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 : 59 ( 2 unt; 0 def)
% Number of atoms : 386 ( 53 equ)
% Maximal formula atoms : 24 ( 6 avg)
% Number of connectives : 328 ( 1 ~; 3 |; 30 &)
% ( 5 <=>; 289 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 12 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 58 ( 58 usr; 4 con; 0-11 aty)
% Number of variables : 412 ( 400 !; 12 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Einftree_2Einftree,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ! [A2] :
( ne(A2)
=> ne(ty_2Einftree_2Einftree(A0,A1,A2)) ) ) ) ).
fof(mem_c_2Einftree_2Efrom__inftree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> 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)))) ) ) ) ).
fof(mem_c_2Einftree_2EiLf,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),arr(A_27a,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))) ) ) ) ).
fof(mem_c_2Einftree_2EiNd,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> 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)))) ) ) ) ).
fof(mem_c_2Einftree_2Einftree__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> 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)))) ) ) ) ) ).
fof(mem_c_2Einftree_2Einftree__rec,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> 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)))) ) ) ) ) ).
fof(mem_c_2Einftree_2Eis__tree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> 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)) ) ) ) ).
fof(mem_c_2Einftree_2Erelrec,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> 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))))) ) ) ) ) ).
fof(mem_c_2Einftree_2Eto__inftree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> 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))) ) ) ) ).
fof(lameq_f1043,axiom,
! [A_27d,A_27b,A_27a,V2a0] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V3a] : 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))) ) ).
fof(lameq_f1044,axiom,
! [A_27b,A_27a,A_27d,V6b] :
( mem(V6b,A_27b)
=> ! [V5f] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V7p] : 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))) ) ) ).
fof(lameq_f1045,axiom,
! [A_27a,A_27b,A_27d,V5f] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V1is__tree_27] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V8d] : ap(f1045(A_27a,A_27b,A_27d,V5f,V1is__tree_27),V8d) = ap(V1is__tree_27,ap(V5f,V8d)) ) ) ).
fof(lameq_f1046,axiom,
! [A_27d,A_27a,A_27b,V2a0] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V1is__tree_27] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V5f] :
( mem(V5f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V6b] : 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))) ) ) ) ).
fof(lameq_f1047,axiom,
! [A_27a,A_27d,A_27b,V1is__tree_27] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V2a0] :
( mem(V2a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V5f] : 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)) ) ) ).
fof(lameq_f1048,axiom,
! [A_27b,A_27a,A_27d,V1is__tree_27] :
( mem(V1is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ! [V2a0] : 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)) ) ).
fof(lameq_f1049,axiom,
! [A_27b,A_27a,A_27d,V0a0] :
( mem(V0a0,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)))
=> ! [V1is__tree_27] : 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)) ) ).
fof(lameq_f1050,axiom,
! [A_27b,A_27a,A_27d,V0a0] : 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)) ).
fof(ax_thm_2Einftree_2Eis__tree__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d) = f1050(A_27b,A_27a,A_27d) ) ) ) ).
fof(lameq_f1051,axiom,
! [A_27b,A_27a,A_27d,V3b] :
( mem(V3b,A_27b)
=> ! [V2f] :
( mem(V2f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V5p] : 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))) ) ) ).
fof(conj_thm_2Einftree_2Eis__tree__rules,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0a] :
( 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] :
( mem(V2f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V3b] :
( mem(V3b,A_27b)
=> ( ! [V4d] :
( 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))) ) ) ) ) ) ) ) ).
fof(lameq_f1052,axiom,
! [A_27b,A_27a,A_27d,V4b] :
( mem(V4b,A_27b)
=> ! [V3f] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V6p] : 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))) ) ) ).
fof(conj_thm_2Einftree_2Eis__tree__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0is__tree_27] :
( mem(V0is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ( ( ! [V1a] :
( 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] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V4b] :
( mem(V4b,A_27b)
=> ( ! [V5d] :
( 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] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Eis__tree__strongind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0is__tree_27] :
( mem(V0is__tree_27,arr(arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b)),bool))
=> ( ( ! [V1a] :
( 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] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V4b] :
( mem(V4b,A_27b)
=> ( ! [V5d] :
( 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] :
( 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)) ) ) ) ) ) ) ) ).
fof(lameq_f1053,axiom,
! [A_27b,A_27a,A_27d,V4b] :
( mem(V4b,A_27b)
=> ! [V3f] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
=> ! [V5p] : 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))) ) ) ).
fof(conj_thm_2Einftree_2Eis__tree__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0a0] :
( 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] :
( mem(V1a,A_27a)
& V0a0 = k(ty_2Elist_2Elist(A_27d),ap(c_2Esum_2EINL(A_27a,A_27b),V1a)) )
| ? [V3f] :
( mem(V3f,arr(A_27d,arr(ty_2Elist_2Elist(A_27d),ty_2Esum_2Esum(A_27a,A_27b))))
& ? [V4b] :
( mem(V4b,A_27b)
& V0a0 = f1053(A_27b,A_27a,A_27d,V4b,V3f)
& ! [V6d] :
( mem(V6d,A_27d)
=> p(ap(c_2Einftree_2Eis__tree(A_27a,A_27b,A_27d),ap(V3f,V6d))) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Einftree_2Einftree__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ? [V0rep] :
( 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)) ) ) ) ) ).
fof(ax_thm_2Einftree_2Einftree__bijections,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0a] :
( 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] :
( 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 ) ) ) ) ) ) ).
fof(ax_thm_2Einftree_2EiLf__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0a] :
( 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))) ) ) ) ) ).
fof(lameq_f1054,axiom,
! [A_27b,A_27a,A_27c,V0b] :
( mem(V0b,A_27b)
=> ! [V1f] :
( mem(V1f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V2p] : 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))) ) ) ).
fof(ax_thm_2Einftree_2EiNd__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0b] :
( mem(V0b,A_27b)
=> ! [V1f] :
( 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)) ) ) ) ) ) ).
fof(lameq_f1055,axiom,
! [A_27a,A_27c,A_27b,V0b] :
( mem(V0b,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27b,ty_2Einftree_2Einftree(A_27c,A_27a,A_27b)))
=> ! [V2p] : 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))) ) ) ).
fof(conj_thm_2Einftree_2EiNd__is__tree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0b] :
( mem(V0b,A_27a)
=> ! [V1f] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Einftree__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0a1] :
( mem(V0a1,A_27a)
=> ! [V1a2] :
( mem(V1a2,A_27a)
=> ! [V2b1] :
( mem(V2b1,A_27b)
=> ! [V3f1] :
( mem(V3f1,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V4b2] :
( mem(V4b2,A_27b)
=> ! [V5f2] :
( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Einftree__distinct,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2f] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Einftree__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27c),bool))
=> ( ( ! [V1a] :
( mem(V1a,A_27a)
=> p(ap(V0P,ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V1a))) )
& ! [V2b] :
( mem(V2b,A_27b)
=> ! [V3f] :
( mem(V3f,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ( ! [V4d] :
( 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] :
( mem(V5t,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))
=> p(ap(V0P,V5t)) ) ) ) ) ) ) ).
fof(lameq_f1056,axiom,
! [A_27b,A_27c,A_27d,A_27a,V8a3] :
( mem(V8a3,A_27b)
=> ! [V5a0] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V9a] : 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))) ) ) ) ).
fof(lameq_f1057,axiom,
! [A_27b,A_27c,A_27a,A_27d,V11df] :
( mem(V11df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V5a0] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27] :
( 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] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12g] :
( mem(V12g,arr(A_27d,A_27b))
=> ! [V13d] : 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)) ) ) ) ) ) ).
fof(lameq_f1058,axiom,
! [A_27a,A_27c,A_27d,A_27b,V6a1] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3] :
( mem(V8a3,A_27b)
=> ! [V4relrec_27] :
( 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] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V10b] :
( mem(V10b,A_27c)
=> ! [V11df] :
( mem(V11df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V12g] : 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)))) ) ) ) ) ) ) ) ).
fof(lameq_f1059,axiom,
! [A_27b,A_27a,A_27c,A_27d,V10b] :
( mem(V10b,A_27c)
=> ! [V7a2] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V5a0] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27] :
( 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] :
( mem(V8a3,A_27b)
=> ! [V6a1] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V11df] : 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)) ) ) ) ) ) ) ).
fof(lameq_f1060,axiom,
! [A_27a,A_27d,A_27b,A_27c,V6a1] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3] :
( mem(V8a3,A_27b)
=> ! [V4relrec_27] :
( 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] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V10b] : 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)) ) ) ) ) ) ).
fof(lameq_f1061,axiom,
! [A_27c,A_27a,A_27d,A_27b,V7a2] :
( mem(V7a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V5a0] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27] :
( 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] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V8a3] : 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)) ) ) ) ) ).
fof(lameq_f1062,axiom,
! [A_27b,A_27c,A_27a,A_27d,V6a1] :
( mem(V6a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V4relrec_27] :
( 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] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V7a2] : 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)) ) ) ) ).
fof(lameq_f1063,axiom,
! [A_27a,A_27b,A_27d,A_27c,V5a0] :
( mem(V5a0,arr(A_27a,A_27b))
=> ! [V4relrec_27] :
( 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] : 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)) ) ) ).
fof(lameq_f1064,axiom,
! [A_27d,A_27c,A_27a,A_27b,V4relrec_27] :
( 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] : 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)) ) ).
fof(lameq_f1065,axiom,
! [A_27d,A_27a,A_27c,A_27b,V3a3] :
( mem(V3a3,A_27b)
=> ! [V1a1] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V0a0] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V2a2] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V4relrec_27] : 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)) ) ) ) ) ).
fof(lameq_f1066,axiom,
! [A_27c,A_27a,A_27d,A_27b,V2a2] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V0a0] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a3] : 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)) ) ) ) ).
fof(lameq_f1067,axiom,
! [A_27b,A_27c,A_27a,A_27d,V0a0] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a2] : 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) ) ) ).
fof(lameq_f1068,axiom,
! [A_27a,A_27b,A_27d,A_27c,V0a0] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1] : ap(f1068(A_27a,A_27b,A_27d,A_27c,V0a0),V1a1) = f1067(A_27b,A_27c,A_27a,A_27d,V0a0,V1a1) ) ).
fof(lameq_f1069,axiom,
! [A_27c,A_27d,A_27a,A_27b,V0a0] : ap(f1069(A_27c,A_27d,A_27a,A_27b),V0a0) = f1068(A_27a,A_27b,A_27d,A_27c,V0a0) ).
fof(ax_thm_2Einftree_2Erelrec__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> c_2Einftree_2Erelrec(A_27a,A_27b,A_27c,A_27d) = f1069(A_27c,A_27d,A_27a,A_27b) ) ) ) ) ).
fof(conj_thm_2Einftree_2Erelrec__rules,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0lf] :
( mem(V0lf,arr(A_27a,A_27b))
=> ! [V1nd] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a] :
( 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] :
( mem(V3lf,arr(A_27a,A_27b))
=> ! [V4nd] :
( mem(V4nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V5b] :
( mem(V5b,A_27c)
=> ! [V6df] :
( mem(V6df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V7g] :
( mem(V7g,arr(A_27d,A_27b))
=> ( ! [V8d] :
( 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))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Erelrec__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0relrec_27] :
( 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] :
( mem(V1lf,arr(A_27a,A_27b))
=> ! [V2nd] :
( mem(V2nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a] :
( 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] :
( mem(V4lf,arr(A_27a,A_27b))
=> ! [V5nd] :
( mem(V5nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V6b] :
( mem(V6b,A_27c)
=> ! [V7df] :
( mem(V7df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V8g] :
( mem(V8g,arr(A_27d,A_27b))
=> ( ! [V9d] :
( 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] :
( mem(V10a0,arr(A_27a,A_27b))
=> ! [V11a1] :
( mem(V11a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12a2] :
( mem(V12a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V13a3] :
( 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)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Erelrec__strongind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0relrec_27] :
( 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] :
( mem(V1lf,arr(A_27a,A_27b))
=> ! [V2nd] :
( mem(V2nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V3a] :
( 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] :
( mem(V4lf,arr(A_27a,A_27b))
=> ! [V5nd] :
( mem(V5nd,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V6b] :
( mem(V6b,A_27c)
=> ! [V7df] :
( mem(V7df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
=> ! [V8g] :
( mem(V8g,arr(A_27d,A_27b))
=> ( ! [V9d] :
( 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] :
( mem(V10a0,arr(A_27a,A_27b))
=> ! [V11a1] :
( mem(V11a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V12a2] :
( mem(V12a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V13a3] :
( 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)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Erelrec__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0a0] :
( mem(V0a0,arr(A_27a,A_27b))
=> ! [V1a1] :
( mem(V1a1,arr(A_27c,arr(arr(A_27d,A_27b),A_27b)))
=> ! [V2a2] :
( mem(V2a2,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d))
=> ! [V3a3] :
( 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] :
( mem(V4a,A_27a)
& V2a2 = ap(c_2Einftree_2EiLf(A_27a,A_27c,A_27d),V4a)
& V3a3 = ap(V0a0,V4a) )
| ? [V5b] :
( mem(V5b,A_27c)
& ? [V6df] :
( mem(V6df,arr(A_27d,ty_2Einftree_2Einftree(A_27a,A_27c,A_27d)))
& ? [V7g] :
( 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] :
( 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))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f1070,axiom,
! [A_27c,A_27b,A_27d,A_27a,V2t] :
( mem(V2t,ty_2Einftree_2Einftree(A_27b,A_27c,A_27d))
=> ! [V0lf] :
( mem(V0lf,arr(A_27b,A_27a))
=> ! [V1nd] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27a),A_27a)))
=> ! [V3r] : 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) ) ) ) ).
fof(ax_thm_2Einftree_2Einftree__rec__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0lf] :
( mem(V0lf,arr(A_27b,A_27a))
=> ! [V1nd] :
( mem(V1nd,arr(A_27c,arr(arr(A_27d,A_27a),A_27a)))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Einftree__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0lf] :
( mem(V0lf,arr(A_27a,A_27d))
=> ! [V1nd] :
( 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] :
( mem(V2f,arr(ty_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d))
& ! [V3a] :
( mem(V3a,A_27a)
=> ap(V2f,ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V3a)) = ap(V0lf,V3a) )
& ! [V4b] :
( mem(V4b,A_27b)
=> ! [V5d] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Einftree_2Einftree__case__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27d))
=> ! [V2f1] :
( 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] :
( mem(V3b,A_27b)
=> ! [V4d] :
( mem(V4d,arr(A_27c,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c)))
=> ! [V5f] :
( mem(V5f,arr(A_27a,A_27d))
=> ! [V6f1] :
( 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) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Einftree_2Einftree__nchotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0t] :
( mem(V0t,ty_2Einftree_2Einftree(A_27a,A_27b,A_27c))
=> ( ? [V1a] :
( mem(V1a,A_27a)
& V0t = ap(c_2Einftree_2EiLf(A_27a,A_27b,A_27c),V1a) )
| ? [V2b] :
( mem(V2b,A_27b)
& ? [V3d] :
( 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) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------