ITP001 Axioms: ITP082^7.ax
%------------------------------------------------------------------------------
% File : ITP082^7 : 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 : container.ax [Gau19]
% : HL4082^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 109 ( 23 unt; 63 typ; 0 def)
% Number of atoms : 148 ( 43 equ; 3 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 654 ( 3 ~; 1 |; 8 &; 611 @)
% ( 12 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg; 611 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 207 ( 207 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 61 usr; 1 con; 0-6 aty)
% Number of variables : 212 ( 12 ^ 135 !; 2 ?; 212 :)
% ( 63 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Efinite__map_2Efmap,type,
tyop_2Efinite__map_2Efmap: $tType > $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
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_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_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_2Ebag_2EBAG__ALL__DISTINCT,type,
c_2Ebag_2EBAG__ALL__DISTINCT:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > $o ) ).
thf(c_2Ebag_2EBAG__CARD,type,
c_2Ebag_2EBAG__CARD:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebag_2EBAG__CHOICE,type,
c_2Ebag_2EBAG__CHOICE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a ) ).
thf(c_2Ebag_2EBAG__EVERY,type,
c_2Ebag_2EBAG__EVERY:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).
thf(c_2Ebag_2EBAG__FILTER,type,
c_2Ebag_2EBAG__FILTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Ebag_2EBAG__IMAGE,type,
c_2Ebag_2EBAG__IMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > ( A_27b > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Ebag_2EBAG__IN,type,
c_2Ebag_2EBAG__IN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).
thf(c_2Ebag_2EBAG__INN,type,
c_2Ebag_2EBAG__INN:
!>[A_27a: $tType] : ( A_27a > tyop_2Enum_2Enum > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).
thf(c_2Ebag_2EBAG__INSERT,type,
c_2Ebag_2EBAG__INSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Econtainer_2EBAG__OF__FMAP,type,
c_2Econtainer_2EBAG__OF__FMAP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c > A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27b @ A_27c ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Ebag_2EBAG__REST,type,
c_2Ebag_2EBAG__REST:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Econtainer_2EBAG__TO__LIST,type,
c_2Econtainer_2EBAG__TO__LIST:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebag_2EBAG__UNION,type,
c_2Ebag_2EBAG__UNION:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebag_2EEMPTY__BAG,type,
c_2Ebag_2EEMPTY__BAG:
!>[A_27a: $tType] : ( A_27a > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2EEVERY,type,
c_2Elist_2EEVERY:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Efinite__map_2EFAPPLY,type,
c_2Efinite__map_2EFAPPLY:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > A_27b ) ).
thf(c_2Efinite__map_2EFDOM,type,
c_2Efinite__map_2EFDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > $o ) ).
thf(c_2Efinite__map_2EFEMPTY,type,
c_2Efinite__map_2EFEMPTY:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ).
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_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebag_2EFINITE__BAG,type,
c_2Ebag_2EFINITE__BAG:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > $o ) ).
thf(c_2Efinite__map_2EFUPDATE,type,
c_2Efinite__map_2EFUPDATE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ 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_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Erich__list_2ELIST__ELEM__COUNT,type,
c_2Erich__list_2ELIST__ELEM__COUNT:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Econtainer_2ELIST__TO__BAG,type,
c_2Econtainer_2ELIST__TO__BAG:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > tyop_2Enum_2Enum ) ).
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_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Esorting_2EPERM,type,
c_2Esorting_2EPERM:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Epred__set_2EREST,type,
c_2Epred__set_2EREST:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
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_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Efinite__map_2Efdomsub,type,
c_2Efinite__map_2Efdomsub:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Econtainer_2Emlt__list,type,
c_2Econtainer_2Emlt__list:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__def,axiom,
! [A_27a: $tType] :
( ( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
& ! [V0h: A_27a,V1t: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Elist_2ECONS @ A_27a @ V0h @ V1t ) )
= ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0h @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__primitive__def,axiom,
! [A_27a: $tType] :
( ( c_2Econtainer_2EBAG__TO__LIST @ A_27a )
= ( c_2Erelation_2EWFREC @ ( A_27a > tyop_2Enum_2Enum ) @ ( tyop_2Elist_2Elist @ A_27a )
@ ( c_2Emin_2E_40 @ ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o )
@ ^ [V0R: ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ ( A_27a > tyop_2Enum_2Enum ) @ V0R )
@ ( c_2Ebool_2E_21 @ ( A_27a > tyop_2Enum_2Enum )
@ ^ [V1bag: A_27a > tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1bag ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( A_27a > tyop_2Enum_2Enum ) @ V1bag @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) @ ( V0R @ ( c_2Ebag_2EBAG__REST @ A_27a @ V1bag ) @ V1bag ) ) ) ) )
@ ^ [V2BAG__TO__LIST: ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Elist_2Elist @ A_27a ),V3a: A_27a > tyop_2Enum_2Enum] : ( c_2Ecombin_2EI @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ebag_2EFINITE__BAG @ A_27a @ V3a ) @ ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Emin_2E_3D @ ( A_27a > tyop_2Enum_2Enum ) @ V3a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V3a ) @ ( V2BAG__TO__LIST @ ( c_2Ebag_2EBAG__REST @ A_27a @ V3a ) ) ) ) @ ( c_2Ebool_2EARB @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(thm_2Econtainer_2Emlt__list__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Econtainer_2Emlt__list @ A_27a @ V0R )
= ( ^ [V1l1: tyop_2Elist_2Elist @ A_27a,V2l2: tyop_2Elist_2Elist @ A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V3h: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Elist_2Elist @ A_27a )
@ ^ [V4t: tyop_2Elist_2Elist @ A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Elist_2Elist @ A_27a )
@ ^ [V5list: tyop_2Elist_2Elist @ A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27a ) @ V1l1 @ ( c_2Elist_2EAPPEND @ A_27a @ V5list @ V4t ) )
@ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27a ) @ V2l2 @ ( c_2Elist_2ECONS @ A_27a @ V3h @ V4t ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V6e: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V6e @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V5list ) ) @ ( V0R @ V6e @ V3h ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Econtainer_2EBAG__OF__FMAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c > A_27a,V1b: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c] :
( ( c_2Econtainer_2EBAG__OF__FMAP @ A_27a @ A_27b @ A_27c @ V0f @ V1b )
= ( ^ [V2x: A_27a] :
( c_2Epred__set_2ECARD @ A_27b
@ ^ [V3k: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V3k @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V1b ) ) @ ( c_2Emin_2E_3D @ A_27a @ V2x @ ( V0f @ V3k @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ V1b @ V3k ) ) ) ) ) ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__THM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Elist_2ESET__TO__LIST @ A_27a @ V0s )
= ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ ( c_2Elist_2ESET__TO__LIST @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) ) ) ) ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__IND,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ! [V1s: A_27a > $o] :
( ( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( V0P @ ( c_2Epred__set_2EREST @ A_27a @ V1s ) ) )
=> ( V0P @ V1s ) )
=> ! [V2v: A_27a > $o] : ( V0P @ V2v ) ) ).
thf(thm_2Econtainer_2ELIST__TO__SET__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: tyop_2Elist_2Elist @ A_27b,V1h: A_27b] :
( ( ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Elist_2ELIST__TO__SET @ A_27b @ ( c_2Elist_2ECONS @ A_27b @ V1h @ V0t ) )
= ( c_2Epred__set_2EINSERT @ A_27b @ V1h @ ( c_2Elist_2ELIST__TO__SET @ A_27b @ V0t ) ) ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__INV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2ESET__TO__LIST @ A_27a @ V0s ) )
= V0s ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__CARD,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Elist_2ESET__TO__LIST @ A_27a @ V0s ) )
= ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__IN__MEM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
= ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2ESET__TO__LIST @ A_27a @ V0s ) ) ) ) ) ).
thf(thm_2Econtainer_2EMEM__SET__TO__LIST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2ESET__TO__LIST @ A_27a @ V0s ) ) )
= ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) ) ) ).
thf(thm_2Econtainer_2ESET__TO__LIST__SING,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Elist_2ESET__TO__LIST @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Elist_2ECONS @ A_27a @ V0x @ ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(thm_2Econtainer_2EUNION__APPEND,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l1 ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l2 ) )
= ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EAPPEND @ A_27a @ V0l1 @ V1l2 ) ) ) ).
thf(thm_2Econtainer_2ELIST__TO__SET__APPEND,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Elist_2EAPPEND @ A_27a @ V0l1 @ V1l2 ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l1 ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l2 ) ) ) ).
thf(thm_2Econtainer_2EFINITE__LIST__TO__SET,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V0l ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__alt,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a,V1x: A_27a] :
( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0l @ V1x )
= ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Elist_2EFILTER @ A_27a @ ( c_2Emin_2E_3D @ A_27a @ V1x ) @ V0l ) ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__THM,axiom,
! [A_27a: $tType,V0bag: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0bag )
=> ( ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0bag )
= ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Emin_2E_3D @ ( A_27a > tyop_2Enum_2Enum ) @ V0bag @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V0bag ) @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ ( c_2Ebag_2EBAG__REST @ A_27a @ V0bag ) ) ) ) ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__IND,axiom,
! [A_27a: $tType,V0P: ( A_27a > tyop_2Enum_2Enum ) > $o] :
( ! [V1bag: A_27a > tyop_2Enum_2Enum] :
( ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1bag )
& ( (~)
@ ( V1bag
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
=> ( V0P @ ( c_2Ebag_2EBAG__REST @ A_27a @ V1bag ) ) )
=> ( V0P @ V1bag ) )
=> ! [V2v: A_27a > tyop_2Enum_2Enum] : ( V0P @ V2v ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__INV,axiom,
! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
=> ( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b ) )
= V0b ) ) ).
thf(thm_2Econtainer_2EBAG__IN__MEM,axiom,
! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
=> ! [V1x: A_27a] :
( ( c_2Ebag_2EBAG__IN @ A_27a @ V1x @ V0b )
= ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b ) ) ) ) ) ).
thf(thm_2Econtainer_2EMEM__BAG__TO__LIST,axiom,
! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
=> ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b ) ) )
= ( c_2Ebag_2EBAG__IN @ A_27a @ V1x @ V0b ) ) ) ).
thf(thm_2Econtainer_2EFINITE__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ A_27a] : ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0ls ) ) ).
thf(thm_2Econtainer_2EEVERY__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ A_27a,V1P: A_27a > $o] :
( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V1P @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0ls ) )
= ( c_2Elist_2EEVERY @ A_27a @ V1P @ V0ls ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__APPEND,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Elist_2EAPPEND @ A_27a @ V0l1 @ V1l2 ) )
= ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0l1 ) @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V1l2 ) ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > A_27a,V1b: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V0f @ V1b ) )
= ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b @ V0f @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27b @ V1b ) ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__FILTER,axiom,
! [A_27a: $tType,V0f: A_27a > $o,V1b: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ ( c_2Elist_2EFILTER @ A_27a @ V0f @ V1b ) )
= ( c_2Ebag_2EBAG__FILTER @ A_27a @ V0f @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V1b ) ) ) ).
thf(thm_2Econtainer_2EINN__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1h: A_27a,V2l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ebag_2EBAG__INN @ A_27a @ V1h @ V0n @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V2l ) )
= ( c_2Earithmetic_2E_3E_3D @ ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Elist_2EFILTER @ A_27a @ ( c_2Emin_2E_3D @ A_27a @ V1h ) @ V2l ) ) @ V0n ) ) ).
thf(thm_2Econtainer_2EIN__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0h: A_27a,V1l: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ebag_2EBAG__IN @ A_27a @ V0h @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V1l ) )
= ( c_2Ebool_2EIN @ A_27a @ V0h @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l ) ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__DISTINCT,axiom,
! [A_27a: $tType,V0b: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0b ) )
= ( c_2Elist_2EALL__DISTINCT @ A_27a @ V0b ) ) ).
thf(thm_2Econtainer_2ELIST__TO__BAG__EQ__EMPTY,axiom,
! [A_27a: $tType,V0l: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0l )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(thm_2Econtainer_2EPERM__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0l1: tyop_2Elist_2Elist @ A_27a,V1l2: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0l1 )
= ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V1l2 ) )
<=> ( c_2Esorting_2EPERM @ A_27a @ V0l1 @ V1l2 ) ) ).
thf(thm_2Econtainer_2ECARD__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0ls ) )
= ( c_2Elist_2ELENGTH @ A_27a @ V0ls ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__CARD,axiom,
! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
=> ( ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b ) )
= ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b ) ) ) ).
thf(thm_2Econtainer_2EBAG__TO__LIST__EQ__NIL,axiom,
! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
=> ( ( ( ( c_2Elist_2ENIL @ A_27a )
= ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ( ( ( c_2Econtainer_2EBAG__TO__LIST @ A_27a @ V0b )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(thm_2Econtainer_2ELIST__ELEM__COUNT__LIST__TO__BAG,axiom,
! [A_27a: $tType,V0ls: tyop_2Elist_2Elist @ A_27a,V1e: A_27a] :
( ( c_2Erich__list_2ELIST__ELEM__COUNT @ A_27a @ V1e @ V0ls )
= ( c_2Econtainer_2ELIST__TO__BAG @ A_27a @ V0ls @ V1e ) ) ).
thf(thm_2Econtainer_2EWF__mlt__list,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EWF @ A_27a @ V0R )
=> ( c_2Erelation_2EWF @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Econtainer_2Emlt__list @ A_27a @ V0R ) ) ) ).
thf(thm_2Econtainer_2EBAG__OF__FMAP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType] :
( ! [V0f: A_27a > A_27b > A_27c] :
( ( c_2Econtainer_2EBAG__OF__FMAP @ A_27c @ A_27a @ A_27b @ V0f @ ( c_2Efinite__map_2EFEMPTY @ A_27a @ A_27b ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27c ) )
& ! [V1f: A_27d > A_27e > A_27f,V2b: tyop_2Efinite__map_2Efmap @ A_27d @ A_27e,V3k: A_27d,V4v: A_27e] :
( ( c_2Econtainer_2EBAG__OF__FMAP @ A_27f @ A_27d @ A_27e @ V1f @ ( c_2Efinite__map_2EFUPDATE @ A_27d @ A_27e @ V2b @ ( c_2Epair_2E_2C @ A_27d @ A_27e @ V3k @ V4v ) ) )
= ( c_2Ebag_2EBAG__INSERT @ A_27f @ ( V1f @ V3k @ V4v ) @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27f @ A_27d @ A_27e @ V1f @ ( c_2Efinite__map_2Efdomsub @ A_27d @ A_27e @ V2b @ V3k ) ) ) ) ) ).
thf(thm_2Econtainer_2EBAG__IN__BAG__OF__FMAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27a,V1f: A_27b > A_27c > A_27a,V2b: tyop_2Efinite__map_2Efmap @ A_27b @ A_27c] :
( ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27a @ A_27b @ A_27c @ V1f @ V2b ) )
<=> ? [V3k: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3k @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c @ V2b ) )
& ( V0x
= ( V1f @ V3k @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c @ V2b @ V3k ) ) ) ) ) ).
thf(thm_2Econtainer_2EFINITE__BAG__OF__FMAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1b: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] : ( c_2Ebag_2EFINITE__BAG @ A_27c @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27c @ A_27a @ A_27b @ V0f @ V1b ) ) ).
%------------------------------------------------------------------------------