ITP001 Axioms: ITP070_5.ax
%------------------------------------------------------------------------------
% File : ITP070_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 : quotient_list_2.ax [Gau20]
% : HL4070_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 1120 ( 15 equ)
% Maximal formula atoms : 18 ( 35 avg)
% Number of connectives : 243 ( 0 ~; 0 |; 12 &)
% ( 8 <=>; 223 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 16 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of FOOLs : 845 ( 845 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 29 ( 28 usr; 2 prp; 0-4 aty)
% Number of functors : 0 ( 0 usr; 0 con; --- aty)
% Number of variables : 247 ( 247 !; 0 ?; 247 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(conj_thm_2Equotient__list_2ELIST__MAP__I,axiom,
! [A_27a: del] : ( ap(c_2Elist_2EMAP(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(ty_2Elist_2Elist(A_27a)) ) ).
tff(conj_thm_2Equotient__list_2ELIST__REL__EQ,axiom,
! [A_27a: del] : ( ap(c_2Elist_2ELIST__REL(A_27a,A_27a),c_2Emin_2E_3D(A_27a)) = c_2Emin_2E_3D(ty_2Elist_2Elist(A_27a)) ) ).
tff(conj_thm_2Equotient__list_2ELIST__REL__REFL,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(ap(V0R,V1x),V2y))
<=> ( ap(V0R,V1x) = ap(V0R,V2y) ) ) ) )
=> ! [V3x: $i] :
( mem(V3x,ty_2Elist_2Elist(A_27a))
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3x),V3x)) ) ) ) ).
tff(conj_thm_2Equotient__list_2ELIST__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Equotient_2EEQUIV(A_27a),V0R))
=> p(ap(c_2Equotient_2EEQUIV(ty_2Elist_2Elist(A_27a)),ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R))) ) ) ).
tff(conj_thm_2Equotient__list_2ELIST__REL__REL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3r: $i] :
( mem(V3r,ty_2Elist_2Elist(A_27a))
=> ! [V4s: $i] :
( mem(V4s,ty_2Elist_2Elist(A_27a))
=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3r),V4s))
<=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3r),V3r))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V4s),V4s))
& ( ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),V3r) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),V4s) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ELIST__QUOTIENT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> p(ap(ap(ap(c_2Equotient_2EQUOTIENT(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27b)),ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R)),ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs)),ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep))) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ECONS__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3t: $i] :
( mem(V3t,ty_2Elist_2Elist(A_27b))
=> ! [V4h: $i] :
( mem(V4h,A_27b)
=> ( ap(ap(c_2Elist_2ECONS(A_27b),V4h),V3t) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),ap(ap(c_2Elist_2ECONS(A_27a),ap(V2rep,V4h)),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3t))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ECONS__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3t1: $i] :
( mem(V3t1,ty_2Elist_2Elist(A_27a))
=> ! [V4t2: $i] :
( mem(V4t2,ty_2Elist_2Elist(A_27a))
=> ! [V5h1: $i] :
( mem(V5h1,A_27a)
=> ! [V6h2: $i] :
( mem(V6h2,A_27a)
=> ( ( p(ap(ap(V0R,V5h1),V6h2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3t1),V4t2)) )
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),ap(ap(c_2Elist_2ECONS(A_27a),V5h1),V3t1)),ap(ap(c_2Elist_2ECONS(A_27a),V6h2),V4t2))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ENIL__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ( c_2Elist_2ENIL(A_27b) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),c_2Elist_2ENIL(A_27a)) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ENIL__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),c_2Elist_2ENIL(A_27a)),c_2Elist_2ENIL(A_27a))) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EMAP__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l: $i] :
( mem(V6l,ty_2Elist_2Elist(A_27c))
=> ! [V7f: $i] :
( mem(V7f,arr(A_27c,A_27d))
=> ( ap(ap(c_2Elist_2EMAP(A_27c,A_27d),V7f),V6l) = ap(ap(c_2Elist_2EMAP(A_27b,A_27d),V4abs2),ap(ap(c_2Elist_2EMAP(A_27a,A_27b),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,A_27d,A_27c,A_27b),V1abs1),V5rep2),V7f)),ap(ap(c_2Elist_2EMAP(A_27c,A_27a),V2rep1),V6l))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EMAP__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l1: $i] :
( mem(V6l1,ty_2Elist_2Elist(A_27a))
=> ! [V7l2: $i] :
( mem(V7l2,ty_2Elist_2Elist(A_27a))
=> ! [V8f1: $i] :
( mem(V8f1,arr(A_27a,A_27b))
=> ! [V9f2: $i] :
( mem(V9f2,arr(A_27a,A_27b))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2),V8f1),V9f2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R1),V6l1),V7l2)) )
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27b,A_27b),V3R2),ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V8f1),V6l1)),ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V9f2),V7l2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ELENGTH__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27b),V3l)) = surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27a),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l))) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ELENGTH__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2))
=> ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27a),V3l1)) = surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27a),V4l2)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EAPPEND__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ! [V4m: $i] :
( mem(V4m,ty_2Elist_2Elist(A_27b))
=> ( ap(ap(c_2Elist_2EAPPEND(A_27b),V3l),V4m) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),ap(ap(c_2Elist_2EAPPEND(A_27a),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l)),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V4m))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EAPPEND__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ! [V5m1: $i] :
( mem(V5m1,ty_2Elist_2Elist(A_27a))
=> ! [V6m2: $i] :
( mem(V6m2,ty_2Elist_2Elist(A_27a))
=> ( ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V5m1),V6m2)) )
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),ap(ap(c_2Elist_2EAPPEND(A_27a),V3l1),V5m1)),ap(ap(c_2Elist_2EAPPEND(A_27a),V4l2),V6m2))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFLAT__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(ty_2Elist_2Elist(A_27b)))
=> ( ap(c_2Elist_2EFLAT(A_27b),V3l) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),ap(c_2Elist_2EFLAT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Elist_2Elist(A_27b),ty_2Elist_2Elist(A_27a)),ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep)),V3l))) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFLAT__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(ty_2Elist_2Elist(A_27a)))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(ty_2Elist_2Elist(A_27a)))
=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a)),ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R)),V3l1),V4l2))
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),ap(c_2Elist_2EFLAT(A_27a),V3l1)),ap(c_2Elist_2EFLAT(A_27a),V4l2))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EREVERSE__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ( ap(c_2Elist_2EREVERSE(A_27b),V3l) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),ap(c_2Elist_2EREVERSE(A_27a),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l))) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EREVERSE__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2))
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),ap(c_2Elist_2EREVERSE(A_27a),V3l1)),ap(c_2Elist_2EREVERSE(A_27a),V4l2))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFILTER__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3P: $i] :
( mem(V3P,arr(A_27b,bool))
=> ! [V4l: $i] :
( mem(V4l,ty_2Elist_2Elist(A_27b))
=> ( ap(ap(c_2Elist_2EFILTER(A_27b),V3P),V4l) = ap(ap(c_2Elist_2EMAP(A_27a,A_27b),V1abs),ap(ap(c_2Elist_2EFILTER(A_27a),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,bool,A_27b,bool),V1abs),c_2Ecombin_2EI(bool)),V3P)),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V4l))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFILTER__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3P1: $i] :
( mem(V3P1,arr(A_27a,bool))
=> ! [V4P2: $i] :
( mem(V4P2,arr(A_27a,bool))
=> ! [V5l1: $i] :
( mem(V5l1,ty_2Elist_2Elist(A_27a))
=> ! [V6l2: $i] :
( mem(V6l2,ty_2Elist_2Elist(A_27a))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,bool),V0R),c_2Emin_2E_3D(bool)),V3P1),V4P2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V5l1),V6l2)) )
=> p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),ap(ap(c_2Elist_2EFILTER(A_27a),V3P1),V5l1)),ap(ap(c_2Elist_2EFILTER(A_27a),V4P2),V6l2))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ENULL__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ( p(ap(c_2Elist_2ENULL(A_27b),V3l))
<=> p(ap(c_2Elist_2ENULL(A_27a),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l))) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ENULL__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ( p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2))
=> ( p(ap(c_2Elist_2ENULL(A_27a),V3l1))
<=> p(ap(c_2Elist_2ENULL(A_27a),V4l2)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EALL__EL__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ! [V4P: $i] :
( mem(V4P,arr(A_27b,bool))
=> ( p(ap(ap(c_2Elist_2EEVERY(A_27b),V4P),V3l))
<=> p(ap(ap(c_2Elist_2EEVERY(A_27a),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,bool,A_27b,bool),V1abs),c_2Ecombin_2EI(bool)),V4P)),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EALL__EL__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ! [V5P1: $i] :
( mem(V5P1,arr(A_27a,bool))
=> ! [V6P2: $i] :
( mem(V6P2,arr(A_27a,bool))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,bool),V0R),c_2Emin_2E_3D(bool)),V5P1),V6P2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2)) )
=> ( p(ap(ap(c_2Elist_2EEVERY(A_27a),V5P1),V3l1))
<=> p(ap(ap(c_2Elist_2EEVERY(A_27a),V6P2),V4l2)) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ESOME__EL__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(A_27b))
=> ! [V4P: $i] :
( mem(V4P,arr(A_27b,bool))
=> ( p(ap(ap(c_2Elist_2EEXISTS(A_27b),V4P),V3l))
<=> p(ap(ap(c_2Elist_2EEXISTS(A_27a),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,bool,A_27b,bool),V1abs),c_2Ecombin_2EI(bool)),V4P)),ap(ap(c_2Elist_2EMAP(A_27b,A_27a),V2rep),V3l))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2ESOME__EL__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs: $i] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ! [V5P1: $i] :
( mem(V5P1,arr(A_27a,bool))
=> ! [V6P2: $i] :
( mem(V6P2,arr(A_27a,bool))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,bool),V0R),c_2Emin_2E_3D(bool)),V5P1),V6P2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R),V3l1),V4l2)) )
=> ( p(ap(ap(c_2Elist_2EEXISTS(A_27a),V5P1),V3l1))
<=> p(ap(ap(c_2Elist_2EEXISTS(A_27a),V6P2),V4l2)) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFOLDL__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l: $i] :
( mem(V6l,ty_2Elist_2Elist(A_27d))
=> ! [V7f: $i] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27c)))
=> ! [V8e: $i] :
( mem(V8e,A_27c)
=> ( ap(ap(ap(c_2Elist_2EFOLDL(A_27d,A_27c),V7f),V8e),V6l) = ap(V1abs1,ap(ap(ap(c_2Elist_2EFOLDL(A_27b,A_27a),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,arr(A_27d,A_27c),A_27c,arr(A_27b,A_27a)),V1abs1),ap(ap(c_2Equotient_2E_2D_2D_3E(A_27b,A_27c,A_27d,A_27a),V4abs2),V2rep1)),V7f)),ap(V2rep1,V8e)),ap(ap(c_2Elist_2EMAP(A_27d,A_27b),V5rep2),V6l))) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFOLDL__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l1: $i] :
( mem(V6l1,ty_2Elist_2Elist(A_27b))
=> ! [V7l2: $i] :
( mem(V7l2,ty_2Elist_2Elist(A_27b))
=> ! [V8f1: $i] :
( mem(V8f1,arr(A_27a,arr(A_27b,A_27a)))
=> ! [V9f2: $i] :
( mem(V9f2,arr(A_27a,arr(A_27b,A_27a)))
=> ! [V10e1: $i] :
( mem(V10e1,A_27a)
=> ! [V11e2: $i] :
( mem(V11e2,A_27a)
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,arr(A_27b,A_27a)),V0R1),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27b,A_27a),V3R2),V0R1)),V8f1),V9f2))
& p(ap(ap(V0R1,V10e1),V11e2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27b,A_27b),V3R2),V6l1),V7l2)) )
=> p(ap(ap(V0R1,ap(ap(ap(c_2Elist_2EFOLDL(A_27b,A_27a),V8f1),V10e1),V6l1)),ap(ap(ap(c_2Elist_2EFOLDL(A_27b,A_27a),V9f2),V11e2),V7l2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFOLDR__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l: $i] :
( mem(V6l,ty_2Elist_2Elist(A_27c))
=> ! [V7f: $i] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27d)))
=> ! [V8e: $i] :
( mem(V8e,A_27d)
=> ( ap(ap(ap(c_2Elist_2EFOLDR(A_27c,A_27d),V7f),V8e),V6l) = ap(V4abs2,ap(ap(ap(c_2Elist_2EFOLDR(A_27a,A_27b),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,arr(A_27d,A_27d),A_27c,arr(A_27b,A_27b)),V1abs1),ap(ap(c_2Equotient_2E_2D_2D_3E(A_27b,A_27d,A_27d,A_27b),V4abs2),V5rep2)),V7f)),ap(V5rep2,V8e)),ap(ap(c_2Elist_2EMAP(A_27c,A_27a),V2rep1),V6l))) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__list_2EFOLDR__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6l1: $i] :
( mem(V6l1,ty_2Elist_2Elist(A_27a))
=> ! [V7l2: $i] :
( mem(V7l2,ty_2Elist_2Elist(A_27a))
=> ! [V8f1: $i] :
( mem(V8f1,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V9f2: $i] :
( mem(V9f2,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V10e1: $i] :
( mem(V10e1,A_27b)
=> ! [V11e2: $i] :
( mem(V11e2,A_27b)
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,arr(A_27b,A_27b)),V0R1),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27b,A_27b),V3R2),V3R2)),V8f1),V9f2))
& p(ap(ap(V3R2,V10e1),V11e2))
& p(ap(ap(ap(c_2Elist_2ELIST__REL(A_27a,A_27a),V0R1),V6l1),V7l2)) )
=> p(ap(ap(V3R2,ap(ap(ap(c_2Elist_2EFOLDR(A_27a,A_27b),V8f1),V10e1),V6l1)),ap(ap(ap(c_2Elist_2EFOLDR(A_27a,A_27b),V9f2),V11e2),V7l2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------