ITP001 Axioms: ITP083^5.ax
%------------------------------------------------------------------------------
% File : ITP083^5 : 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 : fmaptree^2.ax [Gau20]
% : HL4083^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 53 ( 2 unt; 13 typ; 0 def)
% Number of atoms : 1024 ( 30 equ; 0 cnn)
% Maximal formula atoms : 59 ( 19 avg)
% Number of connectives : 2340 ( 0 ~; 0 |; 23 &;2198 @)
% ( 4 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 15 avg;2198 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 34 con; 0-3 aty)
% Number of variables : 213 ( 24 ^ 180 !; 9 ?; 213 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Efmaptree_2Efmaptree,type,
ty_2Efmaptree_2Efmaptree: del > del > del ).
thf(tp_c_2Efmaptree_2EFTNode,type,
c_2Efmaptree_2EFTNode: del > del > $i ).
thf(mem_c_2Efmaptree_2EFTNode,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efmaptree_2Eapply__path,type,
c_2Efmaptree_2Eapply__path: del > del > $i ).
thf(mem_c_2Efmaptree_2Eapply__path,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ).
thf(tp_c_2Efmaptree_2Econstruct,type,
c_2Efmaptree_2Econstruct: del > del > $i ).
thf(mem_c_2Efmaptree_2Econstruct,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(tp_c_2Efmaptree_2Efmtreerec,type,
c_2Efmaptree_2Efmtreerec: del > del > del > $i ).
thf(mem_c_2Efmaptree_2Efmtreerec,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) @ A_27a ) ) ) ).
thf(tp_c_2Efmaptree_2EfromF,type,
c_2Efmaptree_2EfromF: del > del > $i ).
thf(mem_c_2Efmaptree_2EfromF,axiom,
! [A_27key: del,A_27value: del] : ( mem @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) ) ) ).
thf(tp_c_2Efmaptree_2Efupd__at__path,type,
c_2Efmaptree_2Efupd__at__path: del > del > $i ).
thf(mem_c_2Efmaptree_2Efupd__at__path,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2Efmaptree_2Eitem,type,
c_2Efmaptree_2Eitem: del > del > $i ).
thf(mem_c_2Efmaptree_2Eitem,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ A_27b ) ) ).
thf(tp_c_2Efmaptree_2Emap,type,
c_2Efmaptree_2Emap: del > del > $i ).
thf(mem_c_2Efmaptree_2Emap,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Efmaptree_2Erelrec,type,
c_2Efmaptree_2Erelrec: del > del > del > $i ).
thf(mem_c_2Efmaptree_2Erelrec,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) ) ) ).
thf(tp_c_2Efmaptree_2EtoF,type,
c_2Efmaptree_2EtoF: del > del > $i ).
thf(mem_c_2Efmaptree_2EtoF,axiom,
! [A_27key: del,A_27value: del] : ( mem @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) ) ) ).
thf(tp_c_2Efmaptree_2Eupdate__at__path,type,
c_2Efmaptree_2Eupdate__at__path: del > del > $i ).
thf(mem_c_2Efmaptree_2Eupdate__at__path,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2Efmaptree_2Ewf,type,
c_2Efmaptree_2Ewf: del > del > $i ).
thf(mem_c_2Efmaptree_2Ewf,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) ) ).
thf(ax_thm_2Efmaptree_2Econstruct__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1kfm: $i] :
( ( mem @ V1kfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
=> ! [V2kl: $i] :
( ( mem @ V2kl @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V0a ) @ V1kfm ) @ V2kl )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V2kl ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0a ) )
@ ( lam @ A_27b
@ ^ [V3h: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V4t: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1kfm ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1kfm ) @ V3h ) @ V4t ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Ewf__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efmaptree_2Ewf @ A_27a @ A_27b )
= ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ^ [V0a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool )
@ ^ [V1wf_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ^ [V2a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3a: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
@ ^ [V4fm: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2a0 ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V3a ) @ V4fm ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V5k: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V4fm ) ) ) @ ( ap @ V1wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V4fm ) @ V5k ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V1wf_27 @ V2a0 ) ) ) ) )
@ ( ap @ V1wf_27 @ V0a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Ewf__rules,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
=> ( ! [V2k: $i] :
( ( mem @ V2k @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1fm ) ) )
=> ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V1fm ) @ V2k ) ) ) ) )
=> ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V0a ) @ V1fm ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Ewf__ind,axiom,
! [A_27a: del,A_27b: del,V0wf_27: $i] :
( ( mem @ V0wf_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
=> ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) )
=> ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
=> ! [V4a0: $i] :
( ( mem @ V4a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V4a0 ) )
=> ( p @ ( ap @ V0wf_27 @ V4a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Ewf__strongind,axiom,
! [A_27a: del,A_27b: del,V0wf_27: $i] :
( ( mem @ V0wf_27 @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
=> ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) )
& ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) ) )
=> ( p @ ( ap @ V0wf_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
=> ! [V4a0: $i] :
( ( mem @ V4a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V4a0 ) )
=> ( p @ ( ap @ V0wf_27 @ V4a0 ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Ewf__cases,axiom,
! [A_27a: del,A_27b: del,V0a0: $i] :
( ( mem @ V0a0 @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ V0a0 ) )
<=> ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) )
& ( V0a0
= ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27a @ A_27b ) @ V1a ) @ V2fm ) )
& ! [V3k: $i] :
( ( mem @ V3k @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) ) )
=> ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ V2fm ) @ V3k ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Efmaptree__TY__DEF,axiom,
! [A_27key: del,A_27value: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) ) @ ( c_2Efmaptree_2Ewf @ A_27value @ A_27key ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Efmaptree_2Efmap__bij__thm,axiom,
! [A_27key: del,A_27value: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Efmaptree_2Efmaptree @ A_27key @ A_27value ) )
=> ( ( ap @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ ( ap @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Elist_2Elist @ A_27key ) @ ( ty_2Eoption_2Eoption @ A_27value ) ) )
=> ( ( p @ ( ap @ ( c_2Efmaptree_2Ewf @ A_27value @ A_27key ) @ V1r ) )
<=> ( ( ap @ ( c_2Efmaptree_2EtoF @ A_27key @ A_27value ) @ ( ap @ ( c_2Efmaptree_2EfromF @ A_27key @ A_27value ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Efmaptree_2EFTNode__def,axiom,
! [A_27a: del,A_27b: del,V0i: $i] :
( ( mem @ V0i @ A_27b )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0i ) @ V1fm )
= ( ap @ ( c_2Efmaptree_2EfromF @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Econstruct @ A_27b @ A_27a ) @ V0i ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) @ ( c_2Efmaptree_2EtoF @ A_27a @ A_27b ) ) @ V1fm ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2EFTNode__11,axiom,
! [A_27a: del,A_27b: del,V0i1: $i] :
( ( mem @ V0i1 @ A_27b )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
=> ! [V2i2: $i] :
( ( mem @ V2i2 @ A_27b )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0i1 ) @ V1f1 )
= ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V2i2 ) @ V3f2 ) )
<=> ( ( V0i1 = V2i2 )
& ( V1f1 = V3f2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Efmaptree__nchotomy,axiom,
! [A_27a: del,A_27b: del,V0ft: $i] :
( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ? [V1i: $i] :
( ( mem @ V1i @ A_27b )
& ? [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
& ( V0ft
= ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V1i ) @ V2fm ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Eitem__map__def,axiom,
! [A_27a: del,A_27b: del,V0ft: $i] :
( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( V0ft
= ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V0ft ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V0ft ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Eitem__thm,axiom,
! [A_27a: del,A_27b: del,V0i: $i] :
( ( mem @ V0i @ A_27a )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( c_2Efmaptree_2Eitem @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V0i ) @ V1fm ) )
= V0i ) ) ) ).
thf(conj_thm_2Efmaptree_2Emap__thm,axiom,
! [A_27a: del,A_27b: del,V0i: $i] :
( ( mem @ V0i @ A_27a )
=> ! [V1fm: $i] :
( ( mem @ V1fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( c_2Efmaptree_2Emap @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V0i ) @ V1fm ) )
= V1fm ) ) ) ).
thf(ax_thm_2Efmaptree_2Eapply__path__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0ft: $i] :
( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0ft )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V0ft ) ) )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3ft: $i] :
( ( mem @ V3ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V2t ) ) @ V3ft )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft ) ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V2t ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft ) ) @ V1h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Eupdate__at__path__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27b )
=> ! [V1ft: $i] :
( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0a ) @ V1ft )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V1ft ) ) ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27b )
=> ! [V5ft: $i] :
( ( mem @ V5ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4a ) @ V5ft )
= ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Eupdate__at__path @ A_27a @ A_27b ) @ V3t ) @ V4a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ V2h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
@ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
@ ^ [V6ft_27: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2h ) @ V6ft_27 ) ) ) ) ) ) )
@ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Efupd__at__path__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) )
=> ! [V1ft: $i] :
( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0f ) @ V1ft )
= ( ap @ V0f @ V1ft ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) )
=> ! [V5ft: $i] :
( ( mem @ V5ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4f ) @ V5ft )
= ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2h ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Efupd__at__path @ A_27a @ A_27b ) @ V3t ) @ V4f ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ V2h ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
@ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
@ ^ [V6ft_27: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ ( ap @ ( c_2Efmaptree_2Eitem @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFUPDATE @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V5ft ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2h ) @ V6ft_27 ) ) ) ) ) ) )
@ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Eft__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27b )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3k ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2fm ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2fm ) @ V3k ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27a @ A_27b ) @ V1a ) @ V2fm ) ) ) ) ) )
=> ! [V4ft: $i] :
( ( mem @ V4ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V4ft ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Eapplicable__paths__FINITE,axiom,
! [A_27a: del,A_27b: del,V0ft: $i] :
( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1p: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) @ V1p )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
@ ^ [V2ft_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V1p ) @ V0ft ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ V2ft_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Eapply__path__SNOC,axiom,
! [A_27a: del,A_27b: del,V0ft: $i] :
( ( mem @ V0ft @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2p ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ V0ft )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) @ ( ty_2Eoption_2Eoption @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Efmaptree_2Eapply__path @ A_27a @ A_27b ) @ V2p ) @ V0ft ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) )
@ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b )
@ ^ [V3ft_27: $i] : ( ap @ ( ap @ ( c_2Efinite__map_2EFLOOKUP @ A_27a @ ( ty_2Efmaptree_2Efmaptree @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Efmaptree_2Emap @ A_27a @ A_27b ) @ V3ft_27 ) ) @ V1x ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Erelrec__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c )
= ( lam @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) )
@ ^ [V0h: $i] :
( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a )
@ ^ [V1a0: $i] :
( lam @ A_27c
@ ^ [V2a1: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) )
@ ^ [V3relrec_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
@ ( lam @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a )
@ ^ [V4a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27c )
@ ( lam @ A_27c
@ ^ [V5a1: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
@ ^ [V7fm: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
@ ( lam @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c )
@ ^ [V8rfm: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4a0 ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V6i ) @ V7fm ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27c ) @ V5a1 ) @ ( ap @ ( ap @ ( ap @ V0h @ V6i ) @ V8rfm ) @ V7fm ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V8rfm ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V9d: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V9d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) ) ) @ ( ap @ ( ap @ V3relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V7fm ) @ V9d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V8rfm ) @ V9d ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ V3relrec_27 @ V4a0 ) @ V5a1 ) ) ) ) ) ) )
@ ( ap @ ( ap @ V3relrec_27 @ V1a0 ) @ V2a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Erelrec__rules,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
=> ! [V1i: $i] :
( ( mem @ V1i @ A_27a )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ! [V3rfm: $i] :
( ( mem @ V3rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V3rfm )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) )
& ! [V4d: $i] :
( ( mem @ V4d @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V2fm ) @ V4d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V3rfm ) @ V4d ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V1i ) @ V2fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V1i ) @ V3rfm ) @ V2fm ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Erelrec__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
=> ! [V1relrec_27: $i] :
( ( mem @ V1relrec_27 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
=> ( ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3fm: $i] :
( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ! [V4rfm: $i] :
( ( mem @ V4rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V4rfm )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) )
& ! [V5d: $i] :
( ( mem @ V5d @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) ) )
=> ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V4rfm ) @ V3fm ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1relrec_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Erelrec__strongind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
=> ! [V1relrec_27: $i] :
( ( mem @ V1relrec_27 @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ ( arr @ A_27c @ bool ) ) )
=> ( ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3fm: $i] :
( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ! [V4rfm: $i] :
( ( mem @ V4rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
=> ( ( ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V4rfm )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) )
& ! [V5d: $i] :
( ( mem @ V5d @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) )
& ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V3fm ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V4rfm ) @ V5d ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1relrec_27 @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) ) @ ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V4rfm ) @ V3fm ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1relrec_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Erelrec__cases,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ A_27c ) ) ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ A_27c )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V1a0 ) @ V2a1 ) )
<=> ? [V3i: $i] :
( ( mem @ V3i @ A_27a )
& ? [V4fm: $i] :
( ( mem @ V4fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
& ? [V5rfm: $i] :
( ( mem @ V5rfm @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) )
& ( V1a0
= ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V3i ) @ V4fm ) )
& ( V2a1
= ( ap @ ( ap @ ( ap @ V0h @ V3i ) @ V5rfm ) @ V4fm ) )
& ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ A_27c ) @ V5rfm )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) )
& ! [V6d: $i] :
( ( mem @ V6d @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6d ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ V4fm ) @ V6d ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27b @ A_27c ) @ V5rfm ) @ V6d ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efmaptree_2Efmtreerec__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) )
=> ! [V1ft: $i] :
( ( mem @ V1ft @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V1ft )
= ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2r: $i] : ( ap @ ( ap @ ( ap @ ( c_2Efmaptree_2Erelrec @ A_27b @ A_27c @ A_27a ) @ V0h ) @ V1ft ) @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Efmtreerec__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27b @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) @ A_27a ) ) ) )
=> ! [V1i: $i] :
( ( mem @ V1i @ A_27b )
=> ! [V2fm: $i] :
( ( mem @ V2fm @ ( ty_2Efinite__map_2Efmap @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27c @ A_27b ) @ V1i ) @ V2fm ) )
= ( ap @ ( ap @ ( ap @ V0h @ V1i ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27c @ ( ty_2Efmaptree_2Efmaptree @ A_27c @ A_27b ) @ A_27a ) @ ( ap @ ( c_2Efmaptree_2Efmtreerec @ A_27a @ A_27b @ A_27c ) @ V0h ) ) @ V2fm ) ) @ V2fm ) ) ) ) ) ).
thf(conj_thm_2Efmaptree_2Efmtree__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ A_27a @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27b @ A_27c ) @ A_27c ) ) ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ A_27c ) )
& ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ! [V3fm: $i] :
( ( mem @ V3fm @ ( ty_2Efinite__map_2Efmap @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) ) )
=> ( ( ap @ V1f @ ( ap @ ( ap @ ( c_2Efmaptree_2EFTNode @ A_27b @ A_27a ) @ V2i ) @ V3fm ) )
= ( ap @ ( ap @ ( ap @ V0h @ V2i ) @ V3fm ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27b @ ( ty_2Efmaptree_2Efmaptree @ A_27b @ A_27a ) @ A_27c ) @ V1f ) @ V3fm ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------