ITP001 Axioms: ITP068+5.ax
%------------------------------------------------------------------------------
% File : ITP068+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 : path+2.ax [Gau20]
% : HL4068+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 207 ( 1 unt; 0 def)
% Number of atoms : 1448 ( 197 equ)
% Maximal formula atoms : 23 ( 6 avg)
% Number of connectives : 1256 ( 15 ~; 10 |; 157 &)
% ( 53 <=>;1021 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 11 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 120 ( 120 usr; 16 con; 0-7 aty)
% Number of variables : 1055 (1015 !; 40 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Epath_2Epath,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ne(ty_2Epath_2Epath(A0,A1)) ) ) ).
fof(mem_c_2Epath_2EPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2EPL(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),arr(ty_2Enum_2Enum,bool))) ) ) ).
fof(mem_c_2Epath_2ESN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2ESN(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,arr(A_27a,bool))),bool)) ) ) ).
fof(mem_c_2Epath_2Edrop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Edrop(A_27a,A_27b),arr(ty_2Enum_2Enum,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epath_2Eel,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eel(A_27a,A_27b),arr(ty_2Enum_2Enum,arr(ty_2Epath_2Epath(A_27a,A_27b),A_27a))) ) ) ).
fof(mem_c_2Epath_2Eevery,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eevery(A_27a,A_27b),arr(arr(A_27a,bool),arr(ty_2Epath_2Epath(A_27a,A_27b),bool))) ) ) ).
fof(mem_c_2Epath_2Eexists,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eexists(A_27a,A_27b),arr(arr(A_27a,bool),arr(ty_2Epath_2Epath(A_27a,A_27b),bool))) ) ) ).
fof(mem_c_2Epath_2Efilter,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Efilter(A_27a,A_27b),arr(arr(A_27a,bool),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epath_2Efinite,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Efinite(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),bool)) ) ) ).
fof(mem_c_2Epath_2Efirst,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Efirst(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),A_27a)) ) ) ).
fof(mem_c_2Epath_2EfirstP__at,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2EfirstP__at(A_27a,A_27b),arr(arr(A_27a,bool),arr(ty_2Epath_2Epath(A_27a,A_27b),arr(ty_2Enum_2Enum,bool)))) ) ) ).
fof(mem_c_2Epath_2Efirst__label,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Efirst__label(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),A_27b)) ) ) ).
fof(mem_c_2Epath_2EfromPath,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2EfromPath(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))) ) ) ).
fof(mem_c_2Epath_2Eis__stopped,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eis__stopped(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),bool)) ) ) ).
fof(mem_c_2Epath_2Elabels,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Elabels(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Ellist_2Ellist(A_27b))) ) ) ).
fof(mem_c_2Epath_2Elast,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Elast(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),A_27a)) ) ) ).
fof(mem_c_2Epath_2Elength,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Elength(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Eoption_2Eoption(ty_2Enum_2Enum))) ) ) ).
fof(mem_c_2Epath_2Emem,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Emem(A_27a,A_27b),arr(A_27a,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))) ) ) ).
fof(mem_c_2Epath_2Enth__label,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Enth__label(A_27a,A_27b),arr(ty_2Enum_2Enum,arr(ty_2Epath_2Epath(A_27b,A_27a),A_27a))) ) ) ).
fof(mem_c_2Epath_2Eokpath,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eokpath(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,arr(A_27a,bool))),arr(ty_2Epath_2Epath(A_27a,A_27b),bool))) ) ) ).
fof(mem_c_2Epath_2Eokpath__f,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eokpath__f(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,arr(A_27a,bool))),arr(arr(ty_2Epath_2Epath(A_27a,A_27b),bool),arr(ty_2Epath_2Epath(A_27a,A_27b),bool)))) ) ) ).
fof(mem_c_2Epath_2Eparallel__comp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> mem(c_2Epath_2Eparallel__comp(A_27a,A_27b,A_27c,A_27d,A_27e),arr(arr(A_27a,arr(A_27b,arr(A_27c,bool))),arr(arr(A_27d,arr(A_27b,arr(A_27e,bool))),arr(ty_2Epair_2Eprod(A_27a,A_27d),arr(A_27b,arr(ty_2Epair_2Eprod(A_27c,A_27e),bool)))))) ) ) ) ) ) ).
fof(mem_c_2Epath_2Epconcat,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Epconcat(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),arr(A_27b,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b))))) ) ) ).
fof(mem_c_2Epath_2Epcons,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Epcons(A_27a,A_27b),arr(A_27a,arr(A_27b,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b))))) ) ) ).
fof(mem_c_2Epath_2Epgenerate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Epgenerate(A_27a,A_27b),arr(arr(ty_2Enum_2Enum,A_27a),arr(arr(ty_2Enum_2Enum,A_27b),ty_2Epath_2Epath(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epath_2Eplink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eplink(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epath_2Epmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> mem(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,A_27c),arr(arr(A_27b,A_27d),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27c,A_27d))))) ) ) ) ) ).
fof(mem_c_2Epath_2Eseg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Eseg(A_27a,A_27b),arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b))))) ) ) ).
fof(mem_c_2Epath_2Estopped__at,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Estopped__at(A_27a,A_27b),arr(A_27a,ty_2Epath_2Epath(A_27a,A_27b))) ) ) ).
fof(mem_c_2Epath_2Etail,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Etail(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b))) ) ) ).
fof(mem_c_2Epath_2Etake,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2Etake(A_27a,A_27b),arr(ty_2Enum_2Enum,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epath_2EtoPath,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epath_2EtoPath(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ty_2Epath_2Epath(A_27a,A_27b))) ) ) ).
fof(mem_c_2Epath_2Etrace__machine,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epath_2Etrace__machine(A_27a),arr(arr(ty_2Elist_2Elist(A_27a),bool),arr(ty_2Elist_2Elist(A_27a),arr(A_27a,arr(ty_2Elist_2Elist(A_27a),bool))))) ) ).
fof(mem_c_2Epath_2Eunfold,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Epath_2Eunfold(A_27a,A_27b,A_27c),arr(arr(A_27c,A_27a),arr(arr(A_27c,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27c,A_27b))),arr(A_27c,ty_2Epath_2Epath(A_27a,A_27b))))) ) ) ) ).
fof(ax_thm_2Epath_2Epath__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a)))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ty_2Epath_2Epath(A_27a,A_27b)),k(ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),c_2Ebool_2ET)),V0rep)) ) ) ) ).
fof(ax_thm_2Epath_2Epath__absrep__bijections,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0a] :
( mem(V0a,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0a)) = V0a )
& ! [V1r] :
( mem(V1r,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
=> ( p(ap(k(ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),c_2Ebool_2ET),V1r))
<=> ap(c_2Epath_2EfromPath(A_27a,A_27b),ap(c_2Epath_2EtoPath(A_27a,A_27b),V1r)) = V1r ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epath__rep__bijections__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0a] :
( mem(V0a,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0a)) = V0a )
& ! [V1r] :
( mem(V1r,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
=> ap(c_2Epath_2EfromPath(A_27a,A_27b),ap(c_2Epath_2EtoPath(A_27a,A_27b),V1r)) = V1r ) ) ) ) ).
fof(conj_thm_2Epath_2EtoPath__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
=> ! [V1r_27] :
( mem(V1r_27,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
=> ( ap(c_2Epath_2EtoPath(A_27a,A_27b),V0r) = ap(c_2Epath_2EtoPath(A_27a,A_27b),V1r_27)
<=> V0r = V1r_27 ) ) ) ) ) ).
fof(conj_thm_2Epath_2EfromPath__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1a_27] :
( mem(V1a_27,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ap(c_2Epath_2EfromPath(A_27a,A_27b),V0a) = ap(c_2Epath_2EfromPath(A_27a,A_27b),V1a_27)
<=> V0a = V1a_27 ) ) ) ) ) ).
fof(conj_thm_2Epath_2EfromPath__onto,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
=> ? [V1a] :
( mem(V1a,ty_2Epath_2Epath(A_27a,A_27b))
& V0r = ap(c_2Epath_2EfromPath(A_27a,A_27b),V1a) ) ) ) ) ).
fof(conj_thm_2Epath_2EtoPath__onto,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epath_2Epath(A_27a,A_27b))
=> ? [V1r] :
( mem(V1r,ty_2Epair_2Eprod(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))))
& V0a = ap(c_2Epath_2EtoPath(A_27a,A_27b),V1r) ) ) ) ) ).
fof(ax_thm_2Epath_2Efirst__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Efirst(A_27a,A_27b),V0p) = ap(c_2Epair_2EFST(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0p)) ) ) ) ).
fof(ax_thm_2Epath_2Estopped__at__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x) = ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),V0x),c_2Ellist_2ELNIL(ty_2Epair_2Eprod(A_27b,A_27a)))) ) ) ) ).
fof(ax_thm_2Epath_2Epcons__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p) = ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),V0x),ap(ap(c_2Ellist_2ELCONS(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V1r),ap(c_2Epath_2Efirst(A_27a,A_27b),V2p))),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V2p))))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Estopped__at__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x) = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1y)
<=> V0x = V1y ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epcons__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ! [V5q] :
( mem(V5q,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3y),V4s),V5q)
<=> ( V0x = V3y
& V1r = V4s
& V2p = V5q ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Estopped__at__not__pcons,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x) != ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1y),V2r),V3p)
& ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1y),V2r),V3p) != ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epath__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ? [V1x] :
( mem(V1x,A_27a)
& V0p = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x) )
| ? [V2x] :
( mem(V2x,A_27a)
& ? [V3r] :
( mem(V3r,A_27b)
& ? [V4q] :
( mem(V4q,ty_2Epath_2Epath(A_27a,A_27b))
& V0p = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4q) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EFORALL__path,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ( ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> p(ap(V0P,V1p)) )
<=> ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x))) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27b)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
=> p(ap(V0P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EEXISTS__path,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ( ? [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
& p(ap(V0P,V1p)) )
<=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x))) )
| ? [V3x] :
( mem(V3x,A_27a)
& ? [V4r] :
( mem(V4r,A_27b)
& ? [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
& p(ap(V0P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = V0x )
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1x),V2r),V3p)) = V1x ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Efinite__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0sigma] :
( mem(V0sigma,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0sigma))
<=> p(ap(c_2Ellist_2ELFINITE(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0sigma)))) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)))
<=> $true ) )
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1x),V2r),V3p)))
<=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V3p)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Elast__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Elast(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = V0x )
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elast(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1x),V2r),V3p)) = ap(c_2Epath_2Elast(A_27a,A_27b),V3p) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epath__bisimulation,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1p2] :
( mem(V1p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( V0p1 = V1p2
<=> ? [V2R] :
( mem(V2R,arr(ty_2Epath_2Epath(A_27a,A_27b),arr(ty_2Epath_2Epath(A_27a,A_27b),bool)))
& p(ap(ap(V2R,V0p1),V1p2))
& ! [V3q1] :
( mem(V3q1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V4q2] :
( mem(V4q2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(V2R,V3q1),V4q2))
=> ( ? [V5x] :
( mem(V5x,A_27a)
& V3q1 = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V5x)
& V4q2 = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V5x) )
| ? [V6x] :
( mem(V6x,A_27a)
& ? [V7r] :
( mem(V7r,A_27b)
& ? [V8q1_27] :
( mem(V8q1_27,ty_2Epath_2Epath(A_27a,A_27b))
& ? [V9q2_27] :
( mem(V9q2_27,ty_2Epath_2Epath(A_27a,A_27b))
& V3q1 = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V6x),V7r),V8q1_27)
& V4q2 = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V6x),V7r),V9q2_27)
& p(ap(ap(V2R,V8q1_27),V9q2_27)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__path__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ( ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(V0P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x))) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27b)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V4p))
& p(ap(V0P,V4p)) )
=> p(ap(V0P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4p))) ) ) ) ) )
=> ! [V5q] :
( mem(V5q,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V5q))
=> p(ap(V0P,V5q)) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Epmap__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V2p) = ap(c_2Epath_2EtoPath(A_27c,A_27d),ap(ap(ap(c_2Epair_2E_23_23(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a)),A_27c,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27d,A_27c))),V0f),ap(c_2Ellist_2ELMAP(ty_2Epair_2Eprod(A_27b,A_27a),ty_2Epair_2Eprod(A_27d,A_27c)),ap(ap(c_2Epair_2E_23_23(A_27b,A_27a,A_27d,A_27c),V1g),V0f))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V2p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epmap__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27d,A_27c))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27d,A_27b,A_27c),V0f),V1g),ap(c_2Epath_2Estopped__at(A_27a,A_27d),V2x)) = ap(c_2Epath_2Estopped__at(A_27b,A_27c),ap(V0f,V2x)) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27d)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27d))
=> ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27d,A_27b,A_27c),V0f),V1g),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27d),V3x),V4r),V5p)) = ap(ap(ap(c_2Epath_2Epcons(A_27b,A_27c),ap(V0f,V3x)),ap(V1g,V4r)),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27d,A_27b,A_27c),V0f),V1g),V5p)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Efirst(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V2p)) = ap(V0f,ap(c_2Epath_2Efirst(A_27a,A_27b),V2p)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elast__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V2p))
=> ap(c_2Epath_2Elast(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V2p)) = ap(V0f,ap(c_2Epath_2Elast(A_27a,A_27b),V2p)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V2p)))
<=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V2p)) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Etail__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Etail(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p)) = V2p ) ) ) ) ) ).
fof(ax_thm_2Epath_2Efirst__label__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Efirst__label(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p)) = V1r ) ) ) ) ) ).
fof(ax_thm_2Epath_2Elength__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elength(A_27a,A_27b),V0p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(ty_2Enum_2Enum)),ap(c_2Epath_2Efinite(A_27a,A_27b),V0p)),ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,ap(c_2Elist_2ELENGTH(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Eoption_2ETHE(ty_2Elist_2Elist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Ellist_2EtoList(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0p)))))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Eoption_2ENONE(ty_2Enum_2Enum)) ) ) ) ).
fof(conj_thm_2Epath_2Elength__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Elength(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) )
& ! [V1x] :
( mem(V1x,A_27c)
=> ! [V2r] :
( mem(V2r,A_27d)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27c,A_27d))
=> ap(c_2Epath_2Elength(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V1x),V2r),V3p)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(ty_2Enum_2Enum)),ap(c_2Epath_2Efinite(A_27c,A_27d),V3p)),ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,ap(c_2Eoption_2ETHE(ty_2Enum_2Enum),ap(c_2Epath_2Elength(A_27c,A_27d),V3p))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Eoption_2ENONE(ty_2Enum_2Enum)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Ealt__length__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Elength(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) )
& ! [V1x] :
( mem(V1x,A_27c)
=> ! [V2r] :
( mem(V2r,A_27d)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27c,A_27d))
=> ap(c_2Epath_2Elength(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V1x),V2r),V3p)) = ap(ap(c_2Eoption_2EOPTION__MAP(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2ESUC),ap(c_2Epath_2Elength(A_27c,A_27d),V3p)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elength__never__zero,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elength(A_27a,A_27b),V0p) != ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__length,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p))
<=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& ap(c_2Epath_2Elength(A_27a,A_27b),V0p) = ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),V1n) ) )
& ( ~ p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p))
<=> ap(c_2Epath_2Elength(A_27a,A_27b),V0p) = c_2Eoption_2ENONE(ty_2Enum_2Enum) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elength__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27c,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27c))
=> ap(c_2Epath_2Elength(A_27b,A_27d),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27c,A_27b,A_27d),V0f),V1g),V2p)) = ap(c_2Epath_2Elength(A_27a,A_27c),V2p) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eel__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Efirst(A_27a,A_27b),V0p) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(c_2Enum_2ESUC,V1n)),V2p) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V1n),ap(c_2Epath_2Etail(A_27a,A_27b),V2p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eel__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Efirst(A_27a,A_27b),V0p) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),V2p) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2Etail(A_27a,A_27b),V2p)) ) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V3n))),V4p) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Epath_2Etail(A_27a,A_27b),V4p)) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Enth__label__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27b,A_27a))
=> ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Efirst__label(A_27b,A_27a),V0p) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27b,A_27a))
=> ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),ap(c_2Enum_2ESUC,V1n)),V2p) = ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),V1n),ap(c_2Epath_2Etail(A_27b,A_27a),V2p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27b,A_27a))
=> ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Efirst__label(A_27b,A_27a),V0p) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27b,A_27a))
=> ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),V2p) = ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2Etail(A_27b,A_27a),V2p)) ) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27b,A_27a))
=> ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V3n))),V4p) = ap(ap(c_2Epath_2Enth__label(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Epath_2Etail(A_27b,A_27a),V4p)) ) ) ) ) ) ).
fof(lameq_f1679,axiom,
! [A_27c,A_27b,A_27a,V6l] :
( mem(V6l,A_27c)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
=> ! [V7v] : ap(f1679(A_27c,A_27b,A_27a,V6l,V3y,V1g),V7v) = ap(ap(ap(c_2Epath_2Epcons(A_27b,A_27c),V3y),V6l),ap(V1g,V7v)) ) ) ) ).
fof(lameq_f1680,axiom,
! [A_27a,A_27b,A_27c,V3y] :
( mem(V3y,A_27b)
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
=> ! [V6l] : ap(f1680(A_27a,A_27b,A_27c,V3y,V1g),V6l) = f1679(A_27c,A_27b,A_27a,V6l,V3y,V1g) ) ) ).
fof(lameq_f1681,axiom,
! [A_27b,A_27c,A_27a,V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V5v3] : ap(f1681(A_27b,A_27c,A_27a,V1g,V3y),V5v3) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epath_2Epath(A_27b,A_27c),A_27c,A_27a),V5v3),f1680(A_27a,A_27b,A_27c,V3y,V1g)) ) ) ).
fof(lameq_f1682,axiom,
! [A_27b,A_27a,A_27c,V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4v2] : ap(f1682(A_27b,A_27a,A_27c,V1g,V3y),V4v2) = ap(ap(ap(c_2Eoption_2Eoption__CASE(ty_2Epair_2Eprod(A_27c,A_27a),ty_2Epath_2Epath(A_27b,A_27c)),V4v2),ap(c_2Epath_2Estopped__at(A_27b,A_27c),V3y)),f1681(A_27b,A_27c,A_27a,V1g,V3y)) ) ) ).
fof(lameq_f1683,axiom,
! [A_27c,A_27a,A_27b,V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
=> ! [V3y] : ap(f1683(A_27c,A_27a,A_27b,V1g),V3y) = f1682(A_27b,A_27a,A_27c,V1g,V3y) ) ).
fof(conj_thm_2Epath_2Epath__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Epair_2Eprod(A_27b,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27c,A_27a)))))
=> ? [V1g] :
( mem(V1g,arr(A_27a,ty_2Epath_2Epath(A_27b,A_27c)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ap(V1g,V2x) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epath_2Epath(A_27b,A_27c),A_27b,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27c,A_27a))),ap(V0f,V2x)),f1683(A_27c,A_27a,A_27b,V1g)) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Epconcat__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1lab] :
( mem(V1lab,A_27b)
=> ! [V2p2] :
( mem(V2p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V0p1),V1lab),V2p2) = ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2Efirst(A_27a,A_27b),V0p1)),ap(ap(c_2Ellist_2ELAPPEND(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0p1))),ap(ap(c_2Ellist_2ELCONS(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V1lab),ap(c_2Epath_2Efirst(A_27a,A_27b),V2p2))),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V2p2)))))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epconcat__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1lab] :
( mem(V1lab,A_27b)
=> ! [V2p2] :
( mem(V2p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)),V1lab),V2p2) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1lab),V2p2) ) ) )
& ! [V3x] :
( mem(V3x,A_27c)
=> ! [V4r] :
( mem(V4r,A_27d)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27c,A_27d))
=> ! [V6lab] :
( mem(V6lab,A_27d)
=> ! [V7p2] :
( mem(V7p2,ty_2Epath_2Epath(A_27c,A_27d))
=> ap(ap(ap(c_2Epath_2Epconcat(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V3x),V4r),V5p)),V6lab),V7p2) = ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V3x),V4r),ap(ap(ap(c_2Epath_2Epconcat(A_27c,A_27d),V5p),V6lab),V7p2)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epconcat__eq__stopped,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1lab] :
( mem(V1lab,A_27b)
=> ! [V2p2] :
( mem(V2p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V0p1),V1lab),V2p2) != ap(c_2Epath_2Estopped__at(A_27a,A_27b),V3x)
& ap(c_2Epath_2Estopped__at(A_27a,A_27b),V3x) != ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V0p1),V1lab),V2p2) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epconcat__eq__pcons,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V3p1] :
( mem(V3p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V4lab] :
( mem(V4lab,A_27b)
=> ! [V5p2] :
( mem(V5p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V3p1),V4lab),V5p2) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p)
<=> ( ( V4lab = V1r
& V3p1 = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)
& V2p = V5p2 )
| ? [V6p1_27] :
( mem(V6p1_27,ty_2Epath_2Epath(A_27a,A_27b))
& V3p1 = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V6p1_27)
& V2p = ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V6p1_27),V4lab),V5p2) ) ) )
& ( ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2p) = ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V3p1),V4lab),V5p2)
<=> ( ( V4lab = V1r
& V3p1 = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)
& V2p = V5p2 )
| ? [V7p1_27] :
( mem(V7p1_27,ty_2Epath_2Epath(A_27a,A_27b))
& V3p1 = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V7p1_27)
& V2p = ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V7p1_27),V4lab),V5p2) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__pconcat,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1lab] :
( mem(V1lab,A_27b)
=> ! [V2p2] :
( mem(V2p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epconcat(A_27a,A_27b),V0p1),V1lab),V2p2)))
<=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p1))
& p(ap(c_2Epath_2Efinite(A_27a,A_27b),V2p2)) ) ) ) ) ) ) ) ).
fof(lameq_f1684,axiom,
! [A_27a,A_27b,V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] : ap(f1684(A_27a,A_27b,V0p),V1i) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V1i),ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Epath_2Efinite(A_27a,A_27b),V0p)),ap(ap(c_2Eprim__rec_2E_3C,V1i),ap(c_2Eoption_2ETHE(ty_2Enum_2Enum),ap(c_2Epath_2Elength(A_27a,A_27b),V0p))))) ) ).
fof(ax_thm_2Epath_2EPL__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2EPL(A_27a,A_27b),V0p) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1684(A_27a,A_27b,V0p)) ) ) ) ).
fof(conj_thm_2Epath_2Einfinite__PL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ~ p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1i),ap(c_2Epath_2EPL(A_27a,A_27b),V0p))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EPL__pcons,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,A_27b)
=> ! [V2q] :
( mem(V2q,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V0x),V1r),V2q)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2ESUC),ap(c_2Epath_2EPL(A_27a,A_27b),V2q))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EPL__stopped__at,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) ) ) ) ).
fof(conj_thm_2Epath_2EPL__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) )
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3q] :
( mem(V3q,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1x),V2r),V3q)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2ESUC),ap(c_2Epath_2EPL(A_27a,A_27b),V3q))) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EPL__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Epath_2EPL(A_27a,A_27b),V0p))) ) ) ) ).
fof(conj_thm_2Epath_2EPL__downward__closed,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2j),V0i))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2j),ap(c_2Epath_2EPL(A_27a,A_27b),V1p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EPL__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27a))
=> ! [V1g] :
( mem(V1g,arr(A_27d,A_27b))
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27c,A_27d))
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epmap(A_27c,A_27d,A_27a,A_27b),V0f),V1g),V2p)) = ap(c_2Epath_2EPL(A_27c,A_27d),V2p) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eel__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V3p)))
=> ap(ap(c_2Epath_2Eel(A_27c,A_27d),V2i),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V3p)) = ap(V0f,ap(ap(c_2Epath_2Eel(A_27a,A_27b),V2i),V3p)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27d))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,V2i)),ap(c_2Epath_2EPL(A_27a,A_27b),V3p)))
=> ap(ap(c_2Epath_2Enth__label(A_27c,A_27d),V2i),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27d,A_27c),V0f),V1g),V3p)) = ap(V1g,ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V2i),V3p)) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2EfirstP__at__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),V1p),V2i))
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& p(ap(V0P,ap(ap(c_2Epath_2Eel(A_27a,A_27b),V2i),V1p)))
& ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V3j),V2i))
=> ~ p(ap(V0P,ap(ap(c_2Epath_2Eel(A_27a,A_27b),V3j),V1p))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EfirstP__at__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x)),V2n))
<=> ( V2n = c_2Enum_2E0
& p(ap(V0P,V1x)) ) ) ) ) )
& ! [V3P] :
( mem(V3P,arr(A_27a,bool))
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6r] :
( mem(V6r,A_27b)
=> ! [V7p] :
( mem(V7p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V3P),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V5x),V6r),V7p)),V4n))
<=> ( ( V4n = c_2Enum_2E0
& p(ap(V3P,V5x)) )
| ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V4n))
& ~ p(ap(V3P,V5x))
& p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V3P),V7p),ap(ap(c_2Earithmetic_2E_2D,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EfirstP__at__zero,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),V1p),c_2Enum_2E0))
<=> p(ap(V0P,ap(c_2Epath_2Efirst(A_27a,A_27b),V1p))) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eexists__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),V1p))
<=> ? [V2i] :
( mem(V2i,ty_2Enum_2Enum)
& p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),V1p),V2i)) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eevery__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V0P),V1p))
<=> ~ p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Ebool_2E_7E),V0P)),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eexists__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x)))
<=> p(ap(V0P,V1x)) ) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27c)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27c))
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27c),V0P),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27c),V2x),V3r),V4p)))
<=> ( p(ap(V0P,V2x))
| p(ap(ap(c_2Epath_2Eexists(A_27a,A_27c),V0P),V4p)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eevery__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V0P),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x)))
<=> p(ap(V0P,V1x)) ) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27c)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27c))
=> ( p(ap(ap(c_2Epath_2Eevery(A_27a,A_27c),V0P),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27c),V2x),V3r),V4p)))
<=> ( p(ap(V0P,V2x))
& p(ap(ap(c_2Epath_2Eevery(A_27a,A_27c),V0P),V4p)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enot__every,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ~ p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V0P),V1p))
<=> p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Ebool_2E_7E),V0P)),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enot__exists,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ~ p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),V1p))
<=> p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Ebool_2E_7E),V0P)),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eexists__el,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),V1p))
<=> ? [V2i] :
( mem(V2i,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& p(ap(V0P,ap(ap(c_2Epath_2Eel(A_27a,A_27b),V2i),V1p))) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eevery__el,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V0P),V1p))
<=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> p(ap(V0P,ap(ap(c_2Epath_2Eel(A_27a,A_27b),V2i),V1p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eevery__coinduction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x)))
=> p(ap(V1Q,V2x)) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27b)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V0P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p)))
=> ( p(ap(V1Q,V3x))
& p(ap(V0P,V5p)) ) ) ) ) ) )
=> ! [V6p] :
( mem(V6p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V0P,V6p))
=> p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V1Q),V6p)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eexists__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0Q] :
( mem(V0Q,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0Q,V2x))
=> p(ap(V1P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x))) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27b)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V0Q,V3x))
=> p(ap(V1P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p))) ) ) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7r] :
( mem(V7r,A_27b)
=> ! [V8p] :
( mem(V8p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V1P,V8p))
=> p(ap(V1P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V6x),V7r),V8p))) ) ) ) ) )
=> ! [V9p] :
( mem(V9p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0Q),V9p))
=> p(ap(V1P,V9p)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Emem__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,A_27a)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Emem(A_27a,A_27b),V0s),V1p))
<=> ? [V2i] :
( mem(V2i,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& V0s = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V2i),V1p) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Emem__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,A_27a)
=> ( p(ap(ap(c_2Epath_2Emem(A_27a,A_27b),V1s),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)))
<=> V1s = V0x ) ) )
& ! [V2x] :
( mem(V2x,A_27c)
=> ! [V3r] :
( mem(V3r,A_27d)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27c,A_27d))
=> ! [V5s] :
( mem(V5s,A_27c)
=> ( p(ap(ap(c_2Epath_2Emem(A_27c,A_27d),V5s),ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V2x),V3r),V4p)))
<=> ( V5s = V2x
| p(ap(ap(c_2Epath_2Emem(A_27c,A_27d),V5s),V4p)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Edrop__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),c_2Enum_2E0),V0p) = V0p )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Enum_2ESUC,V1n)),V2p) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1n),ap(c_2Epath_2Etail(A_27a,A_27b),V2p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Edrop__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),c_2Enum_2E0),V0p) = V0p )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),V2p) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2Etail(A_27a,A_27b),V2p)) ) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V3n))),V4p) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Epath_2Etail(A_27a,A_27b),V4p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enumeral__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V0n))),V1p) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V0n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2Etail(A_27a,A_27b),V1p)) ) )
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V2n))),V3p) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V2n))),ap(c_2Epath_2Etail(A_27a,A_27b),V3p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1n),V0p)))
<=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p)) ) ) ) ) ) ) ).
fof(lameq_f1685,axiom,
! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2m] : ap(f1685(V1n),V2m) = ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2D,V2m),V1n)) ) ).
fof(conj_thm_2Epath_2Elength__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> ap(c_2Epath_2Elength(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1n),V0p)) = ap(ap(ap(c_2Eoption_2Eoption__CASE(ty_2Enum_2Enum,ty_2Eoption_2Eoption(ty_2Enum_2Enum)),ap(c_2Epath_2Elength(A_27a,A_27b),V0p)),c_2Eoption_2ENONE(ty_2Enum_2Enum)),f1685(V1n)) ) ) ) ) ) ).
fof(lameq_f1686,axiom,
! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ! [V2n] : ap(f1686(V1i),V2n) = ap(ap(c_2Earithmetic_2E_2D,V2n),V1i) ) ).
fof(conj_thm_2Epath_2EPL__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1i),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1i),V0p)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),f1686(V1i)),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EIN__PL__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V2p)))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1j),ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0i),V2p))))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V0i),V1j)),ap(c_2Epath_2EPL(A_27a,A_27b),V2p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0i),V1p)) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),V1p) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__label__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ap(c_2Epath_2Efirst__label(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0i),V1p)) = ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V0i),V1p) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Etail__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V0i),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ap(c_2Epath_2Etail(A_27a,A_27b),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0i),V1p)) = ap(ap(c_2Epath_2Edrop(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2B,V0i),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1p) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eel__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V0i),V1j)),ap(c_2Epath_2EPL(A_27a,A_27b),V2p)))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1j),V2p)) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2B,V0i),V1j)),V2p) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0i),V1j))),ap(c_2Epath_2EPL(A_27a,A_27b),V2p)))
=> ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V0i),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V1j),V2p)) = ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),ap(ap(c_2Earithmetic_2E_2B,V0i),V1j)),V2p) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Etake__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Etake(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Estopped__at(A_27a,A_27b),ap(c_2Epath_2Efirst(A_27a,A_27b),V0p)) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(c_2Enum_2ESUC,V1n)),V2p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(c_2Epath_2Efirst(A_27a,A_27b),V2p)),ap(c_2Epath_2Efirst__label(A_27a,A_27b),V2p)),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V1n),ap(c_2Epath_2Etail(A_27a,A_27b),V2p))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Etake__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Etake(A_27a,A_27b),c_2Enum_2E0),V0p) = ap(c_2Epath_2Estopped__at(A_27a,A_27b),ap(c_2Epath_2Efirst(A_27a,A_27b),V0p)) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),V2p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(c_2Epath_2Efirst(A_27a,A_27b),V2p)),ap(c_2Epath_2Efirst__label(A_27a,A_27b),V2p)),ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2Etail(A_27a,A_27b),V2p))) ) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V3n))),V4p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(c_2Epath_2Efirst(A_27a,A_27b),V4p)),ap(c_2Epath_2Efirst__label(A_27a,A_27b),V4p)),ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Epath_2Etail(A_27a,A_27b),V4p))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V1i),V0p)) = ap(c_2Epath_2Efirst(A_27a,A_27b),V0p) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1i),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V1i),V0p))) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elength__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1i),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> ap(c_2Epath_2Elength(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V1i),V0p)) = ap(c_2Eoption_2ESOME(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V1i),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) ) ) ) ).
fof(lameq_f1687,axiom,
! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ! [V2n] : ap(f1687(V1i),V2n) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V2n),ap(ap(c_2Earithmetic_2E_3C_3D,V2n),V1i)) ) ).
fof(conj_thm_2Epath_2EPL__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1i),ap(c_2Epath_2EPL(A_27a,A_27b),V0p)))
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V1i),V0p)) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1687(V1i)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elast__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ap(c_2Epath_2Elast(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V0i),V1p)) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),V1p) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V2i),V0n))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0n),ap(c_2Epath_2EPL(A_27a,A_27b),V1p))) )
=> ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V2i),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V0n),V1p)) = ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V2i),V1p) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eseg__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V1j),V2p) = ap(ap(c_2Epath_2Etake(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2D,V1j),V0i)),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0i),V2p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Esingleton__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
=> ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V0i),V1p) = ap(c_2Epath_2Estopped__at(A_27a,A_27b),ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Erecursive__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V0i),V1j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1j),ap(c_2Epath_2EPL(A_27a,A_27b),V2p))) )
=> ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V1j),V2p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),V2p)),ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V0i),V2p)),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),ap(ap(c_2Earithmetic_2E_2B,V0i),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1j),V2p)) ) ) ) ) ) ) ).
fof(lameq_f1688,axiom,
! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V3n] : ap(f1688(V1j,V0i),V3n) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V3n),ap(ap(c_2Earithmetic_2E_3C_3D,V3n),ap(ap(c_2Earithmetic_2E_2D,V1j),V0i))) ) ) ).
fof(conj_thm_2Epath_2EPL__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0i),V1j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1j),ap(c_2Epath_2EPL(A_27a,A_27b),V2p))) )
=> ap(c_2Epath_2EPL(A_27a,A_27b),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V1j),V2p)) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1688(V1j,V0i)) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1i),V2j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2j),ap(c_2Epath_2EPL(A_27a,A_27b),V0p))) )
=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V1i),V2j),V0p))) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0i),V1j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1j),ap(c_2Epath_2EPL(A_27a,A_27b),V2p))) )
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V1j),V2p)) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0i),V2p) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elast__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0i),V1j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1j),ap(c_2Epath_2EPL(A_27a,A_27b),V2p))) )
=> ap(c_2Epath_2Elast(A_27a,A_27b),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V0i),V1j),V2p)) = ap(ap(c_2Epath_2Eel(A_27a,A_27b),V1j),V2p) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Elabels__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epath_2Elabels(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)) = c_2Ellist_2ELNIL(A_27b) )
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2r] :
( mem(V2r,A_27b)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elabels(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V1x),V2r),V3p)) = ap(ap(c_2Ellist_2ELCONS(A_27b),V2r),ap(c_2Epath_2Elabels(A_27a,A_27b),V3p)) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2EfirstP__at__unique,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),V1p),V2n))
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(c_2Epath_2EfirstP__at(A_27a,A_27b),V0P),V1p),V3m))
<=> V3m = V2n ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eis__stopped__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Eis__stopped(A_27a,A_27b),V0p))
<=> ? [V1x] :
( mem(V1x,A_27a)
& V0p = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eis__stopped__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Epath_2Eis__stopped(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)))
<=> $true ) )
& ! [V1x] :
( mem(V1x,A_27c)
=> ! [V2r] :
( mem(V2r,A_27d)
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27c,A_27d))
=> ( p(ap(c_2Epath_2Eis__stopped(A_27c,A_27d),ap(ap(ap(c_2Epath_2Epcons(A_27c,A_27d),V1x),V2r),V3p)))
<=> $false ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Efilter__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(V0P,V1x))
=> ap(ap(c_2Epath_2Efilter(A_27a,A_27b),V0P),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x)) = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x) ) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27b)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Efilter(A_27a,A_27b),V0P),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4p)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epath_2Epath(A_27a,A_27b)),ap(V0P,V2x)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Epath_2Epath(A_27a,A_27b)),ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),V4p)),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),ap(ap(c_2Epath_2Efilter(A_27a,A_27b),V0P),V4p))),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x))),ap(ap(c_2Epath_2Efilter(A_27a,A_27b),V0P),V4p)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efilter__every,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eexists(A_27a,A_27b),V0P),V1p))
=> p(ap(ap(c_2Epath_2Eevery(A_27a,A_27b),V0P),ap(ap(c_2Epath_2Efilter(A_27a,A_27b),V0P),V1p))) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Epgenerate__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,A_27b))
=> ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V0f),V1g) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(V0f,c_2Enum_2E0)),ap(V1g,c_2Enum_2E0)),ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,A_27a,ty_2Enum_2Enum),V0f),c_2Enum_2ESUC)),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,A_27b,ty_2Enum_2Enum),V1g),c_2Enum_2ESUC))) ) ) ) ) ).
fof(conj_thm_2Epath_2Epgenerate__infinite,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,A_27b))
=> ~ p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V0f),V1g))) ) ) ) ) ).
fof(conj_thm_2Epath_2Epgenerate__not__stopped,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x) != ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V0f),V1g) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eel__pgenerate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,A_27b))
=> ap(ap(c_2Epath_2Eel(A_27a,A_27b),V0n),ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V1f),V2g)) = ap(V1f,V0n) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__pgenerate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,A_27b))
=> ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V0n),ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V1f),V2g)) = ap(V2g,V0n) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epgenerate__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f1] :
( mem(V0f1,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1g1] :
( mem(V1g1,arr(ty_2Enum_2Enum,A_27b))
=> ! [V2f2] :
( mem(V2f2,arr(ty_2Enum_2Enum,A_27a))
=> ! [V3g2] :
( mem(V3g2,arr(ty_2Enum_2Enum,A_27b))
=> ( ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V0f1),V1g1) = ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V2f2),V3g2)
<=> ( V0f1 = V2f2
& V1g1 = V3g2 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Epgenerate__onto,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ~ p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p))
=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ? [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,A_27b))
& V0p = ap(ap(c_2Epath_2Epgenerate(A_27a,A_27b),V1f),V2g) ) ) ) ) ) ) ).
fof(lameq_f1689,axiom,
! [A_27b,A_27a,V2x] : ap(f1689(A_27b,A_27a),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Epath_2Epath(A_27a,A_27b),bool),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x)),ap(ap(c_2Ebool_2EIN(A_27a),V2x),c_2Epred__set_2EUNIV(A_27a))) ).
fof(lameq_f1690,axiom,
! [A_27b,A_27a,V1X] :
( mem(V1X,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27b)
=> ! [V5p] : ap(f1690(A_27b,A_27a,V1X,V0R,V3x,V4r),V5p) = ap(ap(c_2Epair_2E_2C(ty_2Epath_2Epath(A_27a,A_27b),bool),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(ap(V0R,V3x),V4r),ap(c_2Epath_2Efirst(A_27a,A_27b),V5p))),ap(ap(c_2Ebool_2EIN(ty_2Epath_2Epath(A_27a,A_27b)),V5p),V1X))) ) ) ) ) ).
fof(lameq_f1691,axiom,
! [A_27a,A_27b,V1X] :
( mem(V1X,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] : ap(f1691(A_27a,A_27b,V1X,V0R,V3x),V4r) = f1690(A_27b,A_27a,V1X,V0R,V3x,V4r) ) ) ) ).
fof(lameq_f1692,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1X] :
( mem(V1X,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ! [V3x] : ap(f1692(A_27b,A_27a,V0R,V1X),V3x) = ap(c_2Epair_2EUNCURRY(A_27b,ty_2Epath_2Epath(A_27a,A_27b),ty_2Epair_2Eprod(ty_2Epath_2Epath(A_27a,A_27b),bool)),f1691(A_27a,A_27b,V1X,V0R,V3x)) ) ) ).
fof(ax_thm_2Epath_2Eokpath__f__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1X] :
( mem(V1X,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ap(ap(c_2Epath_2Eokpath__f(A_27a,A_27b),V0R),V1X) = ap(ap(c_2Epred__set_2EUNION(ty_2Epath_2Epath(A_27a,A_27b)),ap(c_2Epred__set_2EGSPEC(ty_2Epath_2Epath(A_27a,A_27b),A_27a),f1689(A_27b,A_27a))),ap(c_2Epred__set_2EGSPEC(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,ty_2Epath_2Epath(A_27a,A_27b)))),ap(c_2Epair_2EUNCURRY(A_27a,ty_2Epair_2Eprod(A_27b,ty_2Epath_2Epath(A_27a,A_27b)),ty_2Epair_2Eprod(ty_2Epath_2Epath(A_27a,A_27b),bool)),f1692(A_27b,A_27a,V0R,V1X)))) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__monotone,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> p(ap(c_2EfixedPoint_2Emonotone(ty_2Epath_2Epath(A_27a,A_27b),ty_2Epath_2Epath(A_27a,A_27b)),ap(c_2Epath_2Eokpath__f(A_27a,A_27b),V0R))) ) ) ) ).
fof(ax_thm_2Epath_2Eokpath__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R) = ap(c_2EfixedPoint_2Egfp(ty_2Epath_2Epath(A_27a,A_27b)),ap(c_2Epath_2Eokpath__f(A_27a,A_27b),V0R)) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__co__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1P] :
( mem(V1P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27b)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V1P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4p)))
=> ( p(ap(ap(ap(V0R,V2x),V3r),ap(c_2Epath_2Efirst(A_27a,A_27b),V4p)))
& p(ap(V1P,V4p)) ) ) ) ) )
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(V1P,V5p))
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V5p)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1x] :
( mem(V1x,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1x))
<=> ( ? [V2x_27] :
( mem(V2x_27,A_27a)
& V1x = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x_27) )
| ? [V3x_27] :
( mem(V3x_27,A_27a)
& ? [V4r] :
( mem(V4r,A_27b)
& ? [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
& V1x = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x_27),V4r),V5p)
& p(ap(ap(ap(V0R,V3x_27),V4r),ap(c_2Epath_2Efirst(A_27a,A_27b),V5p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V5p)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x))) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27b)
=> ! [V4p] :
( mem(V4p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4p)))
<=> ( p(ap(ap(ap(V0R,V2x),V3r),ap(c_2Epath_2Efirst(A_27a,A_27b),V4p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V4p)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__okpath__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epath_2Epath(A_27a,A_27b),bool))
=> ! [V1R] :
( mem(V1R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,ap(c_2Epath_2Estopped__at(A_27a,A_27b),V2x))) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4r] :
( mem(V4r,A_27b)
=> ! [V5p] :
( mem(V5p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V1R),V5p))
& p(ap(c_2Epath_2Efinite(A_27a,A_27b),V5p))
& p(ap(ap(ap(V1R,V3x),V4r),ap(c_2Epath_2Efirst(A_27a,A_27b),V5p)))
& p(ap(V0P,V5p)) )
=> p(ap(V0P,ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V3x),V4r),V5p))) ) ) ) ) )
=> ! [V6sigma] :
( mem(V6sigma,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V1R),V6sigma))
& p(ap(c_2Epath_2Efinite(A_27a,A_27b),V6sigma)) )
=> p(ap(V0P,V6sigma)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__pmap,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27b))
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V3p))
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5r] :
( mem(V5r,A_27b)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ( p(ap(ap(ap(V0R,V4x),V5r),V6y))
=> p(ap(ap(ap(V0R,ap(V1f,V4x)),ap(V2g,V5r)),ap(V1f,V6y))) ) ) ) ) )
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(ap(c_2Epath_2Epmap(A_27a,A_27b,A_27a,A_27b),V1f),V2g),V3p))) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eplink__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eplink(A_27a,A_27b),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V0x)),V1p) = V1p ) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3r] :
( mem(V3r,A_27b)
=> ! [V4p1] :
( mem(V4p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V5p2] :
( mem(V5p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(ap(c_2Epath_2Eplink(A_27a,A_27b),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),V4p1)),V5p2) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2x),V3r),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V4p1),V5p2)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__plink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1p2] :
( mem(V1p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V0p1),V1p2)))
<=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p1))
& p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p2)) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efirst__plink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1p2] :
( mem(V1p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ap(c_2Epath_2Elast(A_27a,A_27b),V0p1) = ap(c_2Epath_2Efirst(A_27a,A_27b),V1p2)
=> ap(c_2Epath_2Efirst(A_27a,A_27b),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V0p1),V1p2)) = ap(c_2Epath_2Efirst(A_27a,A_27b),V0p1) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elast__plink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1p2] :
( mem(V1p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p1))
& p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p2))
& ap(c_2Epath_2Elast(A_27a,A_27b),V0p1) = ap(c_2Epath_2Efirst(A_27a,A_27b),V1p2) )
=> ap(c_2Epath_2Elast(A_27a,A_27b),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V0p1),V1p2)) = ap(c_2Epath_2Elast(A_27a,A_27b),V1p2) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__plink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p1] :
( mem(V1p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2p2] :
( mem(V2p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p1))
& ap(c_2Epath_2Elast(A_27a,A_27b),V1p1) = ap(c_2Epath_2Efirst(A_27a,A_27b),V2p2) )
=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V1p1),V2p2)))
<=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p1))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V2p2)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__take,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p)) )
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V2i),V1p))) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__drop,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p)) )
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V2i),V1p))) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__seg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2i),V3j))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3j),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p)) )
=> p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),ap(ap(ap(c_2Epath_2Eseg(A_27a,A_27b),V2i),V3j),V1p))) ) ) ) ) ) ) ) ).
fof(lameq_f1693,axiom,
! [A_27a,A_27b,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V2y] :
( mem(V2y,A_27a)
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V3l] : ap(f1693(A_27a,A_27b,V0R,V2y,V1x),V3l) = ap(ap(ap(V0R,V2y),V3l),V1x) ) ) ) ).
fof(lameq_f1694,axiom,
! [A_27b,A_27a,V1x] :
( mem(V1x,A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V2y] : ap(f1694(A_27b,A_27a,V1x,V0R),V2y) = ap(c_2Ebool_2E_3F(A_27b),f1693(A_27a,A_27b,V0R,V2y,V1x)) ) ) ).
fof(lameq_f1695,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1x] : ap(f1695(A_27b,A_27a,V0R),V1x) = f1694(A_27b,A_27a,V1x,V0R) ) ).
fof(ax_thm_2Epath_2ESN__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ( p(ap(c_2Epath_2ESN(A_27a,A_27b),V0R))
<=> p(ap(c_2Erelation_2EWF(A_27a),f1695(A_27b,A_27a,V0R))) ) ) ) ) ).
fof(conj_thm_2Epath_2ESN__finite__paths,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(c_2Epath_2ESN(A_27a,A_27b),V0R))
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p)) )
=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__paths__SN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ( ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p))
=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p)) ) )
=> p(ap(c_2Epath_2ESN(A_27a,A_27b),V0R)) ) ) ) ) ).
fof(conj_thm_2Epath_2ESN__finite__paths__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ( p(ap(c_2Epath_2ESN(A_27a,A_27b),V0R))
<=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0R),V1p))
=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V1p)) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elabels__LMAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elabels(A_27a,A_27b),V0p) = ap(ap(c_2Ellist_2ELMAP(ty_2Epair_2Eprod(A_27b,A_27a),A_27b),c_2Epair_2EFST(A_27b,A_27a)),ap(c_2Epair_2ESND(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epath_2EfromPath(A_27a,A_27b),V0p))) ) ) ) ).
fof(conj_thm_2Epath_2Elabels__plink,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p1] :
( mem(V0p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V1p2] :
( mem(V1p2,ty_2Epath_2Epath(A_27a,A_27b))
=> ap(c_2Epath_2Elabels(A_27a,A_27b),ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V0p1),V1p2)) = ap(ap(c_2Ellist_2ELAPPEND(A_27b),ap(c_2Epath_2Elabels(A_27a,A_27b),V0p1)),ap(c_2Epath_2Elabels(A_27a,A_27b),V1p2)) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__labels,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Ellist_2ELFINITE(A_27b),ap(c_2Epath_2Elabels(A_27a,A_27b),V0p)))
<=> p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p)) ) ) ) ) ).
fof(lameq_f1696,axiom,
! [A_27c,A_27a,A_27b,V0proj] :
( mem(V0proj,arr(A_27c,A_27a))
=> ! [V4next__s] :
( mem(V4next__s,A_27c)
=> ! [V5lbl] : ap(f1696(A_27c,A_27a,A_27b,V0proj,V4next__s),V5lbl) = ap(ap(c_2Epair_2E_2C(A_27c,ty_2Epair_2Eprod(A_27b,A_27a)),V4next__s),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V5lbl),ap(V0proj,V4next__s))) ) ) ).
fof(lameq_f1697,axiom,
! [A_27b,A_27a,A_27c,V0proj] :
( mem(V0proj,arr(A_27c,A_27a))
=> ! [V4next__s] : ap(f1697(A_27b,A_27a,A_27c,V0proj),V4next__s) = f1696(A_27c,A_27a,A_27b,V0proj,V4next__s) ) ).
fof(lameq_f1698,axiom,
! [A_27b,A_27a,A_27c,V1f] :
( mem(V1f,arr(A_27c,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27c,A_27b))))
=> ! [V0proj] :
( mem(V0proj,arr(A_27c,A_27a))
=> ! [V3s] : ap(f1698(A_27b,A_27a,A_27c,V1f,V0proj),V3s) = ap(ap(c_2Eoption_2EOPTION__MAP(ty_2Epair_2Eprod(A_27c,A_27b),ty_2Epair_2Eprod(A_27c,ty_2Epair_2Eprod(A_27b,A_27a))),ap(c_2Epair_2EUNCURRY(A_27c,A_27b,ty_2Epair_2Eprod(A_27c,ty_2Epair_2Eprod(A_27b,A_27a))),f1697(A_27b,A_27a,A_27c,V0proj))),ap(V1f,V3s)) ) ) ).
fof(ax_thm_2Epath_2Eunfold__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0proj] :
( mem(V0proj,arr(A_27c,A_27a))
=> ! [V1f] :
( mem(V1f,arr(A_27c,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27c,A_27b))))
=> ! [V2s] :
( mem(V2s,A_27c)
=> ap(ap(ap(c_2Epath_2Eunfold(A_27a,A_27b,A_27c),V0proj),V1f),V2s) = ap(c_2Epath_2EtoPath(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Ellist_2Ellist(ty_2Epair_2Eprod(A_27b,A_27a))),ap(V0proj,V2s)),ap(ap(c_2Ellist_2ELUNFOLD(ty_2Epair_2Eprod(A_27b,A_27a),A_27c),f1698(A_27b,A_27a,A_27c,V1f,V0proj)),V2s))) ) ) ) ) ) ) ).
fof(lameq_f1699,axiom,
! [A_27a,A_27b,A_27c,V2s] :
( mem(V2s,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V4s_27] :
( mem(V4s_27,A_27a)
=> ! [V5l] : ap(f1699(A_27a,A_27b,A_27c,V2s,V1f,V0proj,V4s_27),V5l) = ap(ap(ap(c_2Epath_2Epcons(A_27b,A_27c),ap(V0proj,V2s)),V5l),ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V4s_27)) ) ) ) ) ).
fof(lameq_f1700,axiom,
! [A_27c,A_27b,A_27a,V2s] :
( mem(V2s,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V4s_27] : ap(f1700(A_27c,A_27b,A_27a,V2s,V1f,V0proj),V4s_27) = f1699(A_27a,A_27b,A_27c,V2s,V1f,V0proj,V4s_27) ) ) ) ).
fof(lameq_f1701,axiom,
! [A_27b,A_27a,A_27c,V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ! [V3v] : ap(f1701(A_27b,A_27a,A_27c,V0proj,V1f,V2s),V3v) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epath_2Epath(A_27b,A_27c),A_27a,A_27c),V3v),f1700(A_27c,A_27b,A_27a,V2s,V1f,V0proj)) ) ) ) ).
fof(conj_thm_2Epath_2Eunfold__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V2s) = ap(ap(ap(c_2Eoption_2Eoption__CASE(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epath_2Epath(A_27b,A_27c)),ap(V1f,V2s)),ap(c_2Epath_2Estopped__at(A_27b,A_27c),ap(V0proj,V2s))),f1701(A_27b,A_27a,A_27c,V0proj,V1f,V2s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eunfold__thm2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3v1] :
( mem(V3v1,A_27a)
=> ! [V4v2] :
( mem(V4v2,A_27c)
=> ( ( ap(V1f,V2x) = c_2Eoption_2ENONE(ty_2Epair_2Eprod(A_27a,A_27c))
=> ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V2x) = ap(c_2Epath_2Estopped__at(A_27b,A_27c),ap(V0proj,V2x)) )
& ( ap(V1f,V2x) = ap(c_2Eoption_2ESOME(ty_2Epair_2Eprod(A_27a,A_27c)),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V3v1),V4v2))
=> ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V2x) = ap(ap(ap(c_2Epath_2Epcons(A_27b,A_27c),ap(V0proj,V2x)),V4v2),ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V3v1)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Elabels__unfold,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0proj] :
( mem(V0proj,arr(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ap(c_2Epath_2Elabels(A_27b,A_27c),ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V0proj),V1f),V2s)) = ap(ap(c_2Ellist_2ELUNFOLD(A_27c,A_27a),V1f),V2s) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__unfold,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,arr(A_27b,arr(A_27c,arr(A_27b,bool))))
=> ! [V2proj] :
( mem(V2proj,arr(A_27a,A_27b))
=> ! [V3f] :
( mem(V3f,arr(A_27a,ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,A_27c))))
=> ! [V4s] :
( mem(V4s,A_27a)
=> ( ( p(ap(V0P,V4s))
& ! [V5s] :
( mem(V5s,A_27a)
=> ! [V6s_27] :
( mem(V6s_27,A_27a)
=> ! [V7l] :
( mem(V7l,A_27c)
=> ( ( p(ap(V0P,V5s))
& ap(V3f,V5s) = ap(c_2Eoption_2ESOME(ty_2Epair_2Eprod(A_27a,A_27c)),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V6s_27),V7l)) )
=> p(ap(V0P,V6s_27)) ) ) ) )
& ! [V8s] :
( mem(V8s,A_27a)
=> ! [V9s_27] :
( mem(V9s_27,A_27a)
=> ! [V10l] :
( mem(V10l,A_27c)
=> ( ( p(ap(V0P,V8s))
& ap(V3f,V8s) = ap(c_2Eoption_2ESOME(ty_2Epair_2Eprod(A_27a,A_27c)),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V9s_27),V10l)) )
=> p(ap(ap(ap(V1m,ap(V2proj,V8s)),V10l),ap(V2proj,V9s_27))) ) ) ) ) )
=> p(ap(ap(c_2Epath_2Eokpath(A_27b,A_27c),V1m),ap(ap(ap(c_2Epath_2Eunfold(A_27b,A_27c,A_27a),V2proj),V3f),V4s))) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Etrace__machine__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V1s] :
( mem(V1s,ty_2Elist_2Elist(A_27a))
=> ! [V2l] :
( mem(V2l,A_27a)
=> ! [V3s_27] :
( mem(V3s_27,ty_2Elist_2Elist(A_27a))
=> ( p(ap(ap(ap(ap(c_2Epath_2Etrace__machine(A_27a),V0P),V1s),V2l),V3s_27))
<=> ( p(ap(V0P,ap(ap(c_2Elist_2EAPPEND(A_27a),V1s),ap(ap(c_2Elist_2ECONS(A_27a),V2l),c_2Elist_2ENIL(A_27a)))))
& V3s_27 = ap(ap(c_2Elist_2EAPPEND(A_27a),V1s),ap(ap(c_2Elist_2ECONS(A_27a),V2l),c_2Elist_2ENIL(A_27a))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Etrace__machine__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V1tr] :
( mem(V1tr,ty_2Ellist_2Ellist(A_27a))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3l] :
( mem(V3l,ty_2Elist_2Elist(A_27a))
=> ( ap(ap(c_2Ellist_2ELTAKE(A_27a),V2n),V1tr) = ap(c_2Eoption_2ESOME(ty_2Elist_2Elist(A_27a)),V3l)
=> p(ap(V0P,V3l)) ) ) )
=> ? [V4p] :
( mem(V4p,ty_2Epath_2Epath(ty_2Elist_2Elist(A_27a),A_27a))
& V1tr = ap(c_2Epath_2Elabels(ty_2Elist_2Elist(A_27a),A_27a),V4p)
& p(ap(ap(c_2Epath_2Eokpath(ty_2Elist_2Elist(A_27a),A_27a),ap(c_2Epath_2Etrace__machine(A_27a),V0P)),V4p))
& ap(c_2Epath_2Efirst(ty_2Elist_2Elist(A_27a),A_27a),V4p) = c_2Elist_2ENIL(A_27a) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Etrace__machine__thm2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1l] :
( mem(V1l,ty_2Elist_2Elist(A_27a))
=> ! [V2P] :
( mem(V2P,arr(ty_2Elist_2Elist(A_27a),bool))
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(ty_2Elist_2Elist(A_27a),A_27a))
=> ! [V4init] :
( mem(V4init,A_27b)
=> ( ( p(ap(ap(c_2Epath_2Eokpath(ty_2Elist_2Elist(A_27a),A_27a),ap(c_2Epath_2Etrace__machine(A_27a),V2P)),V3p))
& p(ap(V2P,ap(c_2Epath_2Efirst(ty_2Elist_2Elist(A_27a),A_27a),V3p))) )
=> ( ap(ap(c_2Ellist_2ELTAKE(A_27a),V0n),ap(c_2Epath_2Elabels(ty_2Elist_2Elist(A_27a),A_27a),V3p)) = ap(c_2Eoption_2ESOME(ty_2Elist_2Elist(A_27a)),V1l)
=> p(ap(V2P,ap(ap(c_2Elist_2EAPPEND(A_27a),ap(c_2Epath_2Efirst(ty_2Elist_2Elist(A_27a),A_27a),V3p)),V1l))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2ELTAKE__labels,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2l] :
( mem(V2l,ty_2Elist_2Elist(A_27b))
=> ( ap(ap(c_2Ellist_2ELTAKE(A_27b),V0n),ap(c_2Epath_2Elabels(A_27a,A_27b),V1p)) = ap(c_2Eoption_2ESOME(ty_2Elist_2Elist(A_27b)),V2l)
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0n),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& ap(c_2Ellist_2EtoList(A_27b),ap(c_2Epath_2Elabels(A_27a,A_27b),ap(ap(c_2Epath_2Etake(A_27a,A_27b),V0n),V1p))) = ap(c_2Eoption_2ESOME(ty_2Elist_2Elist(A_27b)),V2l) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Edrop__eq__pcons,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2h] :
( mem(V2h,A_27a)
=> ! [V3l] :
( mem(V3l,A_27b)
=> ! [V4t] :
( mem(V4t,ty_2Epath_2Epath(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0n),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& ap(ap(c_2Epath_2Edrop(A_27a,A_27b),V0n),V1p) = ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),V2h),V3l),V4t) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2EPL(A_27a,A_27b),V1p))) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epath_2Eparallel__comp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> ! [V0m1] :
( mem(V0m1,arr(A_27a,arr(A_27b,arr(A_27c,bool))))
=> ! [V1m2] :
( mem(V1m2,arr(A_27d,arr(A_27b,arr(A_27e,bool))))
=> ! [V2s1] :
( mem(V2s1,A_27a)
=> ! [V3s2] :
( mem(V3s2,A_27d)
=> ! [V4l] :
( mem(V4l,A_27b)
=> ! [V5s1_27] :
( mem(V5s1_27,A_27c)
=> ! [V6s2_27] :
( mem(V6s2_27,A_27e)
=> ( p(ap(ap(ap(ap(ap(c_2Epath_2Eparallel__comp(A_27a,A_27b,A_27c,A_27d,A_27e),V0m1),V1m2),ap(ap(c_2Epair_2E_2C(A_27a,A_27d),V2s1),V3s2)),V4l),ap(ap(c_2Epair_2E_2C(A_27c,A_27e),V5s1_27),V6s2_27)))
<=> ( p(ap(ap(ap(V0m1,V2s1),V4l),V5s1_27))
& p(ap(ap(ap(V1m2,V3s2),V4l),V6s2_27)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Eokpath__parallel__comp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V1m1] :
( mem(V1m1,arr(A_27a,arr(A_27c,arr(A_27a,bool))))
=> ! [V2m2] :
( mem(V2m2,arr(A_27b,arr(A_27c,arr(A_27b,bool))))
=> ( p(ap(ap(c_2Epath_2Eokpath(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(ap(c_2Epath_2Eparallel__comp(A_27a,A_27c,A_27a,A_27b,A_27b),V1m1),V2m2)),V0p))
<=> ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27c),V1m1),ap(ap(ap(c_2Epath_2Epmap(ty_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27a,A_27c),c_2Epair_2EFST(A_27a,A_27b)),i(A_27c)),V0p)))
& p(ap(ap(c_2Epath_2Eokpath(A_27b,A_27c),V2m2),ap(ap(ap(c_2Epath_2Epmap(ty_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27b,A_27c),c_2Epair_2ESND(A_27a,A_27b)),i(A_27c)),V0p))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Ebuild__pcomp__trace,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0m1] :
( mem(V0m1,arr(A_27a,arr(A_27b,arr(A_27a,bool))))
=> ! [V1p1] :
( mem(V1p1,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2m2] :
( mem(V2m2,arr(A_27c,arr(A_27b,arr(A_27c,bool))))
=> ! [V3p2] :
( mem(V3p2,ty_2Epath_2Epath(A_27c,A_27b))
=> ( ( p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27b),V0m1),V1p1))
& p(ap(ap(c_2Epath_2Eokpath(A_27c,A_27b),V2m2),V3p2))
& ap(c_2Epath_2Elabels(A_27a,A_27b),V1p1) = ap(c_2Epath_2Elabels(A_27c,A_27b),V3p2) )
=> ? [V4p] :
( mem(V4p,ty_2Epath_2Epath(ty_2Epair_2Eprod(A_27a,A_27c),A_27b))
& p(ap(ap(c_2Epath_2Eokpath(ty_2Epair_2Eprod(A_27a,A_27c),A_27b),ap(ap(c_2Epath_2Eparallel__comp(A_27a,A_27b,A_27a,A_27c,A_27c),V0m1),V2m2)),V4p))
& ap(c_2Epath_2Elabels(ty_2Epair_2Eprod(A_27a,A_27c),A_27b),V4p) = ap(c_2Epath_2Elabels(A_27a,A_27b),V1p1)
& ap(c_2Epath_2Efirst(ty_2Epair_2Eprod(A_27a,A_27c),A_27b),V4p) = ap(ap(c_2Epair_2E_2C(A_27a,A_27c),ap(c_2Epath_2Efirst(A_27a,A_27b),V1p1)),ap(c_2Epath_2Efirst(A_27c,A_27b),V3p2)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__LNTH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2x] :
( mem(V2x,A_27b)
=> ( ap(ap(c_2Ellist_2ELNTH(A_27b),V0n),ap(c_2Epath_2Elabels(A_27a,A_27b),V1p)) = ap(c_2Eoption_2ESOME(A_27b),V2x)
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epath_2EPL(A_27a,A_27b),V1p)))
& ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V0n),V1p) = V2x ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Enth__label__LTAKE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Epath_2Epath(A_27a,A_27b))
=> ! [V2l] :
( mem(V2l,ty_2Elist_2Elist(A_27b))
=> ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ! [V4v] :
( mem(V4v,A_27c)
=> ( ( ap(ap(c_2Ellist_2ELTAKE(A_27b),V0n),ap(c_2Epath_2Elabels(A_27a,A_27b),V1p)) = ap(c_2Eoption_2ESOME(ty_2Elist_2Elist(A_27b)),V2l)
& p(ap(ap(c_2Eprim__rec_2E_3C,V3i),ap(c_2Elist_2ELENGTH(A_27b),V2l))) )
=> ap(ap(c_2Epath_2Enth__label(A_27b,A_27a),V3i),V1p) = ap(ap(c_2Elist_2EEL(A_27b),V3i),V2l) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Efinite__path__end__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epath_2Epath(A_27a,A_27b))
=> ( p(ap(c_2Epath_2Efinite(A_27a,A_27b),V0p))
=> ( ? [V1x] :
( mem(V1x,A_27a)
& V0p = ap(c_2Epath_2Estopped__at(A_27a,A_27b),V1x) )
| ? [V2p_27] :
( mem(V2p_27,ty_2Epath_2Epath(A_27a,A_27b))
& ? [V3l] :
( mem(V3l,A_27b)
& ? [V4s] :
( mem(V4s,A_27a)
& V0p = ap(ap(c_2Epath_2Eplink(A_27a,A_27b),V2p_27),ap(ap(ap(c_2Epath_2Epcons(A_27a,A_27b),ap(c_2Epath_2Elast(A_27a,A_27b),V2p_27)),V3l),ap(c_2Epath_2Estopped__at(A_27a,A_27b),V4s))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epath_2Esimulation__trace__inclusion,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1M1] :
( mem(V1M1,arr(A_27a,arr(A_27c,arr(A_27a,bool))))
=> ! [V2M2] :
( mem(V2M2,arr(A_27b,arr(A_27c,arr(A_27b,bool))))
=> ! [V3p] :
( mem(V3p,ty_2Epath_2Epath(A_27a,A_27c))
=> ! [V4t__init] :
( mem(V4t__init,A_27b)
=> ( ( ! [V5s1] :
( mem(V5s1,A_27a)
=> ! [V6l] :
( mem(V6l,A_27c)
=> ! [V7s2] :
( mem(V7s2,A_27a)
=> ! [V8t1] :
( mem(V8t1,A_27b)
=> ( ( p(ap(ap(V0R,V5s1),V8t1))
& p(ap(ap(ap(V1M1,V5s1),V6l),V7s2)) )
=> ? [V9t2] :
( mem(V9t2,A_27b)
& p(ap(ap(V0R,V7s2),V9t2))
& p(ap(ap(ap(V2M2,V8t1),V6l),V9t2)) ) ) ) ) ) )
& p(ap(ap(c_2Epath_2Eokpath(A_27a,A_27c),V1M1),V3p))
& p(ap(ap(V0R,ap(c_2Epath_2Efirst(A_27a,A_27c),V3p)),V4t__init)) )
=> ? [V10q] :
( mem(V10q,ty_2Epath_2Epath(A_27b,A_27c))
& p(ap(ap(c_2Epath_2Eokpath(A_27b,A_27c),V2M2),V10q))
& ap(c_2Epath_2Elabels(A_27a,A_27c),V3p) = ap(c_2Epath_2Elabels(A_27b,A_27c),V10q)
& ap(c_2Epath_2Efirst(A_27b,A_27c),V10q) = V4t__init ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------