ITP001 Axioms: ITP081_5.ax
%------------------------------------------------------------------------------
% File : ITP081_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 : alist_2.ax [Gau20]
% : HL4081_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 106 ( 7 unt; 24 typ; 0 def)
% Number of atoms : 1482 ( 91 equ)
% Maximal formula atoms : 13 ( 13 avg)
% Number of connectives : 227 ( 3 ~; 0 |; 16 &)
% ( 6 <=>; 202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 1176 (1176 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 88 ( 24 >; 64 *; 0 +; 0 <<)
% Number of predicates : 66 ( 65 usr; 5 prp; 0-3 aty)
% Number of functors : 24 ( 24 usr; 0 con; 1-7 aty)
% Number of variables : 382 ( 382 !; 0 ?; 382 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Ealist_2EALOOKUP,type,
c_2Ealist_2EALOOKUP: ( del * del ) > $i ).
tff(mem_c_2Ealist_2EALOOKUP,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ealist_2EALOOKUP(A_27a,A_27b),arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)),arr(A_27b,ty_2Eoption_2Eoption(A_27a)))) ).
tff(tp_c_2Ealist_2Ealist__range,type,
c_2Ealist_2Ealist__range: ( del * del ) > $i ).
tff(mem_c_2Ealist_2Ealist__range,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ealist_2Ealist__range(A_27a,A_27b),arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)),arr(A_27a,bool))) ).
tff(tp_c_2Ealist_2Ealist__to__fmap,type,
c_2Ealist_2Ealist__to__fmap: ( del * del ) > $i ).
tff(mem_c_2Ealist_2Ealist__to__fmap,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)),ty_2Efinite__map_2Efmap(A_27a,A_27b))) ).
tff(tp_c_2Ealist_2Efmap__to__alist,type,
c_2Ealist_2Efmap__to__alist: ( del * del ) > $i ).
tff(mem_c_2Ealist_2Efmap__to__alist,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),arr(ty_2Efinite__map_2Efmap(A_27a,A_27b),ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))) ).
tff(lamtp_f2095,type,
f2095: ( del * del * $i ) > $i ).
tff(lameq_f2095,axiom,
! [A_27b: del,A_27a: del,V0s: $i] :
( mem(V0s,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ! [V1k: $i] : ( ap(f2095(A_27b,A_27a,V0s),V1k) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1k),ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,A_27b),V0s),V1k)) ) ) ).
tff(ax_thm_2Ealist_2Efmap__to__alist__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0s) = ap(ap(c_2Elist_2EMAP(A_27a,ty_2Epair_2Eprod(A_27a,A_27b)),f2095(A_27b,A_27a,V0s)),ap(c_2Elist_2ESET__TO__LIST(A_27a),ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),V0s))) ) ) ).
tff(conj_thm_2Ealist_2Efmap__to__alist__FEMPTY,axiom,
! [A_27a: del,A_27b: del] : ( ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),c_2Efinite__map_2EFEMPTY(A_27a,A_27b)) = c_2Elist_2ENIL(ty_2Epair_2Eprod(A_27a,A_27b)) ) ).
tff(lamtp_f2096,type,
f2096: ( del * del * $i * $i ) > $i ).
tff(lameq_f2096,axiom,
! [A_27a: del,A_27b: del,V2v: $i] :
( mem(V2v,A_27b)
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V3f: $i] : ( ap(f2096(A_27a,A_27b,V2v,V1k),V3f) = ap(ap(c_2Efinite__map_2EFUPDATE(A_27a,A_27b),V3f),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1k),V2v)) ) ) ) ).
tff(lamtp_f2097,type,
f2097: ( del * del * $i ) > $i ).
tff(lameq_f2097,axiom,
! [A_27a: del,A_27b: del,V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2v: $i] : ( ap(f2097(A_27a,A_27b,V1k),V2v) = f2096(A_27a,A_27b,V2v,V1k) ) ) ).
tff(lamtp_f2098,type,
f2098: ( del * del ) > $i ).
tff(lameq_f2098,axiom,
! [A_27b: del,A_27a: del,V1k: $i] : ( ap(f2098(A_27b,A_27a),V1k) = f2097(A_27a,A_27b,V1k) ) ).
tff(ax_thm_2Ealist_2Ealist__to__fmap__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0s) = ap(ap(ap(c_2Elist_2EFOLDR(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Efinite__map_2Efmap(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,arr(ty_2Efinite__map_2Efmap(A_27a,A_27b),ty_2Efinite__map_2Efmap(A_27a,A_27b))),f2098(A_27b,A_27a))),c_2Efinite__map_2EFEMPTY(A_27a,A_27b)),V0s) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0k: $i] :
( mem(V0k,A_27c)
=> ! [V1v: $i] :
( mem(V1v,A_27d)
=> ! [V2t: $i] :
( mem(V2t,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27c,A_27d)))
=> ( ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),c_2Elist_2ENIL(ty_2Epair_2Eprod(A_27a,A_27b))) = c_2Efinite__map_2EFEMPTY(A_27a,A_27b) )
& ( ap(c_2Ealist_2Ealist__to__fmap(A_27c,A_27d),ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(A_27c,A_27d)),ap(ap(c_2Epair_2E_2C(A_27c,A_27d),V0k),V1v)),V2t)) = ap(ap(c_2Efinite__map_2EFUPDATE(A_27c,A_27d),ap(c_2Ealist_2Ealist__to__fmap(A_27c,A_27d),V2t)),ap(ap(c_2Epair_2E_2C(A_27c,A_27d),V0k),V1v)) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)),arr(A_27b,bool)))
=> ( ( ! [V1q: $i] :
( mem(V1q,A_27b)
=> p(ap(ap(V0P,c_2Elist_2ENIL(ty_2Epair_2Eprod(A_27b,A_27a))),V1q)) )
& ! [V2x: $i] :
( mem(V2x,A_27b)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4t: $i] :
( mem(V4t,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V5q: $i] :
( mem(V5q,A_27b)
=> ( ( ( V2x != V5q )
=> p(ap(ap(V0P,V4t),V5q)) )
=> p(ap(ap(V0P,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2x),V3y)),V4t)),V5q)) ) ) ) ) ) )
=> ! [V6v: $i] :
( mem(V6v,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V7v1: $i] :
( mem(V7v1,A_27b)
=> p(ap(ap(V0P,V6v),V7v1)) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0q: $i] :
( mem(V0q,A_27b)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),c_2Elist_2ENIL(ty_2Epair_2Eprod(A_27b,A_27a))),V0q) = c_2Eoption_2ENONE(A_27a) ) )
& ! [V1y: $i] :
( mem(V1y,A_27a)
=> ! [V2x: $i] :
( mem(V2x,A_27b)
=> ! [V3t: $i] :
( mem(V3t,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V4q: $i] :
( mem(V4q,A_27b)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2x),V1y)),V3t)),V4q) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Emin_2E_3D(A_27b),V2x),V4q)),ap(c_2Eoption_2ESOME(A_27a),V1y)),ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),V3t),V4q)) ) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__FAILS,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),V0l),V1x) = c_2Eoption_2ENONE(A_27a) )
<=> ! [V2k: $i] :
( mem(V2k,A_27b)
=> ! [V3v: $i] :
( mem(V3v,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2k),V3v)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27b,A_27a)),V0l)))
=> ( V2k != V1x ) ) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__NONE,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0l),V1x) = c_2Eoption_2ENONE(A_27b) )
<=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0l)))) ) ) ) ).
tff(lamtp_f2099,type,
f2099: ( del * del * $i ) > $i ).
tff(lameq_f2099,axiom,
! [A_27b: del,A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3k: $i] : ( ap(f2099(A_27b,A_27a,V2f),V3k) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V3k),ap(V2f,V3k)) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__TABULATE,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1l: $i] :
( mem(V1l,ty_2Elist_2Elist(A_27a))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V1l)))
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EMAP(A_27a,ty_2Epair_2Eprod(A_27a,A_27b)),f2099(A_27b,A_27a,V2f)),V1l)),V0x) = ap(c_2Eoption_2ESOME(A_27b),ap(V2f,V0x)) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__EQ__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1fm: $i] :
( mem(V1fm,ty_2Efinite__map_2Efmap(A_27c,A_27d))
=> ( ( ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al)) = ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0al) )
& ( ap(c_2Ealist_2EALOOKUP(A_27d,A_27c),ap(c_2Ealist_2Efmap__to__alist(A_27c,A_27d),V1fm)) = ap(c_2Efinite__map_2EFLOOKUP(A_27c,A_27d),V1fm) ) ) ) ) ).
tff(conj_thm_2Ealist_2EMEM__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ! [V2fm: $i] :
( mem(V2fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V2fm))))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),V2fm)))
& ( ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,A_27b),V2fm),V0x) = V1y ) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EMEM__fmap__to__alist__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1fm: $i] :
( mem(V1fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),V0p),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V1fm))))
<=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),V1fm),ap(c_2Epair_2EFST(A_27a,A_27b),V0p)) = ap(c_2Eoption_2ESOME(A_27b),ap(c_2Epair_2ESND(A_27a,A_27b),V0p)) ) ) ) ) ).
tff(conj_thm_2Ealist_2EMEM__pair__fmap__to__alist__FLOOKUP,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ! [V2fm: $i] :
( mem(V2fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V2fm))))
<=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),V2fm),V0x) = ap(c_2Eoption_2ESOME(A_27b),V1y) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2ELENGTH__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( mem(V0fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm))) = surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),V0fm))) ) ) ).
tff(conj_thm_2Ealist_2Efmap__to__alist__to__fmap,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( mem(V0fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm)) = V0fm ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__MEM,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0al),V1k) = ap(c_2Eoption_2ESOME(A_27b),V2v) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1k),V2v)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V0al))) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__SOME__FAPPLY__alist__to__fmap,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0al),V1k) = ap(c_2Eoption_2ESOME(A_27b),V2v) )
=> ( ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al)),V1k) = V2v ) ) ) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__FAPPLY__MEM,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1z: $i] :
( mem(V1z,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1z),ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1z),ap(ap(c_2Efinite__map_2EFAPPLY(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al)),V1z))),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V0al))) ) ) ) ).
tff(lamtp_f2100,type,
f2100: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f2100,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( mem(V2x,A_27c)
=> ! [V3y: $i] : ( ap(f2100(A_27b,A_27c,A_27a,V0f,V2x),V3y) = ap(ap(c_2Epair_2E_2C(A_27c,A_27b),V2x),ap(V0f,V3y)) ) ) ) ).
tff(lamtp_f2101,type,
f2101: ( del * del * del * $i ) > $i ).
tff(lameq_f2101,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x: $i] : ( ap(f2101(A_27a,A_27b,A_27c,V0f),V2x) = f2100(A_27b,A_27c,A_27a,V0f,V2x) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1al: $i] :
( mem(V1al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27c,A_27a)))
=> ( ap(c_2Ealist_2EALOOKUP(A_27b,A_27c),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27c,A_27a),ty_2Epair_2Eprod(A_27c,A_27b)),ap(c_2Epair_2EUNCURRY(A_27c,A_27a,ty_2Epair_2Eprod(A_27c,A_27b)),f2101(A_27a,A_27b,A_27c,V0f))),V1al)) = ap(ap(c_2Ecombin_2Eo(A_27c,ty_2Eoption_2Eoption(A_27b),ty_2Eoption_2Eoption(A_27a)),ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V0f)),ap(c_2Ealist_2EALOOKUP(A_27a,A_27c),V1al)) ) ) ) ).
tff(conj_thm_2Ealist_2EFDOM__alist__to__fmap,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al)) = ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0al)) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__prefix,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l1: $i] :
( mem(V1l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V2l2: $i] :
( mem(V2l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V1l1) = ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V2l2) )
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V0ls),V1l1)) = ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V0ls),V2l2)) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__APPEND,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V0l1),V1l2)) = ap(ap(c_2Efinite__map_2EFUNION(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0l1)),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V1l2)) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__prefix,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( mem(V0v,A_27b)
=> ! [V1ls: $i] :
( mem(V1ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V2k: $i] :
( mem(V2k,A_27a)
=> ! [V3ls2: $i] :
( mem(V3ls2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1ls),V2k) = ap(c_2Eoption_2ESOME(A_27b),V0v) )
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V1ls),V3ls2)),V2k) = ap(c_2Eoption_2ESOME(A_27b),V0v) ) )
& ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1ls),V2k) = c_2Eoption_2ENONE(A_27b) )
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V1ls),V3ls2)),V2k) = ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V3ls2),V2k) ) ) ) ) ) ) ) ).
tff(lamtp_f2102,type,
f2102: del > $i ).
tff(lameq_f2102,axiom,
! [A_27b: del,V3v: $i] : ( ap(f2102(A_27b),V3v) = ap(c_2Eoption_2ESOME(A_27b),V3v) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__APPEND,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V2k: $i] :
( mem(V2k,A_27a)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V0l1),V1l2)),V2k) = ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27b,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0l1),V2k)),ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1l2),V2k)),f2102(A_27b)) ) ) ) ) ).
tff(conj_thm_2Ealist_2EFUPDATE__LIST__EQ__APPEND__REVERSE,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1fm: $i] :
( mem(V1fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),V1fm),V0ls) = ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27a,A_27b)),V0ls)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V1fm))) ) ) ) ).
tff(conj_thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__SOME,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1k: $i] :
( mem(V1k,A_27b)
=> ! [V2v: $i] :
( mem(V2v,A_27a)
=> ! [V3fm: $i] :
( mem(V3fm,ty_2Efinite__map_2Efmap(A_27b,A_27a))
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),V0ls),V1k) = ap(c_2Eoption_2ESOME(A_27a),V2v) )
=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27b,A_27a),ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27b,A_27a),V3fm),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27b,A_27a)),V0ls))),V1k) = ap(c_2Eoption_2ESOME(A_27a),V2v) ) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__NONE,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1k: $i] :
( mem(V1k,A_27b)
=> ! [V2fm: $i] :
( mem(V2fm,ty_2Efinite__map_2Efmap(A_27b,A_27a))
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),V0ls),V1k) = c_2Eoption_2ENONE(A_27a) )
=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27b,A_27a),ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27b,A_27a),V2fm),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27b,A_27a)),V0ls))),V1k) = ap(ap(c_2Efinite__map_2EFLOOKUP(A_27b,A_27a),V2fm),V1k) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EFUNION__alist__to__fmap,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1fm: $i] :
( mem(V1fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(ap(c_2Efinite__map_2EFUNION(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0ls)),V1fm) = ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),V1fm),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27a,A_27b)),V0ls)) ) ) ) ).
tff(lamtp_f2103,type,
f2103: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f2103,axiom,
! [A_27d: del,A_27b: del,A_27a: del,A_27c: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V4y: $i] : ( ap(f2103(A_27d,A_27b,A_27a,A_27c,V0f1,V3x,V1f2),V4y) = ap(ap(c_2Epair_2E_2C(A_27b,A_27d),ap(V0f1,V3x)),ap(V1f2,V4y)) ) ) ) ) ).
tff(lamtp_f2104,type,
f2104: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f2104,axiom,
! [A_27c: del,A_27b: del,A_27d: del,A_27a: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V3x: $i] : ( ap(f2104(A_27c,A_27b,A_27d,A_27a,V0f1,V1f2),V3x) = f2103(A_27d,A_27b,A_27a,A_27c,V0f1,V3x,V1f2) ) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V2al: $i] :
( mem(V2al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27c)))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f1),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST(A_27a,A_27c)),V2al))),c_2Epred__set_2EUNIV(A_27b)))
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27b,A_27d),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(c_2Epair_2EUNCURRY(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27d)),f2104(A_27c,A_27b,A_27d,A_27a,V0f1,V1f2))),V2al)) = ap(ap(c_2Efinite__map_2EMAP__KEYS(A_27a,A_27b,A_27d),V0f1),ap(ap(c_2Efinite__map_2Eo__f(A_27a,A_27c,A_27d),V1f2),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27c),V2al))) ) ) ) ) ) ).
tff(lamtp_f2105,type,
f2105: ( del * del * $i ) > $i ).
tff(lameq_f2105,axiom,
! [A_27b: del,A_27a: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] : ( ap(f2105(A_27b,A_27a,V0al),V1k) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1k),ap(c_2Eoption_2ETHE(A_27b),ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0al),V1k))) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al)) = ap(ap(c_2Elist_2EMAP(A_27a,ty_2Epair_2Eprod(A_27a,A_27b)),f2105(A_27b,A_27a,V0al)),ap(c_2Elist_2ESET__TO__LIST(A_27a),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0al)))) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__to__alist__PERM,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0al)))
=> p(ap(ap(c_2Esorting_2EPERM(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0al))),V0al)) ) ) ).
tff(lamtp_f2106,type,
f2106: ( del * del * $i * $i ) > $i ).
tff(lameq_f2106,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2106(A_27a,A_27b,V0ls,V1k),inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Emin_2E_3D(A_27a),ap(ap(c_2Elist_2EEL(A_27a),inj__ty_2Enum_2Enum(V2n)),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0ls))),V1k) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__LEAST__EL,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0ls),V1k) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ebool_2EIN(A_27a),V1k),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0ls)))),ap(c_2Eoption_2ESOME(A_27b),ap(ap(c_2Elist_2EEL(A_27b),ap(c_2Ewhile_2ELEAST,f2106(A_27a,A_27b,V0ls,V1k))),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27b),c_2Epair_2ESND(A_27a,A_27b)),V0ls)))),c_2Eoption_2ENONE(A_27b)) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__ALL__DISTINCT__MEM,axiom,
! [A_27a: del,A_27b: del,V0al: $i] :
( mem(V0al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0al)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1k),V2v)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V0al))) )
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0al),V1k) = ap(c_2Eoption_2ESOME(A_27b),V2v) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALL__DISTINCT__fmap__to__alist__keys,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( mem(V0fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm)))) ) ).
tff(conj_thm_2Ealist_2Efmap__to__alist__inj,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( mem(V0f1,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ( ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0f1) = ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V1f2) )
=> ( V0f1 = V1f2 ) ) ) ) ).
tff(conj_thm_2Ealist_2Efmap__to__alist__preserves__FDOM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fm1: $i] :
( mem(V0fm1,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ! [V1fm2: $i] :
( mem(V1fm2,ty_2Efinite__map_2Efmap(A_27a,A_27c))
=> ( ( ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),V0fm1) = ap(c_2Efinite__map_2EFDOM(A_27a,A_27c),V1fm2) )
=> ( ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm1)) = ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST(A_27a,A_27c)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27c),V1fm2)) ) ) ) ) ).
tff(conj_thm_2Ealist_2EPERM__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,V0fm1: $i] :
( mem(V0fm1,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ! [V1fm2: $i] :
( mem(V1fm2,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( p(ap(ap(c_2Esorting_2EPERM(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm1)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V1fm2)))
<=> ( V0fm1 = V1fm2 ) ) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__PERM,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( p(ap(ap(c_2Esorting_2EPERM(ty_2Epair_2Eprod(A_27a,A_27b)),V0l1),V1l2))
& p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0l1))) )
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0l1) = ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V1l2) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__ALL__DISTINCT__EL,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),ap(c_2Elist_2ELENGTH(ty_2Epair_2Eprod(A_27a,A_27b)),V0ls)))
& p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0ls))) )
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0ls),ap(c_2Epair_2EFST(A_27a,A_27b),ap(ap(c_2Elist_2EEL(ty_2Epair_2Eprod(A_27a,A_27b)),inj__ty_2Enum_2Enum(V1n)),V0ls))) = ap(c_2Eoption_2ESOME(A_27b),ap(c_2Epair_2ESND(A_27a,A_27b),ap(ap(c_2Elist_2EEL(ty_2Epair_2Eprod(A_27a,A_27b)),inj__ty_2Enum_2Enum(V1n)),V0ls))) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__ZIP__MAP__SND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(A_27a))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(A_27b))
=> ! [V2k: $i] :
( mem(V2k,A_27c)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27b,A_27d))
=> ( ( surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27a),V0l1)) = surj__ty_2Enum_2Enum(ap(c_2Elist_2ELENGTH(A_27b),V1l2)) )
=> ( ap(c_2Ealist_2EALOOKUP(A_27d,A_27a),ap(c_2Elist_2EZIP(A_27a,A_27d),ap(ap(c_2Epair_2E_2C(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27d)),V0l1),ap(ap(c_2Elist_2EMAP(A_27b,A_27d),V3f),V1l2)))) = ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Eoption_2Eoption(A_27d),ty_2Eoption_2Eoption(A_27b)),ap(c_2Eoption_2EOPTION__MAP(A_27b,A_27d),V3f)),ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(c_2Elist_2EZIP(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27b)),V0l1),V1l2)))) ) ) ) ) ) ) ).
tff(lamtp_f2107,type,
f2107: ( del * del * $i ) > $i ).
tff(lameq_f2107,axiom,
! [A_27b: del,A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3k: $i] : ( ap(f2107(A_27b,A_27a,V0P),V3k) = k(A_27b,ap(V0P,V3k)) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__FILTER,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1ls: $i] :
( mem(V1ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EFILTER(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f2107(A_27b,A_27a,V0P))),V1ls)),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27b)),ap(V0P,V2x)),ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1ls),V2x)),c_2Eoption_2ENONE(A_27b)) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__APPEND__same,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V2l: $i] :
( mem(V2l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0l1) = ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1l2) )
=> ( ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V0l1),V2l)) = ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(c_2Elist_2EAPPEND(ty_2Epair_2Eprod(A_27a,A_27b)),V1l2),V2l)) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__IN__FRANGE,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0ls),V1k) = ap(c_2Eoption_2ESOME(A_27b),V2v) )
=> p(ap(ap(c_2Ebool_2EIN(A_27b),V2v),ap(c_2Efinite__map_2EFRANGE(A_27a,A_27b),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27b),V0ls)))) ) ) ) ) ).
tff(conj_thm_2Ealist_2EFRANGE__alist__to__fmap__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Efinite__map_2EFRANGE(A_27b,A_27a),ap(c_2Ealist_2Ealist__to__fmap(A_27b,A_27a),V0ls))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27b,A_27a),A_27a),c_2Epair_2ESND(A_27b,A_27a)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27b,A_27a)),V0ls)))) ) ).
tff(conj_thm_2Ealist_2EIN__FRANGE__alist__to__fmap__suff,axiom,
! [A_27a: del,A_27b: del,V0ls: $i] :
( mem(V0ls,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2v: $i] :
( mem(V2v,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2v),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27b,A_27a),A_27a),c_2Epair_2ESND(A_27b,A_27a)),V0ls))))
=> p(ap(V1P,V2v)) ) )
=> ! [V3v: $i] :
( mem(V3v,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3v),ap(c_2Efinite__map_2EFRANGE(A_27b,A_27a),ap(c_2Ealist_2Ealist__to__fmap(A_27b,A_27a),V0ls))))
=> p(ap(V1P,V3v)) ) ) ) ) ) ).
tff(lamtp_f2108,type,
f2108: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f2108,axiom,
! [A_27d: del,A_27b: del,A_27a: del,A_27c: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V6y: $i] : ( ap(f2108(A_27d,A_27b,A_27a,A_27c,V0f1,V5x,V1f2),V6y) = ap(ap(c_2Epair_2E_2C(A_27b,A_27d),ap(V0f1,V5x)),ap(V1f2,V6y)) ) ) ) ) ).
tff(lamtp_f2109,type,
f2109: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f2109,axiom,
! [A_27c: del,A_27b: del,A_27d: del,A_27a: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V5x: $i] : ( ap(f2109(A_27c,A_27b,A_27d,A_27a,V0f1,V1f2),V5x) = f2108(A_27d,A_27b,A_27a,A_27c,V0f1,V5x,V1f2) ) ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__MAP__matchable,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27c,A_27d))
=> ! [V2al: $i] :
( mem(V2al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27c)))
=> ! [V3mal: $i] :
( mem(V3mal,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27d)))
=> ! [V4v: $i] :
( mem(V4v,ty_2Efinite__map_2Efmap(A_27b,A_27d))
=> ( ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f1),ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST(A_27a,A_27c)),V2al))),c_2Epred__set_2EUNIV(A_27b)))
& ( V3mal = ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(c_2Epair_2EUNCURRY(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27d)),f2109(A_27c,A_27b,A_27d,A_27a,V0f1,V1f2))),V2al) )
& ( V4v = ap(ap(c_2Efinite__map_2EMAP__KEYS(A_27a,A_27b,A_27d),V0f1),ap(ap(c_2Efinite__map_2Eo__f(A_27a,A_27c,A_27d),V1f2),ap(c_2Ealist_2Ealist__to__fmap(A_27a,A_27c),V2al))) ) )
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27b,A_27d),V3mal) = V4v ) ) ) ) ) ) ) ).
tff(lamtp_f2110,type,
f2110: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f2110,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2k: $i] :
( mem(V2k,A_27c)
=> ! [V3v: $i] : ( ap(f2110(A_27b,A_27c,A_27a,V0f,V2k),V3v) = ap(ap(c_2Epair_2E_2C(A_27c,A_27b),V2k),ap(V0f,V3v)) ) ) ) ).
tff(lamtp_f2111,type,
f2111: ( del * del * del * $i ) > $i ).
tff(lameq_f2111,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2k: $i] : ( ap(f2111(A_27a,A_27b,A_27c,V0f),V2k) = f2110(A_27b,A_27c,A_27a,V0f,V2k) ) ) ).
tff(conj_thm_2Ealist_2EMAP__values__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1fm: $i] :
( mem(V1fm,ty_2Efinite__map_2Efmap(A_27c,A_27a))
=> ( ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27c,A_27a),ty_2Epair_2Eprod(A_27c,A_27b)),ap(c_2Epair_2EUNCURRY(A_27c,A_27a,ty_2Epair_2Eprod(A_27c,A_27b)),f2111(A_27a,A_27b,A_27c,V0f))),ap(c_2Ealist_2Efmap__to__alist(A_27c,A_27a),V1fm)) = ap(c_2Ealist_2Efmap__to__alist(A_27c,A_27b),ap(ap(c_2Efinite__map_2Eo__f(A_27c,A_27a,A_27b),V0f),V1fm)) ) ) ) ).
tff(conj_thm_2Ealist_2EMAP__KEYS__I,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( mem(V0fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(ap(c_2Efinite__map_2EMAP__KEYS(A_27a,A_27a,A_27b),c_2Ecombin_2EI(A_27a)),V0fm) = V0fm ) ) ).
tff(conj_thm_2Ealist_2Ealist__to__fmap__MAP__values,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1al: $i] :
( mem(V1al,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27c,A_27a)))
=> ( ap(c_2Ealist_2Ealist__to__fmap(A_27c,A_27b),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27c,A_27a),ty_2Epair_2Eprod(A_27c,A_27b)),ap(c_2Epair_2EUNCURRY(A_27c,A_27a,ty_2Epair_2Eprod(A_27c,A_27b)),f2111(A_27a,A_27b,A_27c,V0f))),V1al)) = ap(ap(c_2Efinite__map_2Eo__f(A_27c,A_27a,A_27b),V0f),ap(c_2Ealist_2Ealist__to__fmap(A_27c,A_27a),V1al)) ) ) ) ).
tff(conj_thm_2Ealist_2Eset__MAP__FST__fmap__to__alist,axiom,
! [A_27a: del,A_27b: del,V0fm: $i] :
( mem(V0fm,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(c_2Elist_2ELIST__TO__SET(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),ap(c_2Ealist_2Efmap__to__alist(A_27a,A_27b),V0fm))) = ap(c_2Efinite__map_2EFDOM(A_27a,A_27b),V0fm) ) ) ).
tff(conj_thm_2Ealist_2Ealookup__distinct__reverse,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0l)))
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27a,A_27b)),V0l)),V1k) = ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0l),V1k) ) ) ) ) ).
tff(conj_thm_2Ealist_2Eflookup__fupdate__list,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1k: $i] :
( mem(V1k,A_27a)
=> ! [V2m: $i] :
( mem(V2m,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),V2m),V0l)),V1k) = ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27b,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(c_2Elist_2EREVERSE(ty_2Epair_2Eprod(A_27a,A_27b)),V0l)),V1k)),ap(ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),V2m),V1k)),f2102(A_27b)) ) ) ) ) ).
tff(conj_thm_2Ealist_2Efupdate__list__funion,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Efinite__map_2Efmap(A_27a,A_27b))
=> ! [V1l: $i] :
( mem(V1l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),V0m),V1l) = ap(ap(c_2Efinite__map_2EFUNION(A_27a,A_27b),ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),c_2Efinite__map_2EFEMPTY(A_27a,A_27b)),V1l)),V0m) ) ) ) ).
tff(conj_thm_2Ealist_2Emem__to__flookup,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ! [V2l: $i] :
( mem(V2l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V2l)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V2l))) )
=> ( ap(ap(c_2Efinite__map_2EFLOOKUP(A_27a,A_27b),ap(ap(c_2Efinite__map_2EFUPDATE__LIST(A_27a,A_27b),c_2Efinite__map_2EFEMPTY(A_27a,A_27b)),V2l)),V0x) = ap(c_2Eoption_2ESOME(A_27b),V1y) ) ) ) ) ) ).
tff(lamtp_f2112,type,
f2112: ( del * del * $i ) > $i ).
tff(lameq_f2112,axiom,
! [A_27c: del,A_27b: del,V2x: $i] :
( mem(V2x,A_27b)
=> ! [V3x_27: $i] : ( ap(f2112(A_27c,A_27b,V2x),V3x_27) = k(A_27c,ap(ap(c_2Emin_2E_3D(A_27b),V2x),V3x_27)) ) ) ).
tff(conj_thm_2Ealist_2Ealookup__filter,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,A_27a)
=> ! [V1l: $i] :
( mem(V1l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27c)))
=> ! [V2x: $i] :
( mem(V2x,A_27b)
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27c,A_27b),V1l),V2x) = ap(ap(c_2Ealist_2EALOOKUP(A_27c,A_27b),ap(ap(c_2Elist_2EFILTER(ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Epair_2EUNCURRY(A_27b,A_27c,bool),f2112(A_27c,A_27b,V2x))),V1l)),V2x) ) ) ) ) ).
tff(lamtp_f2113,type,
f2113: ( del * del * $i * $i ) > $i ).
tff(lameq_f2113,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1v: $i] :
( mem(V1v,A_27a)
=> ! [V2k: $i] : ( ap(f2113(A_27a,A_27b,V0m,V1v),V2k) = ap(ap(c_2Emin_2E_3D(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Ealist_2EALOOKUP(A_27a,A_27b),V0m),V2k)),ap(c_2Eoption_2ESOME(A_27a),V1v)) ) ) ) ).
tff(lamtp_f2114,type,
f2114: ( del * del * $i ) > $i ).
tff(lameq_f2114,axiom,
! [A_27b: del,A_27a: del,V0m: $i] :
( mem(V0m,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ! [V1v: $i] : ( ap(f2114(A_27b,A_27a,V0m),V1v) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),ap(c_2Ebool_2E_3F(A_27b),f2113(A_27a,A_27b,V0m,V1v))) ) ) ).
tff(ax_thm_2Ealist_2Ealist__range__def,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a)))
=> ( ap(c_2Ealist_2Ealist__range(A_27a,A_27b),V0m) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f2114(A_27b,A_27a,V0m)) ) ) ).
tff(conj_thm_2Ealist_2Ealookup__stable__sorted,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1sort: $i] :
( mem(V1sort,arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)),arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)),ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3l: $i] :
( mem(V3l,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R))
& p(ap(c_2Erelation_2Etotal(A_27a),V0R))
& p(ap(ap(c_2Esorting_2ESTABLE(ty_2Epair_2Eprod(A_27a,A_27b)),V1sort),ap(ap(c_2Erelation_2Einv__image(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),V0R),c_2Epair_2EFST(A_27a,A_27b)))) )
=> ( ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),ap(ap(V1sort,ap(ap(c_2Erelation_2Einv__image(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),V0R),c_2Epair_2EFST(A_27a,A_27b))),V3l)),V2x) = ap(ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V3l),V2x) ) ) ) ) ) ) ).
tff(conj_thm_2Ealist_2EALOOKUP__ALL__DISTINCT__PERM__same,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ! [V1l2: $i] :
( mem(V1l2,ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27a,A_27b)))
=> ( ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0l1)))
& p(ap(ap(c_2Esorting_2EPERM(A_27a),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V0l1)),ap(ap(c_2Elist_2EMAP(ty_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST(A_27a,A_27b)),V1l2)))
& ( ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V0l1) = ap(c_2Elist_2ELIST__TO__SET(ty_2Epair_2Eprod(A_27a,A_27b)),V1l2) ) )
=> ( ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V0l1) = ap(c_2Ealist_2EALOOKUP(A_27b,A_27a),V1l2) ) ) ) ) ).
%------------------------------------------------------------------------------