > $o] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s1 @ V2s2 ) @ V0x )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s1 @ V0x ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s2 @ V0x ) ) ) ).
%------------------------------------------------------------------------------