ITP001 Axioms: ITP048_5.ax
%------------------------------------------------------------------------------
% File : ITP048_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 : inftree_2.ax [Gau20]
% : HL4048_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 12 unt; 37 typ; 0 def)
% Number of atoms : 1119 ( 53 equ)
% Maximal formula atoms : 20 ( 11 avg)
% Number of connectives : 224 ( 1 ~; 3 |; 30 &)
% ( 5 <=>; 185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 11 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 838 ( 838 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 200 ( 37 >; 163 *; 0 +; 0 <<)
% Number of predicates : 25 ( 24 usr; 5 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 0 con; 3-11 aty)
% Number of variables : 409 ( 397 !; 12 ?; 409 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2Einftree_2Einftree,type,
ty_2Einftree_2Einftree: ( del * del * del ) > del ).
tff(tp_c_2Einftree_2Efrom__inftree,type,
c_2Einftree_2Efrom__inftree: ( del * del * del ) > $i ).
tff(mem_c_2Einftree_2Efrom__inftree,axiom,
! [A_27a: del,A_27b: del,A_27d: del] : mem(c_2Einftree_2Efrom__inftree(A_27a,A_27b,A_27d),arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27d),arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))) ).
tff(tp_c_2Einftree_2EiLf,type,
c_2Einftree_2EiLf: ( del * del * del ) > $i ).
tff(mem_c_2Einftree_2EiLf,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),arr(A_27a,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))) ).
tff(tp_c_2Einftree_2EiNd,type,
c_2Einftree_2EiNd: ( del * del * del ) > $i ).
tff(mem_c_2Einftree_2EiNd,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),arr(A_27b,arr(arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))) ).
tff(tp_c_2Einftree_2Einftree__CASE,type,
c_2Einftree_2Einftree__CASE: ( del * del * del * del ) > $i ).
tff(mem_c_2Einftree_2Einftree__CASE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Einftree_2Einftree__CASE(A_27a,A_27b,A_27c,A_27d),arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27c),arr(arr(A_27a,A_27d),arr(arr(A_27b,arr(arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)),A_27d)))) ).
tff(tp_c_2Einftree_2Einftree__rec,type,
c_2Einftree_2Einftree__rec: ( del * del * del * del ) > $i ).
tff(mem_c_2Einftree_2Einftree__rec,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Einftree_2Einftree__rec(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27b,A_27a),arr(arr(A_27c,arr(arr(A_27d,A_27a),A_27a)),arr(ty_2Einftree_2Einftree(A_27b,A_27c,A_27d),A_27a)))) ).
tff(tp_c_2Einftree_2Eis__tree,type,
c_2Einftree_2Eis__tree: ( del * del * del ) > $i ).
tff(mem_c_2Einftree_2Eis__tree,axiom,
! [A_27a: del,A_27b: del,A_27d: del] : mem(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool)) ).
tff(tp_c_2Einftree_2Erelrec,type,
c_2Einftree_2Erelrec: ( del * del * del * del ) > $i ).
tff(mem_c_2Einftree_2Erelrec,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool))))) ).
tff(tp_c_2Einftree_2Eto__inftree,type,
c_2Einftree_2Eto__inftree: ( del * del * del ) > $i ).
tff(mem_c_2Einftree_2Eto__inftree,axiom,
! [A_27a: del,A_27b: del,A_27d: del] : mem(c_2Einftree_2Eto__inftree(A_27a,A_27b,A_27d),arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),ty_2Einftree_2Einftree(A_27a,A_27b,A_27d))) ).
tff(lamtp_f1043,type,
f1043: ( del * del * del * $i ) > $i ).
tff(lameq_f1043,axiom,
! [A_27d: del,A_27b: del,A_27a: del,V2a0: $i] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V3a: $i] : ( ap(f1043(A_27d,A_27b,A_27a,V2a0),V3a) = ap(ap(c_2Emin_2E_3D(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))),V2a0),k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V3a))) ) ) ).
tff(lamtp_f1044,type,
f1044: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1044,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V6b: $i] :
( mem(V6b,A_27b)
=> ! [V5f: $i] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V7p: $i] : ( ap(f1044(A_27b,A_27a,A_27d,V6b,V5f),V7p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27d)),V7p),c_2Elist_2ENIL(A_27d))),ap(c_2Esum_2EINR(A_27a,A_27b),V6b)),ap(ap(V5f,ap(c_2Elist_2EHD(A_27d),V7p)),ap(c_2Elist_2ETL(A_27d),V7p))) ) ) ) ).
tff(lamtp_f1045,type,
f1045: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1045,axiom,
! [A_27a: del,A_27b: del,A_27d: del,V5f: $i] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V1is__tree_27: $i] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V8d: $i] : ( ap(f1045(A_27a,A_27b,A_27d,V5f,V1is__tree_27),V8d) = ap(V1is__tree_27,ap(V5f,V8d)) ) ) ) ).
tff(lamtp_f1046,type,
f1046: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f1046,axiom,
! [A_27d: del,A_27a: del,A_27b: del,V2a0: $i] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V1is__tree_27: $i] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V5f: $i] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V6b: $i] : ( ap(f1046(A_27d,A_27a,A_27b,V2a0,V1is__tree_27,V5f),V6b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))),V2a0),f1044(A_27b,A_27a,A_27d,V6b,V5f))),ap(c_2Ebool_2E_21(A_27d),f1045(A_27a,A_27b,A_27d,V5f,V1is__tree_27))) ) ) ) ) ).
tff(lamtp_f1047,type,
f1047: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1047,axiom,
! [A_27a: del,A_27d: del,A_27b: del,V1is__tree_27: $i] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V2a0: $i] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V5f: $i] : ( ap(f1047(A_27a,A_27d,A_27b,V1is__tree_27,V2a0),V5f) = ap(c_2Ebool_2E_3F(A_27b),f1046(A_27d,A_27a,A_27b,V2a0,V1is__tree_27,V5f)) ) ) ) ).
tff(lamtp_f1048,type,
f1048: ( del * del * del * $i ) > $i ).
tff(lameq_f1048,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V1is__tree_27: $i] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V2a0: $i] : ( ap(f1048(A_27b,A_27a,A_27d,V1is__tree_27),V2a0) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(A_27a),f1043(A_27d,A_27b,A_27a,V2a0))),ap(c_2Ebool_2E_3F(arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))),f1047(A_27a,A_27d,A_27b,V1is__tree_27,V2a0)))),ap(V1is__tree_27,V2a0)) ) ) ).
tff(lamtp_f1049,type,
f1049: ( del * del * del * $i ) > $i ).
tff(lameq_f1049,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V0a0: $i] :
( mem(V0a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V1is__tree_27: $i] : ( ap(f1049(A_27b,A_27a,A_27d,V0a0),V1is__tree_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))),f1048(A_27b,A_27a,A_27d,V1is__tree_27))),ap(V1is__tree_27,V0a0)) ) ) ).
tff(lamtp_f1050,type,
f1050: ( del * del * del ) > $i ).
tff(lameq_f1050,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V0a0: $i] : ( ap(f1050(A_27b,A_27a,A_27d),V0a0) = ap(c_2Ebool_2E_21(arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool)),f1049(A_27b,A_27a,A_27d,V0a0)) ) ).
tff(ax_thm_2Einftree_2Eis__tree__def,axiom,
! [A_27a: del,A_27b: del,A_27d: del] : ( c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d) = f1050(A_27b,A_27a,A_27d) ) ).
tff(lamtp_f1051,type,
f1051: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1051,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V3b: $i] :
( mem(V3b,A_27b)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V5p: $i] : ( ap(f1051(A_27b,A_27a,A_27d,V3b,V2f),V5p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27d)),V5p),c_2Elist_2ENIL(A_27d))),ap(c_2Esum_2EINR(A_27a,A_27b),V3b)),ap(ap(V2f,ap(c_2Elist_2EHD(A_27d),V5p)),ap(c_2Elist_2ETL(A_27d),V5p))) ) ) ) ).
tff(conj_thm_2Einftree_2Eis__tree__rules,axiom,
! [A_27a: del,A_27b: del,A_27d: del] :
( ! [V0a: $i] :
( mem(V0a,A_27a)
=> p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V0a)))) )
& ! [V2f: $i] :
( mem(V2f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ( ! [V4d: $i] :
( mem(V4d,A_27d)
=> p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),ap(V2f,V4d))) )
=> p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),f1051(A_27b,A_27a,A_27d,V3b,V2f))) ) ) ) ) ).
tff(lamtp_f1052,type,
f1052: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1052,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V4b: $i] :
( mem(V4b,A_27b)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V6p: $i] : ( ap(f1052(A_27b,A_27a,A_27d,V4b,V3f),V6p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27d)),V6p),c_2Elist_2ENIL(A_27d))),ap(c_2Esum_2EINR(A_27a,A_27b),V4b)),ap(ap(V3f,ap(c_2Elist_2EHD(A_27d),V6p)),ap(c_2Elist_2ETL(A_27d),V6p))) ) ) ) ).
tff(conj_thm_2Einftree_2Eis__tree__ind,axiom,
! [A_27a: del,A_27b: del,A_27d: del,V0is__tree_27: $i] :
( mem(V0is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ( ( ! [V1a: $i] :
( mem(V1a,A_27a)
=> p(ap(V0is__tree_27,k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V1a)))) )
& ! [V3f: $i] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V4b: $i] :
( mem(V4b,A_27b)
=> ( ! [V5d: $i] :
( mem(V5d,A_27d)
=> p(ap(V0is__tree_27,ap(V3f,V5d))) )
=> p(ap(V0is__tree_27,f1052(A_27b,A_27a,A_27d,V4b,V3f))) ) ) ) )
=> ! [V7a0: $i] :
( mem(V7a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ( p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),V7a0))
=> p(ap(V0is__tree_27,V7a0)) ) ) ) ) ).
tff(conj_thm_2Einftree_2Eis__tree__strongind,axiom,
! [A_27a: del,A_27b: del,A_27d: del,V0is__tree_27: $i] :
( mem(V0is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ( ( ! [V1a: $i] :
( mem(V1a,A_27a)
=> p(ap(V0is__tree_27,k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V1a)))) )
& ! [V3f: $i] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V4b: $i] :
( mem(V4b,A_27b)
=> ( ! [V5d: $i] :
( mem(V5d,A_27d)
=> ( p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),ap(V3f,V5d)))
& p(ap(V0is__tree_27,ap(V3f,V5d))) ) )
=> p(ap(V0is__tree_27,f1052(A_27b,A_27a,A_27d,V4b,V3f))) ) ) ) )
=> ! [V7a0: $i] :
( mem(V7a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ( p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),V7a0))
=> p(ap(V0is__tree_27,V7a0)) ) ) ) ) ).
tff(lamtp_f1053,type,
f1053: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1053,axiom,
! [A_27b: del,A_27a: del,A_27d: del,V4b: $i] :
( mem(V4b,A_27b)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V5p: $i] : ( ap(f1053(A_27b,A_27a,A_27d,V4b,V3f),V5p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27d)),V5p),c_2Elist_2ENIL(A_27d))),ap(c_2Esum_2EINR(A_27a,A_27b),V4b)),ap(ap(V3f,ap(c_2Elist_2EHD(A_27d),V5p)),ap(c_2Elist_2ETL(A_27d),V5p))) ) ) ) ).
tff(conj_thm_2Einftree_2Eis__tree__cases,axiom,
! [A_27a: del,A_27b: del,A_27d: del,V0a0: $i] :
( mem(V0a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ( p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),V0a0))
<=> ( ? [V1a: $i] :
( mem(V1a,A_27a)
& ( V0a0 = k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V1a)) ) )
| ? [V3f: $i] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
& ? [V4b: $i] :
( mem(V4b,A_27b)
& ( V0a0 = f1053(A_27b,A_27a,A_27d,V4b,V3f) )
& ! [V6d: $i] :
( mem(V6d,A_27d)
=> p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),ap(V3f,V6d))) ) ) ) ) ) ) ).
tff(ax_thm_2Einftree_2Einftree__TY__DEF,axiom,
! [A_27a: del,A_27b: del,A_27d: del] :
? [V0rep: $i] :
( mem(V0rep,arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27d),arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),ty_2Einftree_2Einftree(A_27a,A_27b,A_27d)),c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d)),V0rep)) ) ).
tff(ax_thm_2Einftree_2Einftree__bijections,axiom,
! [A_27a: del,A_27b: del,A_27d: del] :
( ! [V0a: $i] :
( mem(V0a,ty_2Einftree_2Einftree(A_27a,A_27b,A_27d))
=> ( ap(c_2Einftree_2Eto__inftree(A_27a,A_27b,A_27d),ap(c_2Einftree_2Efrom__inftree(A_27a,A_27b,A_27d),V0a)) = V0a ) )
& ! [V1r: $i] :
( mem(V1r,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ( p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),V1r))
<=> ( ap(c_2Einftree_2Efrom__inftree(A_27a,A_27b,A_27d),ap(c_2Einftree_2Eto__inftree(A_27a,A_27b,A_27d),V1r)) = V1r ) ) ) ) ).
tff(ax_thm_2Einftree_2EiLf__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( mem(V0a,A_27a)
=> ( ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V0a) = ap(c_2Einftree_2Eto__inftree(A_27a,A_27b,A_27c),k(ty_2Elist_2Elist(A_27c),ap(c_2Esum_2EINL(A_27a,A_27b),V0a))) ) ) ).
tff(lamtp_f1054,type,
f1054: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1054,axiom,
! [A_27b: del,A_27a: del,A_27c: del,V0b: $i] :
( mem(V0b,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V2p: $i] : ( ap(f1054(A_27b,A_27a,A_27c,V0b,V1f),V2p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27a,A_27b)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27c)),V2p),c_2Elist_2ENIL(A_27c))),ap(c_2Esum_2EINR(A_27a,A_27b),V0b)),ap(ap(c_2Einftree_2Efrom__inftree(A_27a,A_27b,A_27c),ap(V1f,ap(c_2Elist_2EHD(A_27c),V2p))),ap(c_2Elist_2ETL(A_27c),V2p))) ) ) ) ).
tff(ax_thm_2Einftree_2EiNd__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0b: $i] :
( mem(V0b,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V0b),V1f) = ap(c_2Einftree_2Eto__inftree(A_27a,A_27b,A_27c),f1054(A_27b,A_27a,A_27c,V0b,V1f)) ) ) ) ).
tff(lamtp_f1055,type,
f1055: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f1055,axiom,
! [A_27a: del,A_27c: del,A_27b: del,V0b: $i] :
( mem(V0b,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,ty_2Einftree_2Einftree(A_27c,A_27a,A_27b)))
=> ! [V2p: $i] : ( ap(f1055(A_27a,A_27c,A_27b,V0b,V1f),V2p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Esum_2Esum(A_27c,A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Elist_2Elist(A_27b)),V2p),c_2Elist_2ENIL(A_27b))),ap(c_2Esum_2EINR(A_27c,A_27a),V0b)),ap(ap(c_2Einftree_2Efrom__inftree(A_27c,A_27a,A_27b),ap(V1f,ap(c_2Elist_2EHD(A_27b),V2p))),ap(c_2Elist_2ETL(A_27b),V2p))) ) ) ) ).
tff(conj_thm_2Einftree_2EiNd__is__tree,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0b: $i] :
( mem(V0b,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,ty_2Einftree_2Einftree(A_27c,A_27a,A_27b)))
=> p(ap(c_2Einftree_2Eis__tree(A_27c,A_27a,A_27b),f1055(A_27a,A_27c,A_27b,V0b,V1f))) ) ) ).
tff(conj_thm_2Einftree_2Einftree__11,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a1: $i] :
( mem(V0a1,A_27a)
=> ! [V1a2: $i] :
( mem(V1a2,A_27a)
=> ! [V2b1: $i] :
( mem(V2b1,A_27b)
=> ! [V3f1: $i] :
( mem(V3f1,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V4b2: $i] :
( mem(V4b2,A_27b)
=> ! [V5f2: $i] :
( mem(V5f2,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ( ( ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V0a1) = ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V1a2) )
<=> ( V0a1 = V1a2 ) )
& ( ( ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V2b1),V3f1) = ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V4b2),V5f2) )
<=> ( ( V2b1 = V4b2 )
& ( V3f1 = V5f2 ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Einftree_2Einftree__distinct,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V0a) != ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V1b),V2f) ) ) ) ) ).
tff(conj_thm_2Einftree_2Einftree__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( mem(V0P,arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27c),bool))
=> ( ( ! [V1a: $i] :
( mem(V1a,A_27a)
=> p(ap(V0P,ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V1a))) )
& ! [V2b: $i] :
( mem(V2b,A_27b)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ! [V4d: $i] :
( mem(V4d,A_27c)
=> p(ap(V0P,ap(V3f,V4d))) )
=> p(ap(V0P,ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V2b),V3f))) ) ) ) )
=> ! [V5t: $i] :
( mem(V5t,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))
=> p(ap(V0P,V5t)) ) ) ) ).
tff(lamtp_f1056,type,
f1056: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f1056,axiom,
! [A_27b: del,A_27c: del,A_27d: del,A_27a: del,V8a3: $i] :
( mem(V8a3,A_27b)
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2: $i] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V9a: $i] : ( ap(f1056(A_27b,A_27c,A_27d,A_27a,V8a3,V5a0,V7a2),V9a) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V7a2),ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V9a))),ap(ap(c_2Emin_2E_3D(A_27b),V8a3),ap(V5a0,V9a))) ) ) ) ) ).
tff(lamtp_f1057,type,
f1057: ( del * del * del * del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f1057,axiom,
! [A_27b: del,A_27c: del,A_27a: del,A_27d: del,V11df: $i] :
( mem(V11df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12g: $i] :
( mem(V12g,arr(A_27d,A_27b))
=> ! [V13d: $i] : ( ap(f1057(A_27b,A_27c,A_27a,A_27d,V11df,V5a0,V4relrec_27,V6a1,V12g),V13d) = ap(ap(ap(ap(V4relrec_27,V5a0),V6a1),ap(V11df,V13d)),ap(V12g,V13d)) ) ) ) ) ) ) ).
tff(lamtp_f1058,type,
f1058: ( del * del * del * del * $i * $i * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f1058,axiom,
! [A_27a: del,A_27c: del,A_27d: del,A_27b: del,V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3: $i] :
( mem(V8a3,A_27b)
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2: $i] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V10b: $i] :
( mem(V10b,A_27c)
=> ! [V11df: $i] :
( mem(V11df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V12g: $i] : ( ap(f1058(A_27a,A_27c,A_27d,A_27b,V6a1,V8a3,V4relrec_27,V5a0,V7a2,V10b,V11df),V12g) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V7a2),ap(ap(c_2Einftree_2EiNd(A_27a,A_27c,A_27d),V10b),V11df))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27b),V8a3),ap(ap(V6a1,V10b),V12g))),ap(c_2Ebool_2E_21(A_27d),f1057(A_27b,A_27c,A_27a,A_27d,V11df,V5a0,V4relrec_27,V6a1,V12g)))) ) ) ) ) ) ) ) ) ).
tff(lamtp_f1059,type,
f1059: ( del * del * del * del * $i * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f1059,axiom,
! [A_27b: del,A_27a: del,A_27c: del,A_27d: del,V10b: $i] :
( mem(V10b,A_27c)
=> ! [V7a2: $i] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V8a3: $i] :
( mem(V8a3,A_27b)
=> ! [V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V11df: $i] : ( ap(f1059(A_27b,A_27a,A_27c,A_27d,V10b,V7a2,V5a0,V4relrec_27,V8a3,V6a1),V11df) = ap(c_2Ebool_2E_3F(arr(A_27d,A_27b)),f1058(A_27a,A_27c,A_27d,A_27b,V6a1,V8a3,V4relrec_27,V5a0,V7a2,V10b,V11df)) ) ) ) ) ) ) ) ).
tff(lamtp_f1060,type,
f1060: ( del * del * del * del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f1060,axiom,
! [A_27a: del,A_27d: del,A_27b: del,A_27c: del,V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3: $i] :
( mem(V8a3,A_27b)
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2: $i] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V10b: $i] : ( ap(f1060(A_27a,A_27d,A_27b,A_27c,V6a1,V8a3,V4relrec_27,V5a0,V7a2),V10b) = ap(c_2Ebool_2E_3F(arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))),f1059(A_27b,A_27a,A_27c,A_27d,V10b,V7a2,V5a0,V4relrec_27,V8a3,V6a1)) ) ) ) ) ) ) ).
tff(lamtp_f1061,type,
f1061: ( del * del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f1061,axiom,
! [A_27c: del,A_27a: del,A_27d: del,A_27b: del,V7a2: $i] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3: $i] : ( ap(f1061(A_27c,A_27a,A_27d,A_27b,V7a2,V5a0,V4relrec_27,V6a1),V8a3) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(A_27a),f1056(A_27b,A_27c,A_27d,A_27a,V8a3,V5a0,V7a2))),ap(c_2Ebool_2E_3F(A_27c),f1060(A_27a,A_27d,A_27b,A_27c,V6a1,V8a3,V4relrec_27,V5a0,V7a2)))),ap(ap(ap(ap(V4relrec_27,V5a0),V6a1),V7a2),V8a3)) ) ) ) ) ) ).
tff(lamtp_f1062,type,
f1062: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f1062,axiom,
! [A_27b: del,A_27c: del,A_27a: del,A_27d: del,V6a1: $i] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2: $i] : ( ap(f1062(A_27b,A_27c,A_27a,A_27d,V6a1,V4relrec_27,V5a0),V7a2) = ap(c_2Ebool_2E_21(A_27b),f1061(A_27c,A_27a,A_27d,A_27b,V7a2,V5a0,V4relrec_27,V6a1)) ) ) ) ) ).
tff(lamtp_f1063,type,
f1063: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f1063,axiom,
! [A_27a: del,A_27b: del,A_27d: del,A_27c: del,V5a0: $i] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V6a1: $i] : ( ap(f1063(A_27a,A_27b,A_27d,A_27c,V5a0,V4relrec_27),V6a1) = ap(c_2Ebool_2E_21(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)),f1062(A_27b,A_27c,A_27a,A_27d,V6a1,V4relrec_27,V5a0)) ) ) ) ).
tff(lamtp_f1064,type,
f1064: ( del * del * del * del * $i ) > $i ).
tff(lameq_f1064,axiom,
! [A_27d: del,A_27c: del,A_27a: del,A_27b: del,V4relrec_27: $i] :
( mem(V4relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ! [V5a0: $i] : ( ap(f1064(A_27d,A_27c,A_27a,A_27b,V4relrec_27),V5a0) = ap(c_2Ebool_2E_21(arr(A_27c,arr(arr(A_27d,A_27b),A_27b))),f1063(A_27a,A_27b,A_27d,A_27c,V5a0,V4relrec_27)) ) ) ).
tff(lamtp_f1065,type,
f1065: ( del * del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f1065,axiom,
! [A_27d: del,A_27a: del,A_27c: del,A_27b: del,V3a3: $i] :
( mem(V3a3,A_27b)
=> ! [V1a1: $i] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V0a0: $i] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V2a2: $i] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V4relrec_27: $i] : ( ap(f1065(A_27d,A_27a,A_27c,A_27b,V3a3,V1a1,V0a0,V2a2),V4relrec_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(arr(A_27a,A_27b)),f1064(A_27d,A_27c,A_27a,A_27b,V4relrec_27))),ap(ap(ap(ap(V4relrec_27,V0a0),V1a1),V2a2),V3a3)) ) ) ) ) ) ).
tff(lamtp_f1066,type,
f1066: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f1066,axiom,
! [A_27c: del,A_27a: del,A_27d: del,A_27b: del,V2a2: $i] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V0a0: $i] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1: $i] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a3: $i] : ( ap(f1066(A_27c,A_27a,A_27d,A_27b,V2a2,V0a0,V1a1),V3a3) = ap(c_2Ebool_2E_21(arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool))))),f1065(A_27d,A_27a,A_27c,A_27b,V3a3,V1a1,V0a0,V2a2)) ) ) ) ) ).
tff(lamtp_f1067,type,
f1067: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f1067,axiom,
! [A_27b: del,A_27c: del,A_27a: del,A_27d: del,V0a0: $i] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1: $i] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a2: $i] : ( ap(f1067(A_27b,A_27c,A_27a,A_27d,V0a0,V1a1),V2a2) = f1066(A_27c,A_27a,A_27d,A_27b,V2a2,V0a0,V1a1) ) ) ) ).
tff(lamtp_f1068,type,
f1068: ( del * del * del * del * $i ) > $i ).
tff(lameq_f1068,axiom,
! [A_27a: del,A_27b: del,A_27d: del,A_27c: del,V0a0: $i] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1: $i] : ( ap(f1068(A_27a,A_27b,A_27d,A_27c,V0a0),V1a1) = f1067(A_27b,A_27c,A_27a,A_27d,V0a0,V1a1) ) ) ).
tff(lamtp_f1069,type,
f1069: ( del * del * del * del ) > $i ).
tff(lameq_f1069,axiom,
! [A_27c: del,A_27d: del,A_27a: del,A_27b: del,V0a0: $i] : ( ap(f1069(A_27c,A_27d,A_27a,A_27b),V0a0) = f1068(A_27a,A_27b,A_27d,A_27c,V0a0) ) ).
tff(ax_thm_2Einftree_2Erelrec__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d) = f1069(A_27c,A_27d,A_27a,A_27b) ) ).
tff(conj_thm_2Einftree_2Erelrec__rules,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0lf: $i] :
( mem(V0lf,arr(A_27a,A_27b))
=> ! [V1nd: $i] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V0lf),V1nd),ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V2a)),ap(V0lf,V2a))) ) ) )
& ! [V3lf: $i] :
( mem(V3lf,arr(A_27a,A_27b))
=> ! [V4nd: $i] :
( mem(V4nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V5b: $i] :
( mem(V5b,A_27c)
=> ! [V6df: $i] :
( mem(V6df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V7g: $i] :
( mem(V7g,arr(A_27d,A_27b))
=> ( ! [V8d: $i] :
( mem(V8d,A_27d)
=> p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V3lf),V4nd),ap(V6df,V8d)),ap(V7g,V8d))) )
=> p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V3lf),V4nd),ap(ap(c_2Einftree_2EiNd(A_27a,A_27c,A_27d),V5b),V6df)),ap(ap(V4nd,V5b),V7g))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Einftree_2Erelrec__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0relrec_27: $i] :
( mem(V0relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ( ( ! [V1lf: $i] :
( mem(V1lf,arr(A_27a,A_27b))
=> ! [V2nd: $i] :
( mem(V2nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a: $i] :
( mem(V3a,A_27a)
=> p(ap(ap(ap(ap(V0relrec_27,V1lf),V2nd),ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V3a)),ap(V1lf,V3a))) ) ) )
& ! [V4lf: $i] :
( mem(V4lf,arr(A_27a,A_27b))
=> ! [V5nd: $i] :
( mem(V5nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V6b: $i] :
( mem(V6b,A_27c)
=> ! [V7df: $i] :
( mem(V7df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V8g: $i] :
( mem(V8g,arr(A_27d,A_27b))
=> ( ! [V9d: $i] :
( mem(V9d,A_27d)
=> p(ap(ap(ap(ap(V0relrec_27,V4lf),V5nd),ap(V7df,V9d)),ap(V8g,V9d))) )
=> p(ap(ap(ap(ap(V0relrec_27,V4lf),V5nd),ap(ap(c_2Einftree_2EiNd(A_27a,A_27c,A_27d),V6b),V7df)),ap(ap(V5nd,V6b),V8g))) ) ) ) ) ) ) )
=> ! [V10a0: $i] :
( mem(V10a0,arr(A_27a,A_27b))
=> ! [V11a1: $i] :
( mem(V11a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12a2: $i] :
( mem(V12a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V13a3: $i] :
( mem(V13a3,A_27b)
=> ( p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V10a0),V11a1),V12a2),V13a3))
=> p(ap(ap(ap(ap(V0relrec_27,V10a0),V11a1),V12a2),V13a3)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Einftree_2Erelrec__strongind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0relrec_27: $i] :
( mem(V0relrec_27,arr(arr(A_27a,A_27b),arr(arr(A_27c,arr(arr(A_27d,A_27b),A_27b)),arr(ty_2Einftree_2Einftree(A_27a,A_27c,A_27d),arr(A_27b,bool)))))
=> ( ( ! [V1lf: $i] :
( mem(V1lf,arr(A_27a,A_27b))
=> ! [V2nd: $i] :
( mem(V2nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a: $i] :
( mem(V3a,A_27a)
=> p(ap(ap(ap(ap(V0relrec_27,V1lf),V2nd),ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V3a)),ap(V1lf,V3a))) ) ) )
& ! [V4lf: $i] :
( mem(V4lf,arr(A_27a,A_27b))
=> ! [V5nd: $i] :
( mem(V5nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V6b: $i] :
( mem(V6b,A_27c)
=> ! [V7df: $i] :
( mem(V7df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V8g: $i] :
( mem(V8g,arr(A_27d,A_27b))
=> ( ! [V9d: $i] :
( mem(V9d,A_27d)
=> ( p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V4lf),V5nd),ap(V7df,V9d)),ap(V8g,V9d)))
& p(ap(ap(ap(ap(V0relrec_27,V4lf),V5nd),ap(V7df,V9d)),ap(V8g,V9d))) ) )
=> p(ap(ap(ap(ap(V0relrec_27,V4lf),V5nd),ap(ap(c_2Einftree_2EiNd(A_27a,A_27c,A_27d),V6b),V7df)),ap(ap(V5nd,V6b),V8g))) ) ) ) ) ) ) )
=> ! [V10a0: $i] :
( mem(V10a0,arr(A_27a,A_27b))
=> ! [V11a1: $i] :
( mem(V11a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12a2: $i] :
( mem(V12a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V13a3: $i] :
( mem(V13a3,A_27b)
=> ( p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V10a0),V11a1),V12a2),V13a3))
=> p(ap(ap(ap(ap(V0relrec_27,V10a0),V11a1),V12a2),V13a3)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Einftree_2Erelrec__cases,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0a0: $i] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1: $i] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a2: $i] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V3a3: $i] :
( mem(V3a3,A_27b)
=> ( p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V0a0),V1a1),V2a2),V3a3))
<=> ( ? [V4a: $i] :
( mem(V4a,A_27a)
& ( V2a2 = ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V4a) )
& ( V3a3 = ap(V0a0,V4a) ) )
| ? [V5b: $i] :
( mem(V5b,A_27c)
& ? [V6df: $i] :
( mem(V6df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
& ? [V7g: $i] :
( mem(V7g,arr(A_27d,A_27b))
& ( V2a2 = ap(ap(c_2Einftree_2EiNd(A_27a,A_27c,A_27d),V5b),V6df) )
& ( V3a3 = ap(ap(V1a1,V5b),V7g) )
& ! [V8d: $i] :
( mem(V8d,A_27d)
=> p(ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d),V0a0),V1a1),ap(V6df,V8d)),ap(V7g,V8d))) ) ) ) ) ) ) ) ) ) ) ).
tff(lamtp_f1070,type,
f1070: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f1070,axiom,
! [A_27c: del,A_27b: del,A_27d: del,A_27a: del,V2t: $i] :
( mem(V2t,ty_2Einftree_2Einftree(A_27b,A_27c,A_27d))
=> ! [V0lf: $i] :
( mem(V0lf,arr(A_27b,A_27a))
=> ! [V1nd: $i] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27a),A_27a)))
=> ! [V3r: $i] : ( ap(f1070(A_27c,A_27b,A_27d,A_27a,V2t,V0lf,V1nd),V3r) = ap(ap(ap(ap(c_2Einftree_2Erelrec(A_27b,A_27a,A_27c,A_27d),V0lf),V1nd),V2t),V3r) ) ) ) ) ).
tff(ax_thm_2Einftree_2Einftree__rec__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0lf: $i] :
( mem(V0lf,arr(A_27b,A_27a))
=> ! [V1nd: $i] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27a),A_27a)))
=> ! [V2t: $i] :
( mem(V2t,ty_2Einftree_2Einftree(A_27b,A_27c,A_27d))
=> ( ap(ap(ap(c_2Einftree_2Einftree__rec(A_27a,A_27b,A_27c,A_27d),V0lf),V1nd),V2t) = ap(c_2Emin_2E_40(A_27a),f1070(A_27c,A_27b,A_27d,A_27a,V2t,V0lf,V1nd)) ) ) ) ) ).
tff(conj_thm_2Einftree_2Einftree__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0lf: $i] :
( mem(V0lf,arr(A_27a,A_27d))
=> ! [V1nd: $i] :
( mem(V1nd,arr(A_27b,arr(arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),arr(arr(A_27c,A_27d),A_27d))))
=> ? [V2f: $i] :
( mem(V2f,arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d))
& ! [V3a: $i] :
( mem(V3a,A_27a)
=> ( ap(V2f,ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V3a)) = ap(V0lf,V3a) ) )
& ! [V4b: $i] :
( mem(V4b,A_27b)
=> ! [V5d: $i] :
( mem(V5d,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ap(V2f,ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V4b),V5d)) = ap(ap(ap(V1nd,V4b),V5d),ap(ap(c_2Ecombin_2Eo(A_27c,A_27d,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V2f),V5d)) ) ) ) ) ) ) ).
tff(ax_thm_2Einftree_2Einftree__case__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27d))
=> ! [V2f1: $i] :
( mem(V2f1,arr(A_27b,arr(arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)))
=> ( ap(ap(ap(c_2Einftree_2Einftree__CASE(A_27a,A_27b,A_27c,A_27d),ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V0a)),V1f),V2f1) = ap(V1f,V0a) ) ) ) )
& ! [V3b: $i] :
( mem(V3b,A_27b)
=> ! [V4d: $i] :
( mem(V4d,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V5f: $i] :
( mem(V5f,arr(A_27a,A_27d))
=> ! [V6f1: $i] :
( mem(V6f1,arr(A_27b,arr(arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)))
=> ( ap(ap(ap(c_2Einftree_2Einftree__CASE(A_27a,A_27b,A_27c,A_27d),ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V3b),V4d)),V5f),V6f1) = ap(ap(V6f1,V3b),V4d) ) ) ) ) ) ) ).
tff(conj_thm_2Einftree_2Einftree__nchotomy,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t: $i] :
( mem(V0t,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))
=> ( ? [V1a: $i] :
( mem(V1a,A_27a)
& ( V0t = ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V1a) ) )
| ? [V2b: $i] :
( mem(V2b,A_27b)
& ? [V3d: $i] :
( mem(V3d,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
& ( V0t = ap(ap(c_2Einftree_2EiNd(A_27a,A_27b,A_27c),V2b),V3d) ) ) ) ) ) ).
%------------------------------------------------------------------------------