ITP001 Axioms: ITP070^7.ax
%------------------------------------------------------------------------------
% File : ITP070^7 : 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 : quotient_list.ax [Gau19]
% : HL4070^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 69 ( 7 unt; 30 typ; 0 def)
% Number of atoms : 120 ( 22 equ; 1 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 857 ( 1 ~; 1 |; 13 &; 784 @)
% ( 9 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 13 avg; 784 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 251 ( 251 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 28 usr; 1 con; 0-8 aty)
% Number of variables : 294 ( 0 ^ 262 !; 1 ?; 294 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( A_27c > A_27b ) > A_27a > A_27d ) ).
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_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > $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_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_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Elist_2EEVERY,type,
c_2Elist_2EEVERY:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EEXISTS,type,
c_2Elist_2EEXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
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_2Elist_2EFLAT,type,
c_2Elist_2EFLAT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Elist_2Elist @ A_27a ) ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2ELIST__REL,type,
c_2Elist_2ELIST__REL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > $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_2Elist_2ENULL,type,
c_2Elist_2ENULL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Elist_2EREVERSE,type,
c_2Elist_2EREVERSE:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $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_2Equotient__list_2ELIST__MAP__I,axiom,
! [A_27a: $tType] :
( ( c_2Elist_2EMAP @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ).
thf(thm_2Equotient__list_2ELIST__REL__EQ,axiom,
! [A_27a: $tType] :
( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) )
= ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ).
thf(thm_2Equotient__list_2ELIST__REL__REFL,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
<=> ( ( V0R @ V1x )
= ( V0R @ V2y ) ) )
=> ! [V3x: tyop_2Elist_2Elist @ A_27a] : ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3x @ V3x ) ) ).
thf(thm_2Equotient__list_2ELIST__EQUIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0R )
=> ( c_2Equotient_2EEQUIV @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Equotient__list_2ELIST__REL__REL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3r: tyop_2Elist_2Elist @ A_27a,V4s: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3r @ V4s )
<=> ( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3r @ V3r )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V4s @ V4s )
& ( ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ V3r )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ V4s ) ) ) ) ) ).
thf(thm_2Equotient__list_2ELIST__QUOTIENT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Equotient_2EQUOTIENT @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R ) @ ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep ) ) ) ).
thf(thm_2Equotient__list_2ECONS__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3t: tyop_2Elist_2Elist @ A_27b,V4h: A_27b] :
( ( c_2Elist_2ECONS @ A_27b @ V4h @ V3t )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2ECONS @ A_27a @ ( V2rep @ V4h ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3t ) ) ) ) ) ).
thf(thm_2Equotient__list_2ECONS__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3t1: tyop_2Elist_2Elist @ A_27a,V4t2: tyop_2Elist_2Elist @ A_27a,V5h1: A_27a,V6h2: A_27a] :
( ( ( V0R @ V5h1 @ V6h2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3t1 @ V4t2 ) )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2ECONS @ A_27a @ V5h1 @ V3t1 ) @ ( c_2Elist_2ECONS @ A_27a @ V6h2 @ V4t2 ) ) ) ) ).
thf(thm_2Equotient__list_2ENIL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( ( c_2Elist_2ENIL @ A_27b )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(thm_2Equotient__list_2ENIL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ).
thf(thm_2Equotient__list_2EMAP__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l: tyop_2Elist_2Elist @ A_27c,V7f: A_27c > A_27d] :
( ( c_2Elist_2EMAP @ A_27c @ A_27d @ V7f @ V6l )
= ( c_2Elist_2EMAP @ A_27b @ A_27d @ V4abs2 @ ( c_2Elist_2EMAP @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V7f ) @ ( c_2Elist_2EMAP @ A_27c @ A_27a @ V2rep1 @ V6l ) ) ) ) ) ) ).
thf(thm_2Equotient__list_2EMAP__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l1: tyop_2Elist_2Elist @ A_27a,V7l2: tyop_2Elist_2Elist @ A_27a,V8f1: A_27a > A_27b,V9f2: A_27a > A_27b] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V8f1 @ V9f2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R1 @ V6l1 @ V7l2 ) )
=> ( c_2Elist_2ELIST__REL @ A_27b @ A_27b @ V3R2 @ ( c_2Elist_2EMAP @ A_27a @ A_27b @ V8f1 @ V6l1 ) @ ( c_2Elist_2EMAP @ A_27a @ A_27b @ V9f2 @ V7l2 ) ) ) ) ) ).
thf(thm_2Equotient__list_2ELENGTH__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Elist_2ELENGTH @ A_27b @ V3l )
= ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) ) ) ) ).
thf(thm_2Equotient__list_2ELENGTH__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 )
=> ( ( c_2Elist_2ELENGTH @ A_27a @ V3l1 )
= ( c_2Elist_2ELENGTH @ A_27a @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2EAPPEND__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b,V4m: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Elist_2EAPPEND @ A_27b @ V3l @ V4m )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2EAPPEND @ A_27a @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V4m ) ) ) ) ) ).
thf(thm_2Equotient__list_2EAPPEND__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a,V5m1: tyop_2Elist_2Elist @ A_27a,V6m2: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V5m1 @ V6m2 ) )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2EAPPEND @ A_27a @ V3l1 @ V5m1 ) @ ( c_2Elist_2EAPPEND @ A_27a @ V4l2 @ V6m2 ) ) ) ) ).
thf(thm_2Equotient__list_2EFLAT__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ ( tyop_2Elist_2Elist @ A_27b )] :
( ( c_2Elist_2EFLAT @ A_27b @ V3l )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2EFLAT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Elist_2Elist @ A_27b ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep ) @ V3l ) ) ) ) ) ).
thf(thm_2Equotient__list_2EFLAT__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ ( tyop_2Elist_2Elist @ A_27a ),V4l2: tyop_2Elist_2Elist @ ( tyop_2Elist_2Elist @ A_27a )] :
( ( c_2Elist_2ELIST__REL @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R ) @ V3l1 @ V4l2 )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2EFLAT @ A_27a @ V3l1 ) @ ( c_2Elist_2EFLAT @ A_27a @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2EREVERSE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Elist_2EREVERSE @ A_27b @ V3l )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2EREVERSE @ A_27a @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) ) ) ) ) ).
thf(thm_2Equotient__list_2EREVERSE__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2EREVERSE @ A_27a @ V3l1 ) @ ( c_2Elist_2EREVERSE @ A_27a @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2EFILTER__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3P: A_27b > $o,V4l: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Elist_2EFILTER @ A_27b @ V3P @ V4l )
= ( c_2Elist_2EMAP @ A_27a @ A_27b @ V1abs @ ( c_2Elist_2EFILTER @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3P ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V4l ) ) ) ) ) ).
thf(thm_2Equotient__list_2EFILTER__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3P1: A_27a > $o,V4P2: A_27a > $o,V5l1: tyop_2Elist_2Elist @ A_27a,V6l2: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3P1 @ V4P2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V5l1 @ V6l2 ) )
=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ ( c_2Elist_2EFILTER @ A_27a @ V3P1 @ V5l1 ) @ ( c_2Elist_2EFILTER @ A_27a @ V4P2 @ V6l2 ) ) ) ) ).
thf(thm_2Equotient__list_2ENULL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b] :
( ( c_2Elist_2ENULL @ A_27b @ V3l )
= ( c_2Elist_2ENULL @ A_27a @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) ) ) ) ).
thf(thm_2Equotient__list_2ENULL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 )
=> ( ( c_2Elist_2ENULL @ A_27a @ V3l1 )
= ( c_2Elist_2ENULL @ A_27a @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2EALL__EL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b,V4P: A_27b > $o] :
( ( c_2Elist_2EEVERY @ A_27b @ V4P @ V3l )
= ( c_2Elist_2EEVERY @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4P ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) ) ) ) ).
thf(thm_2Equotient__list_2EALL__EL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a,V5P1: A_27a > $o,V6P2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5P1 @ V6P2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 ) )
=> ( ( c_2Elist_2EEVERY @ A_27a @ V5P1 @ V3l1 )
= ( c_2Elist_2EEVERY @ A_27a @ V6P2 @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2ESOME__EL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l: tyop_2Elist_2Elist @ A_27b,V4P: A_27b > $o] :
( ( c_2Elist_2EEXISTS @ A_27b @ V4P @ V3l )
= ( c_2Elist_2EEXISTS @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4P ) @ ( c_2Elist_2EMAP @ A_27b @ A_27a @ V2rep @ V3l ) ) ) ) ).
thf(thm_2Equotient__list_2ESOME__EL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3l1: tyop_2Elist_2Elist @ A_27a,V4l2: tyop_2Elist_2Elist @ A_27a,V5P1: A_27a > $o,V6P2: A_27a > $o] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V5P1 @ V6P2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R @ V3l1 @ V4l2 ) )
=> ( ( c_2Elist_2EEXISTS @ A_27a @ V5P1 @ V3l1 )
= ( c_2Elist_2EEXISTS @ A_27a @ V6P2 @ V4l2 ) ) ) ) ).
thf(thm_2Equotient__list_2EFOLDL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l: tyop_2Elist_2Elist @ A_27d,V7f: A_27c > A_27d > A_27c,V8e: A_27c] :
( ( c_2Elist_2EFOLDL @ A_27d @ A_27c @ V7f @ V8e @ V6l )
= ( V1abs1 @ ( c_2Elist_2EFOLDL @ A_27b @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( A_27d > A_27c ) @ A_27c @ ( A_27b > A_27a ) @ V1abs1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27c @ A_27d @ A_27a @ V4abs2 @ V2rep1 ) @ V7f ) @ ( V2rep1 @ V8e ) @ ( c_2Elist_2EMAP @ A_27d @ A_27b @ V5rep2 @ V6l ) ) ) ) ) ) ).
thf(thm_2Equotient__list_2EFOLDL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l1: tyop_2Elist_2Elist @ A_27b,V7l2: tyop_2Elist_2Elist @ A_27b,V8f1: A_27a > A_27b > A_27a,V9f2: A_27a > A_27b > A_27a,V10e1: A_27a,V11e2: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27b > A_27a ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27a @ V3R2 @ V0R1 ) @ V8f1 @ V9f2 )
& ( V0R1 @ V10e1 @ V11e2 )
& ( c_2Elist_2ELIST__REL @ A_27b @ A_27b @ V3R2 @ V6l1 @ V7l2 ) )
=> ( V0R1 @ ( c_2Elist_2EFOLDL @ A_27b @ A_27a @ V8f1 @ V10e1 @ V6l1 ) @ ( c_2Elist_2EFOLDL @ A_27b @ A_27a @ V9f2 @ V11e2 @ V7l2 ) ) ) ) ) ).
thf(thm_2Equotient__list_2EFOLDR__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l: tyop_2Elist_2Elist @ A_27c,V7f: A_27c > A_27d > A_27d,V8e: A_27d] :
( ( c_2Elist_2EFOLDR @ A_27c @ A_27d @ V7f @ V8e @ V6l )
= ( V4abs2 @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( A_27d > A_27d ) @ A_27c @ ( A_27b > A_27b ) @ V1abs1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27d @ A_27d @ A_27b @ V4abs2 @ V5rep2 ) @ V7f ) @ ( V5rep2 @ V8e ) @ ( c_2Elist_2EMAP @ A_27c @ A_27a @ V2rep1 @ V6l ) ) ) ) ) ) ).
thf(thm_2Equotient__list_2EFOLDR__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6l1: tyop_2Elist_2Elist @ A_27a,V7l2: tyop_2Elist_2Elist @ A_27a,V8f1: A_27a > A_27b > A_27b,V9f2: A_27a > A_27b > A_27b,V10e1: A_27b,V11e2: A_27b] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27b > A_27b ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27b @ V3R2 @ V3R2 ) @ V8f1 @ V9f2 )
& ( V3R2 @ V10e1 @ V11e2 )
& ( c_2Elist_2ELIST__REL @ A_27a @ A_27a @ V0R1 @ V6l1 @ V7l2 ) )
=> ( V3R2 @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b @ V8f1 @ V10e1 @ V6l1 ) @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b @ V9f2 @ V11e2 @ V7l2 ) ) ) ) ) ).
%------------------------------------------------------------------------------