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 : 217 ( 0 unt; 34 typ; 0 def)
% Number of atoms : 3613 ( 174 equ; 0 cnn)
% Maximal formula atoms : 53 ( 16 avg)
% Number of connectives : 7980 ( 15 ~; 10 |; 153 &;7259 @)
% ( 53 <=>; 490 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 15 avg;7259 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 73 ( 73 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 102 usr; 70 con; 0-5 aty)
% Number of variables : 962 ( 27 ^ 895 !; 40 ?; 962 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Epath_2Epath,type,
ty_2Epath_2Epath: del > del > del ).
thf(tp_c_2Epath_2EPL,type,
c_2Epath_2EPL: del > del > $i ).
thf(mem_c_2Epath_2EPL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Epath_2ESN,type,
c_2Epath_2ESN: del > del > $i ).
thf(mem_c_2Epath_2ESN,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2ESN @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) @ bool ) ) ).
thf(tp_c_2Epath_2Edrop,type,
c_2Epath_2Edrop: del > del > $i ).
thf(mem_c_2Epath_2Edrop,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Eel,type,
c_2Epath_2Eel: del > del > $i ).
thf(mem_c_2Epath_2Eel,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ A_27a ) ) ) ).
thf(tp_c_2Epath_2Eevery,type,
c_2Epath_2Eevery: del > del > $i ).
thf(mem_c_2Epath_2Eevery,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Eevery @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Epath_2Eexists,type,
c_2Epath_2Eexists: del > del > $i ).
thf(mem_c_2Epath_2Eexists,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Eexists @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Epath_2Efilter,type,
c_2Epath_2Efilter: del > del > $i ).
thf(mem_c_2Epath_2Efilter,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Efinite,type,
c_2Epath_2Efinite: del > del > $i ).
thf(mem_c_2Epath_2Efinite,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) ) ).
thf(tp_c_2Epath_2Efirst,type,
c_2Epath_2Efirst: del > del > $i ).
thf(mem_c_2Epath_2Efirst,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ A_27a ) ) ).
thf(tp_c_2Epath_2EfirstP__at,type,
c_2Epath_2EfirstP__at: del > del > $i ).
thf(mem_c_2Epath_2EfirstP__at,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Efirst__label,type,
c_2Epath_2Efirst__label: del > del > $i ).
thf(mem_c_2Epath_2Efirst__label,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Efirst__label @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ A_27b ) ) ).
thf(tp_c_2Epath_2EfromPath,type,
c_2Epath_2EfromPath: del > del > $i ).
thf(mem_c_2Epath_2EfromPath,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Eis__stopped,type,
c_2Epath_2Eis__stopped: del > del > $i ).
thf(mem_c_2Epath_2Eis__stopped,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Eis__stopped @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) ) ).
thf(tp_c_2Epath_2Elabels,type,
c_2Epath_2Elabels: del > del > $i ).
thf(mem_c_2Epath_2Elabels,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Elabels @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ ( ty_2Ellist_2Ellist @ A_27b ) ) ) ).
thf(tp_c_2Epath_2Elast,type,
c_2Epath_2Elast: del > del > $i ).
thf(mem_c_2Epath_2Elast,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Elast @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ A_27a ) ) ).
thf(tp_c_2Epath_2Elength,type,
c_2Epath_2Elength: del > del > $i ).
thf(mem_c_2Epath_2Elength,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Epath_2Emem,type,
c_2Epath_2Emem: del > del > $i ).
thf(mem_c_2Epath_2Emem,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Emem @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Epath_2Enth__label,type,
c_2Epath_2Enth__label: del > del > $i ).
thf(mem_c_2Epath_2Enth__label,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epath_2Epath @ A_27b @ A_27a ) @ A_27a ) ) ) ).
thf(tp_c_2Epath_2Eokpath,type,
c_2Epath_2Eokpath: del > del > $i ).
thf(mem_c_2Epath_2Eokpath,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ).
thf(tp_c_2Epath_2Eokpath__f,type,
c_2Epath_2Eokpath__f: del > del > $i ).
thf(mem_c_2Epath_2Eokpath__f,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Eparallel__comp,type,
c_2Epath_2Eparallel__comp: del > del > del > del > del > $i ).
thf(mem_c_2Epath_2Eparallel__comp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del] : ( 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 ) ) ) ) ) ) ).
thf(tp_c_2Epath_2Epconcat,type,
c_2Epath_2Epconcat: del > del > $i ).
thf(mem_c_2Epath_2Epconcat,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Epcons,type,
c_2Epath_2Epcons: del > del > $i ).
thf(mem_c_2Epath_2Epcons,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Epgenerate,type,
c_2Epath_2Epgenerate: del > del > $i ).
thf(mem_c_2Epath_2Epgenerate,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Eplink,type,
c_2Epath_2Eplink: del > del > $i ).
thf(mem_c_2Epath_2Eplink,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2Epmap,type,
c_2Epath_2Epmap: del > del > del > del > $i ).
thf(mem_c_2Epath_2Epmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Eseg,type,
c_2Epath_2Eseg: del > del > $i ).
thf(mem_c_2Epath_2Eseg,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Estopped__at,type,
c_2Epath_2Estopped__at: del > del > $i ).
thf(mem_c_2Epath_2Estopped__at,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Epath_2Etail,type,
c_2Epath_2Etail: del > del > $i ).
thf(mem_c_2Epath_2Etail,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Epath_2Etake,type,
c_2Epath_2Etake: del > del > $i ).
thf(mem_c_2Epath_2Etake,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Epath_2EtoPath,type,
c_2Epath_2EtoPath: del > del > $i ).
thf(mem_c_2Epath_2EtoPath,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ).
thf(tp_c_2Epath_2Etrace__machine,type,
c_2Epath_2Etrace__machine: del > $i ).
thf(mem_c_2Epath_2Etrace__machine,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Epath_2Eunfold,type,
c_2Epath_2Eunfold: del > del > del > $i ).
thf(mem_c_2Epath_2Eunfold,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( 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 ) ) ) ) ) ).
thf(ax_thm_2Epath_2Epath__TY__DEF,axiom,
! [A_27a: del,A_27b: del] :
? [V0rep: $i] :
( ( 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 ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Ellist_2Ellist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
@ ^ [V1x: $i] : c_2Ebool_2ET ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Epath_2Epath__absrep__bijections,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( 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: $i] :
( ( mem @ V1r @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Ellist_2Ellist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ( ( p
@ ( ap
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Ellist_2Ellist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
@ ^ [V2x: $i] : c_2Ebool_2ET )
@ V1r ) )
<=> ( ( ap @ ( c_2Epath_2EfromPath @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2EtoPath @ A_27a @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Epath_2Epath__rep__bijections__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ).
thf(conj_thm_2Epath_2EtoPath__11,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Ellist_2Ellist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ! [V1r_27: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Epath_2EfromPath__11,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1a_27: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Epath_2EfromPath__onto,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Ellist_2Ellist @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ? [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
& ( V0r
= ( ap @ ( c_2Epath_2EfromPath @ A_27a @ A_27b ) @ V1a ) ) ) ) ).
thf(conj_thm_2Epath_2EtoPath__onto,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ? [V1r: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Efirst__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Estopped__at__def,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Epcons__def,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Estopped__at__11,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Epath_2Epcons__11,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ! [V5q: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Estopped__at__not__pcons,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epath__cases,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0p
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V1x ) ) )
| ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ? [V3r: $i] :
( ( mem @ V3r @ A_27b )
& ? [V4q: $i] :
( ( mem @ V4q @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
& ( V0p
= ( ap @ ( ap @ ( ap @ ( c_2Epath_2Epcons @ A_27a @ A_27b ) @ V2x ) @ V3r ) @ V4q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EFORALL__path,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V1p ) ) )
<=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27b )
=> ! [V5p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EEXISTS__path,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ( ? [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
& ( p @ ( ap @ V0P @ V1p ) ) )
<=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x ) ) ) )
| ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ? [V4r: $i] :
( ( mem @ V4r @ A_27b )
& ? [V5p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V0x ) )
= V0x ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ).
thf(ax_thm_2Epath_2Efinite__def,axiom,
! [A_27a: del,A_27b: del,V0sigma: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( 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: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Elast__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Epath_2Elast @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V0x ) )
= V0x ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epath__bisimulation,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1p2: $i] :
( ( mem @ V1p2 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( V0p1 = V1p2 )
<=> ? [V2R: $i] :
( ( 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: $i] :
( ( mem @ V3q1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V4q2: $i] :
( ( mem @ V4q2 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ V2R @ V3q1 ) @ V4q2 ) )
=> ( ? [V5x: $i] :
( ( 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: $i] :
( ( mem @ V6x @ A_27a )
& ? [V7r: $i] :
( ( mem @ V7r @ A_27b )
& ? [V8q1_27: $i] :
( ( mem @ V8q1_27 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
& ? [V9q2_27: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__path__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V1x ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27b )
=> ! [V4p: $i] :
( ( 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: $i] :
( ( mem @ V5q @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V5q ) )
=> ( p @ ( ap @ V0P @ V5q ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Epmap__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epmap__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27d @ A_27c ) )
=> ( ! [V2x: $i] :
( ( 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: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27d )
=> ! [V5p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elast__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Etail__def,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Efirst__label__def,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Elength__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ V0p ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( 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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elength__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V0x ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27c )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27d )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27c @ A_27d ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2Epcons @ A_27c @ A_27d ) @ V1x ) @ V2r ) @ V3p ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( 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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Ealt__length__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V0x ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27c )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27d )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27c @ A_27d ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2Epcons @ A_27c @ A_27d ) @ V1x ) @ V2r ) @ V3p ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ c_2Enum_2ESUC ) @ ( ap @ ( c_2Epath_2Elength @ A_27c @ A_27d ) @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elength__never__zero,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ V0p ) )
!= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__length,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) )
<=> ? [V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ V0p ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ( ~ ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) )
<=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ V0p ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( c_2Eoption_2ENONE @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elength__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ A_27d ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27c ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( 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 ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27c ) @ V2p ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eel__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V0p ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2p )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eel__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V0p ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4p )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V4p ) ) ) ) ) ).
thf(ax_thm_2Epath_2Enth__label__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Efirst__label @ A_27b @ A_27a ) @ V0p ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2p )
= ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27b @ A_27a ) @ V2p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Efirst__label @ A_27b @ A_27a ) @ V0p ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27b @ A_27a ) @ V2p ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4p )
= ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27a @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27b @ A_27a ) @ V4p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epath__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) )
=> ? [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( ty_2Epath_2Epath @ A_27b @ A_27c ) ) )
& ! [V2x: $i] :
( ( 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 ) )
@ ( lam @ A_27b
@ ^ [V3y: $i] :
( lam @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) )
@ ^ [V4v2: $i] :
( 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 ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27c @ A_27a )
@ ^ [V5v3: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Epath_2Epath @ A_27b @ A_27c ) @ A_27c @ A_27a ) @ V5v3 )
@ ( lam @ A_27c
@ ^ [V6l: $i] :
( lam @ A_27a
@ ^ [V7v: $i] : ( ap @ ( ap @ ( ap @ ( c_2Epath_2Epcons @ A_27b @ A_27c ) @ V3y ) @ V6l ) @ ( ap @ V1g @ V7v ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Epconcat__def,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1lab: $i] :
( ( mem @ V1lab @ A_27b )
=> ! [V2p2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epconcat__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1lab: $i] :
( ( mem @ V1lab @ A_27b )
=> ! [V2p2: $i] :
( ( 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: $i] :
( ( mem @ V3x @ A_27c )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27d )
=> ! [V5p: $i] :
( ( mem @ V5p @ ( ty_2Epath_2Epath @ A_27c @ A_27d ) )
=> ! [V6lab: $i] :
( ( mem @ V6lab @ A_27d )
=> ! [V7p2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epconcat__eq__stopped,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1lab: $i] :
( ( mem @ V1lab @ A_27b )
=> ! [V2p2: $i] :
( ( mem @ V2p2 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epconcat__eq__pcons,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V3p1: $i] :
( ( mem @ V3p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V4lab: $i] :
( ( mem @ V4lab @ A_27b )
=> ! [V5p2: $i] :
( ( 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: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__pconcat,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1lab: $i] :
( ( mem @ V1lab @ A_27b )
=> ! [V2p2: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2EPL__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( 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 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] : ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Einfinite__PL,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) )
=> ! [V1i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__pcons,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ A_27b )
=> ! [V2q: $i] :
( ( 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 ) @ ( inj__ty_2Enum_2Enum @ fo__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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__stopped__at,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( 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 ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( 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 ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3q: $i] :
( ( 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 ) @ ( inj__ty_2Enum_2Enum @ fo__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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__0,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V0p ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__downward__closed,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) )
=> ! [V2j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27d @ A_27b ) )
=> ! [V2p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eel__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V3p ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27c @ A_27d ) @ ( inj__ty_2Enum_2Enum @ 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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V3p ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__pmap,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27d ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V3p ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27c @ A_27d ) @ ( inj__ty_2Enum_2Enum @ 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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V3p ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2EfirstP__at__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2EfirstP__at @ A_27a @ A_27b ) @ V0P ) @ V1p ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) )
& ! [V3j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) )
=> ~ ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V1p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EfirstP__at__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2n: tp__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 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
<=> ( ( V2n = fo__c_2Enum_2E0 )
& ( p @ ( ap @ V0P @ V1x ) ) ) ) ) )
& ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ! [V4n: tp__ty_2Enum_2Enum,V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6r: $i] :
( ( mem @ V6r @ A_27b )
=> ! [V7p: $i] :
( ( 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 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
<=> ( ( ( V4n = fo__c_2Enum_2E0 )
& ( p @ ( ap @ V3P @ V5x ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EfirstP__at__zero,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2EfirstP__at @ A_27a @ A_27b ) @ V0P ) @ V1p ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V1p ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eexists__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Eexists @ A_27a @ A_27b ) @ V0P ) @ V1p ) )
<=> ? [V2i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2EfirstP__at @ A_27a @ A_27b ) @ V0P ) @ V1p ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eevery__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eexists__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ! [V1x: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27c )
=> ! [V4p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eevery__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ! [V1x: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27c )
=> ! [V4p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enot__every,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enot__exists,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eexists__el,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Eexists @ A_27a @ A_27b ) @ V0P ) @ V1p ) )
<=> ? [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eevery__el,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Eevery @ A_27a @ A_27b ) @ V0P ) @ V1p ) )
<=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eevery__coinduction,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x ) ) )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27b )
=> ! [V5p: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eexists__induction,axiom,
! [A_27a: del,A_27b: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0Q @ V2x ) )
=> ( p @ ( ap @ V1P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27b )
=> ! [V5p: $i] :
( ( 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: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7r: $i] :
( ( mem @ V7r @ A_27b )
=> ! [V8p: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Emem__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ A_27a )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Emem @ A_27a @ A_27b ) @ V0s ) @ V1p ) )
<=> ? [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Emem__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27c )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27d )
=> ! [V4p: $i] :
( ( mem @ V4p @ ( ty_2Epath_2Epath @ A_27c @ A_27d ) )
=> ! [V5s: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Edrop__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= V0p ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2p )
= ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Edrop__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= V0p ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4p )
= ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V4p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enumeral__drop,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V1p ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ V3p )
= ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V3p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__drop,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0p ) ) )
<=> ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elength__drop,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V0p ) ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0p ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum
@ ( 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 ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2m: $i] : ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2m ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__drop,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0p ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) )
@ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epath_2EIN__PL__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V2p ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V2p ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__label__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) )
= ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) ) ) ) ).
thf(conj_thm_2Epath_2Etail__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) )
= ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1p ) ) ) ) ).
thf(conj_thm_2Epath_2Eel__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V2p ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) ) @ V2p ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__drop,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V2p ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) )
= ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) ) @ V2p ) ) ) ) ).
thf(ax_thm_2Epath_2Etake__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V0p ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ 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 ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Etake__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0p )
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V0p ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V2p ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4p: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( c_2Epath_2Etail @ A_27a @ A_27b ) @ V4p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__take,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0p ) )
= ( ap @ ( c_2Epath_2Efirst @ A_27a @ A_27b ) @ V0p ) ) ) ).
thf(conj_thm_2Epath_2Efinite__take,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elength__take,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V0p ) ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Epath_2Elength @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0p ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Enum_2Enum @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__take,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0p ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V2n ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V2n ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elast__take,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__take,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1p ) )
= ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ).
thf(ax_thm_2Epath_2Eseg__def,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epath_2Eseg @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p )
= ( ap @ ( ap @ ( c_2Epath_2Etake @ A_27a @ A_27b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) @ ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V2p ) ) ) ) ).
thf(conj_thm_2Epath_2Esingleton__seg,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epath_2Eseg @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p )
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Erecursive__seg,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V2p ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epath_2Eseg @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p )
= ( ap @ ( ap @ ( ap @ ( c_2Epath_2Epcons @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V2p ) ) @ ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V2p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2Eseg @ A_27a @ A_27b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) ) ) ) ) ).
thf(conj_thm_2Epath_2EPL__seg,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( 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 @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__seg,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__seg,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V2p ) ) ) ) ).
thf(conj_thm_2Epath_2Elast__seg,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum,V2p: $i] :
( ( mem @ V2p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) )
= ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2p ) ) ) ) ).
thf(ax_thm_2Epath_2Elabels__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( 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: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27b )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2EfirstP__at__unique,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2EfirstP__at @ A_27a @ A_27b ) @ V0P ) @ V1p ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ! [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epath_2EfirstP__at @ A_27a @ A_27b ) @ V0P ) @ V1p ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
<=> ( V3m = V2n ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eis__stopped__def,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Epath_2Eis__stopped @ A_27a @ A_27b ) @ V0p ) )
<=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0p
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eis__stopped__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0x: $i] :
( ( 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: $i] :
( ( mem @ V1x @ A_27c )
=> ! [V2r: $i] :
( ( mem @ V2r @ A_27d )
=> ! [V3p: $i] :
( ( 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 ) ) ) ) ) ).
thf(ax_thm_2Epath_2Efilter__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ! [V1x: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27b )
=> ! [V4p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efilter__every,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Epgenerate__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1g: $i] :
( ( 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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ fo__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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epgenerate__infinite,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1g: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epgenerate__not__stopped,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ! [V2x: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eel__pgenerate,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Eel @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Epath_2Epgenerate @ A_27a @ A_27b ) @ V1f ) @ V2g ) )
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__pgenerate,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Epath_2Epgenerate @ A_27a @ A_27b ) @ V1f ) @ V2g ) )
= ( ap @ V2g @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epgenerate__11,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1g1: $i] :
( ( mem @ V1g1 @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V3g2: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Epgenerate__onto,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ? [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
& ( V0p
= ( ap @ ( ap @ ( c_2Epath_2Epgenerate @ A_27a @ A_27b ) @ V1f ) @ V2g ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eokpath__f__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1X: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( 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 ) ) ) ) ) )
@ ( 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 ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27b
@ ^ [V4r: $i] :
( lam @ ( ty_2Epath_2Epath @ A_27a @ A_27b )
@ ^ [V5p: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__monotone,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Eokpath__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__co__ind,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27b )
=> ! [V4p: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__cases,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Eokpath @ A_27a @ A_27b ) @ V0R ) @ V1x ) )
<=> ( ? [V2x_27: $i] :
( ( mem @ V2x_27 @ A_27a )
& ( V1x
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x_27 ) ) )
| ? [V3x_27: $i] :
( ( mem @ V3x_27 @ A_27a )
& ? [V4r: $i] :
( ( mem @ V4r @ A_27b )
& ? [V5p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__thm,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ( ! [V1x: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27b )
=> ! [V4p: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__okpath__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4r: $i] :
( ( mem @ V4r @ A_27b )
=> ! [V5p: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__pmap,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27b ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epath_2Eokpath @ A_27a @ A_27b ) @ V0R ) @ V3p ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5r: $i] :
( ( mem @ V5r @ A_27b )
=> ! [V6y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eplink__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1p: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27b )
=> ! [V4p1: $i] :
( ( mem @ V4p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V5p2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__plink,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1p2: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efirst__plink,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1p2: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elast__plink,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1p2: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__plink,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p1: $i] :
( ( mem @ V1p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2p2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__take,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__drop,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__seg,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V1p ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2ESN__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V3l: $i] : ( ap @ ( ap @ ( ap @ V0R @ V2y ) @ V3l ) @ V1x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2ESN__finite__paths,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__paths__SN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ( ! [V1p: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Epath_2ESN__finite__paths__EQ,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epath_2ESN @ A_27a @ A_27b ) @ V0R ) )
<=> ! [V1p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elabels__LMAP,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elabels__plink,axiom,
! [A_27a: del,A_27b: del,V0p1: $i] :
( ( mem @ V0p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V1p2: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__labels,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Epath_2Eunfold__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0proj: $i] :
( ( mem @ V0proj @ ( arr @ A_27c @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) )
=> ! [V2s: $i] :
( ( 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 )
@ ( lam @ A_27c
@ ^ [V3s: $i] :
( 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 ) ) )
@ ( lam @ A_27c
@ ^ [V4next__s: $i] :
( lam @ A_27b
@ ^ [V5lbl: $i] : ( 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 ) ) ) ) ) ) )
@ ( ap @ V1f @ V3s ) ) ) )
@ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eunfold__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0proj: $i] :
( ( mem @ V0proj @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) )
=> ! [V2s: $i] :
( ( 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 ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27c )
@ ^ [V3v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Epath_2Epath @ A_27b @ A_27c ) @ A_27a @ A_27c ) @ V3v )
@ ( lam @ A_27a
@ ^ [V4s_27: $i] :
( lam @ A_27c
@ ^ [V5l: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eunfold__thm2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0proj: $i] :
( ( mem @ V0proj @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27a )
=> ! [V4v2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Elabels__unfold,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0proj: $i] :
( ( mem @ V0proj @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) )
=> ! [V2s: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__unfold,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27b @ ( arr @ A_27c @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V2proj: $i] :
( ( mem @ V2proj @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4s ) )
& ! [V5s: $i] :
( ( mem @ V5s @ A_27a )
=> ! [V6s_27: $i] :
( ( mem @ V6s_27 @ A_27a )
=> ! [V7l: $i] :
( ( 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: $i] :
( ( mem @ V8s @ A_27a )
=> ! [V9s_27: $i] :
( ( mem @ V9s_27 @ A_27a )
=> ! [V10l: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Etrace__machine__def,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ A_27a )
=> ! [V3s_27: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Etrace__machine__thm,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ! [V1tr: $i] :
( ( mem @ V1tr @ ( ty_2Ellist_2Ellist @ A_27a ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum,V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Ellist_2ELTAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1tr )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Elist_2Elist @ A_27a ) ) @ V3l ) )
=> ( p @ ( ap @ V0P @ V3l ) ) ) )
=> ? [V4p: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Etrace__machine__thm2,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) )
=> ! [V4init: $i] :
( ( 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 ) @ ( inj__ty_2Enum_2Enum @ 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2ELTAKE__labels,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Ellist_2ELTAKE @ A_27b ) @ ( inj__ty_2Enum_2Enum @ 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 ) @ ( inj__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 ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1p ) ) )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Elist_2Elist @ A_27b ) ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Edrop__eq__pcons,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3l: $i] :
( ( mem @ V3l @ A_27b )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) )
& ( ( ap @ ( ap @ ( c_2Epath_2Edrop @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epath_2Eparallel__comp__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,V0m1: $i] :
( ( mem @ V0m1 @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ bool ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( arr @ A_27d @ ( arr @ A_27b @ ( arr @ A_27e @ bool ) ) ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ A_27a )
=> ! [V3s2: $i] :
( ( mem @ V3s2 @ A_27d )
=> ! [V4l: $i] :
( ( mem @ V4l @ A_27b )
=> ! [V5s1_27: $i] :
( ( mem @ V5s1_27 @ A_27c )
=> ! [V6s2_27: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Eokpath__parallel__comp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( arr @ A_27a @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V2m2: $i] :
( ( 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 ) )
@ ( lam @ A_27c
@ ^ [V3x: $i] : V3x ) )
@ 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 ) )
@ ( lam @ A_27c
@ ^ [V4x: $i] : V4x ) )
@ V0p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Ebuild__pcomp__trace,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m1: $i] :
( ( mem @ V0m1 @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1p1: $i] :
( ( mem @ V1p1 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( arr @ A_27c @ ( arr @ A_27b @ ( arr @ A_27c @ bool ) ) ) )
=> ! [V3p2: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__LNTH,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Ellist_2ELNTH @ A_27b ) @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epath_2EPL @ A_27a @ A_27b ) @ V1p ) ) )
& ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1p )
= V2x ) ) ) ) ) ).
thf(conj_thm_2Epath_2Enth__label__LTAKE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( ( mem @ V1p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3i: tp__ty_2Enum_2Enum,V4v: $i] :
( ( mem @ V4v @ A_27c )
=> ( ( ( ( ap @ ( ap @ ( c_2Ellist_2ELTAKE @ A_27b ) @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epath_2Enth__label @ A_27b @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1p )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Efinite__path__end__cases,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Epath_2Efinite @ A_27a @ A_27b ) @ V0p ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0p
= ( ap @ ( c_2Epath_2Estopped__at @ A_27a @ A_27b ) @ V1x ) ) )
| ? [V2p_27: $i] :
( ( mem @ V2p_27 @ ( ty_2Epath_2Epath @ A_27a @ A_27b ) )
& ? [V3l: $i] :
( ( mem @ V3l @ A_27b )
& ? [V4s: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epath_2Esimulation__trace__inclusion,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1M1: $i] :
( ( mem @ V1M1 @ ( arr @ A_27a @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V2M2: $i] :
( ( mem @ V2M2 @ ( arr @ A_27b @ ( arr @ A_27c @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epath_2Epath @ A_27a @ A_27c ) )
=> ! [V4t__init: $i] :
( ( mem @ V4t__init @ A_27b )
=> ( ( ! [V5s1: $i] :
( ( mem @ V5s1 @ A_27a )
=> ! [V6l: $i] :
( ( mem @ V6l @ A_27c )
=> ! [V7s2: $i] :
( ( mem @ V7s2 @ A_27a )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V5s1 ) @ V8t1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ V1M1 @ V5s1 ) @ V6l ) @ V7s2 ) ) )
=> ? [V9t2: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------