ITP001 Axioms: ITP082^5.ax
%------------------------------------------------------------------------------
% File : ITP082^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : container^2.ax [Gau20]
% : HL4082^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 1 unt; 4 typ; 0 def)
% Number of atoms : 784 ( 34 equ; 0 cnn)
% Maximal formula atoms : 23 ( 16 avg)
% Number of connectives : 1312 ( 2 ~; 0 |; 8 &;1204 @)
% ( 13 <=>; 85 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg;1204 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 65 con; 0-3 aty)
% Number of variables : 139 ( 12 ^ 126 !; 1 ?; 139 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Econtainer_2EBAG__OF__FMAP,type,
c_2Econtainer_2EBAG__OF__FMAP: del > del > del > $i ).
thf(mem_c_2Econtainer_2EBAG__OF__FMAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Econtainer_2EBAG__TO__LIST,type,
c_2Econtainer_2EBAG__TO__LIST: del > $i ).
thf(mem_c_2Econtainer_2EBAG__TO__LIST,axiom,
! [A_27a: del] : ( mem @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Econtainer_2ELIST__TO__BAG,type,
c_2Econtainer_2ELIST__TO__BAG: del > $i ).
thf(mem_c_2Econtainer_2ELIST__TO__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Econtainer_2Emlt__list,type,
c_2Econtainer_2Emlt__list: del > $i ).
thf(mem_c_2Econtainer_2Emlt__list,axiom,
! [A_27a: del] : ( mem @ ( c_2Econtainer_2Emlt__list @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__THM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) ) )
=> ( p @ ( ap @ V0P @ V1s ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ V0P @ V2v ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__SET__THM,axiom,
! [A_27a: del,A_27b: del,V0h: $i] :
( ( mem @ V0h @ A_27b )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__INV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) )
= V0s ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__CARD,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__IN__MEM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EMEM__SET__TO__LIST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ V0s ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ESET__TO__LIST__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Econtainer_2EUNION__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__SET__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EFINITE__LIST__TO__SET,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) ) ) ).
thf(ax_thm_2Econtainer_2ELIST__TO__BAG__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
& ! [V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0h ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__alt,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0l ) @ V1x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) ) @ V0l ) ) ) ) ) ) ).
thf(ax_thm_2Econtainer_2EBAG__TO__LIST__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Econtainer_2EBAG__TO__LIST @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
@ ( lam @ ( arr @ A_27a @ ty_2Enum_2Enum )
@ ^ [V1bag: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1bag ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1bag ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V1bag ) ) @ V1bag ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ^ [V2BAG__TO__LIST: $i] :
( lam @ ( arr @ A_27a @ ty_2Enum_2Enum )
@ ^ [V3a: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V3a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V3a ) ) @ ( ap @ V2BAG__TO__LIST @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V3a ) ) ) ) ) @ ( c_2Ebool_2EARB @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__TO__LIST__THM,axiom,
! [A_27a: del,V0bag: $i] :
( ( mem @ V0bag @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0bag ) )
=> ( ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0bag )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0bag ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V0bag ) ) @ ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0bag ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__TO__LIST__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ! [V1bag: $i] :
( ( mem @ V1bag @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1bag ) )
& ( V1bag
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V1bag ) ) ) )
=> ( p @ ( ap @ V0P @ V1bag ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ V0P @ V2v ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__TO__LIST__INV,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b ) )
= V0b ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__IN__MEM,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1x ) @ V0b ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EMEM__BAG__TO__LIST,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1x ) @ V0b ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EFINITE__LIST__TO__BAG,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Econtainer_2EEVERY__LIST__TO__BAG,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1ls ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ V1ls ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V0f ) @ V1b ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27b ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__FILTER,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0f ) @ V1b ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0f ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EINN__LIST__TO__BAG,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1h ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V2l ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1h ) ) @ V2l ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EIN__LIST__TO__BAG,axiom,
! [A_27a: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0h ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1l ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0h ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__DISTINCT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0b ) ) )
<=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0b ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__TO__BAG__EQ__EMPTY,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0l )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
<=> ( V0l
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Econtainer_2EPERM__LIST__TO__BAG,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ECARD__LIST__TO__BAG,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V0ls ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__TO__LIST__CARD,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__TO__LIST__EQ__NIL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ( ( c_2Elist_2ENIL @ A_27a )
= ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ( ( ( ap @ ( c_2Econtainer_2EBAG__TO__LIST @ A_27a ) @ V0b )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2ELIST__ELEM__COUNT__LIST__TO__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Erich__list_2ELIST__ELEM__COUNT @ A_27a ) @ V0e ) @ V1ls ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Econtainer_2ELIST__TO__BAG @ A_27a ) @ V1ls ) @ V0e ) ) ) ) ) ).
thf(ax_thm_2Econtainer_2Emlt__list__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Econtainer_2Emlt__list @ A_27a ) @ V0R )
= ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1l1: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V2l2: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3h: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4t: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V5list: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5list ) @ V4t ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V2l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V4t ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6e: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V5list ) ) ) @ ( ap @ ( ap @ V0R @ V6e ) @ V3h ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EWF__mlt__list,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Econtainer_2Emlt__list @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Econtainer_2EBAG__OF__FMAP__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1b )
= ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( lam @ A_27b
@ ^ [V3k: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ ( ap @ ( ap @ V0f @ V3k ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V1b ) @ V3k ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__OF__FMAP__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( ap @ ( 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: $i] :
( ( mem @ V1f @ ( arr @ A_27d @ ( arr @ A_27e @ A_27f ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Efinite__map_2Efmap @ A_27d @ A_27e ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ A_27d )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27e )
=> ( ( ap @ ( ap @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27f @ A_27d @ A_27e ) @ V1f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27d @ A_27e ) @ V2b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27d @ A_27e ) @ V3k ) @ V4v ) ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27f ) @ ( ap @ ( ap @ V1f @ V3k ) @ V4v ) ) @ ( ap @ ( ap @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27f @ A_27d @ A_27e ) @ V1f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Efdomsub @ A_27d @ A_27e ) @ V2b ) @ V3k ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EBAG__IN__BAG__OF__FMAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27a @ A_27b @ A_27c ) @ V1f ) @ V2b ) ) )
<=> ? [V3k: $i] :
( ( mem @ V3k @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V2b ) ) )
& ( V0x
= ( ap @ ( ap @ V1f @ V3k ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V2b ) @ V3k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Econtainer_2EFINITE__BAG__OF__FMAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27c ) @ ( ap @ ( ap @ ( c_2Econtainer_2EBAG__OF__FMAP @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1b ) ) ) ) ) ).
%------------------------------------------------------------------------------