ITP001 Axioms: ITP101+5.ax
%------------------------------------------------------------------------------
% File : ITP101+5 : TPTP v9.0.0. 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 : patricia+2.ax [Gau20]
% : HL4101+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 208 ( 15 unt; 0 def)
% Number of atoms : 1311 ( 199 equ)
% Maximal formula atoms : 59 ( 6 avg)
% Number of connectives : 1129 ( 26 ~; 12 |; 139 &)
% ( 42 <=>; 910 =>; 0 <=; 0 <~>)
% Maximal formula depth : 63 ( 9 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 155 ( 155 usr; 33 con; 0-10 aty)
% Number of variables : 939 ( 923 !; 16 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Epatricia_2Eptree,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Epatricia_2Eptree(A0)) ) ).
fof(mem_c_2Epatricia_2EADD,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EADD(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a),ty_2Epatricia_2Eptree(A_27a)))) ) ).
fof(mem_c_2Epatricia_2EADD__LIST,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EADD__LIST(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a)),ty_2Epatricia_2Eptree(A_27a)))) ) ).
fof(mem_c_2Epatricia_2EBRANCH,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EBRANCH(A_27a),arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),ty_2Epatricia_2Eptree(A_27a))) ) ).
fof(mem_c_2Epatricia_2EBRANCHING__BIT,axiom,
mem(c_2Epatricia_2EBRANCHING__BIT,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Epatricia_2EBranch,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EBranch(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))))) ) ).
fof(mem_c_2Epatricia_2EDEPTH,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EDEPTH(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Enum_2Enum)) ) ).
fof(mem_c_2Epatricia_2EEVERY__LEAF,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EEVERY__LEAF(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Epatricia_2Eptree(A_27a),bool))) ) ).
fof(mem_c_2Epatricia_2EEXISTS__LEAF,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EEXISTS__LEAF(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Epatricia_2Eptree(A_27a),bool))) ) ).
fof(mem_c_2Epatricia_2EEmpty,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EEmpty(A_27a),ty_2Epatricia_2Eptree(A_27a)) ) ).
fof(mem_c_2Epatricia_2EFIND,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EFIND(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Enum_2Enum,A_27a))) ) ).
fof(mem_c_2Epatricia_2EINSERT__PTREE,axiom,
mem(c_2Epatricia_2EINSERT__PTREE,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),ty_2Epatricia_2Eptree(ty_2Eone_2Eone)))) ).
fof(mem_c_2Epatricia_2EIN__PTREE,axiom,
mem(c_2Epatricia_2EIN__PTREE,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),bool))) ).
fof(mem_c_2Epatricia_2EIS__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EIS__EMPTY(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),bool)) ) ).
fof(mem_c_2Epatricia_2EIS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EIS__PTREE(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),bool)) ) ).
fof(mem_c_2Epatricia_2EJOIN,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EJOIN(A_27a),arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)))),ty_2Epatricia_2Eptree(A_27a))) ) ).
fof(mem_c_2Epatricia_2EKEYS,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EKEYS(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Elist_2Elist(ty_2Enum_2Enum))) ) ).
fof(mem_c_2Epatricia_2ELeaf,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2ELeaf(A_27a),arr(ty_2Enum_2Enum,arr(A_27a,ty_2Epatricia_2Eptree(A_27a)))) ) ).
fof(mem_c_2Epatricia_2ENUMSET__OF__PTREE,axiom,
mem(c_2Epatricia_2ENUMSET__OF__PTREE,arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Epatricia_2EPEEK,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EPEEK(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a)))) ) ).
fof(mem_c_2Epatricia_2EPTREE__OF__NUMSET,axiom,
mem(c_2Epatricia_2EPTREE__OF__NUMSET,arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),arr(arr(ty_2Enum_2Enum,bool),ty_2Epatricia_2Eptree(ty_2Eone_2Eone)))) ).
fof(mem_c_2Epatricia_2EREMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2EREMOVE(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)))) ) ).
fof(mem_c_2Epatricia_2ESIZE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2ESIZE(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Enum_2Enum)) ) ).
fof(mem_c_2Epatricia_2ETRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),arr(arr(A_27b,A_27a),arr(ty_2Epatricia_2Eptree(A_27b),ty_2Epatricia_2Eptree(A_27a)))) ) ) ).
fof(mem_c_2Epatricia_2ETRAVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2ETRAVERSE(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Elist_2Elist(ty_2Enum_2Enum))) ) ).
fof(mem_c_2Epatricia_2ETRAVERSE__AUX,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2ETRAVERSE__AUX(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Elist_2Elist(ty_2Enum_2Enum),ty_2Elist_2Elist(ty_2Enum_2Enum)))) ) ).
fof(mem_c_2Epatricia_2EUNION__PTREE,axiom,
mem(c_2Epatricia_2EUNION__PTREE,arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),arr(ty_2Epatricia_2Eptree(ty_2Eone_2Eone),ty_2Epatricia_2Eptree(ty_2Eone_2Eone)))) ).
fof(mem_c_2Epatricia_2Eptree__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),arr(ty_2Epatricia_2Eptree(A_27a),arr(A_27b,arr(arr(ty_2Enum_2Enum,arr(A_27a,A_27b)),arr(arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))),A_27b))))) ) ) ).
fof(mem_c_2Epatricia_2Eptree__size,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epatricia_2Eptree__size(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Enum_2Enum))) ) ).
fof(lameq_f2521,axiom,
! [A_27a,V7a0] :
( mem(V7a0,ty_2Enum_2Enum)
=> ! [V8a1] : ap(f2521(A_27a,V7a0),V8a1) = ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),ap(c_2Enum_2ESUC,c_2Enum_2E0)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)),V7a0),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Enum_2Enum),V8a1),c_2Ebool_2EARB(ty_2Enum_2Enum)))),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))) ) ).
fof(lameq_f2522,axiom,
! [A_27a,V7a0] : ap(f2522(A_27a),V7a0) = f2521(A_27a,V7a0) ).
fof(lameq_f2523,axiom,
! [A_27a,V5a0] :
( mem(V5a0,ty_2Enum_2Enum)
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V6a1] : ap(f2523(A_27a,V5a0,V3a0_27),V6a1) = ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),V3a0_27),ap(ap(f2522(A_27a),V5a0),V6a1)) ) ) ).
fof(lameq_f2524,axiom,
! [A_27a,V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V5a0] : ap(f2524(A_27a,V3a0_27),V5a0) = ap(c_2Ebool_2E_3F(A_27a),f2523(A_27a,V5a0,V3a0_27)) ) ).
fof(lameq_f2525,axiom,
! [A_27a,V14a0] :
( mem(V14a0,ty_2Enum_2Enum)
=> ! [V15a1] :
( mem(V15a1,ty_2Enum_2Enum)
=> ! [V16a2] :
( mem(V16a2,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V17a3] : ap(f2525(A_27a,V14a0,V15a1,V16a2),V17a3) = ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)),V14a0),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Enum_2Enum),c_2Ebool_2EARB(A_27a)),V15a1))),ap(ap(c_2Eind__type_2EFCONS(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),V16a2),ap(ap(c_2Eind__type_2EFCONS(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),V17a3),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))))) ) ) ) ).
fof(lameq_f2526,axiom,
! [A_27a,V14a0] :
( mem(V14a0,ty_2Enum_2Enum)
=> ! [V15a1] :
( mem(V15a1,ty_2Enum_2Enum)
=> ! [V16a2] : ap(f2526(A_27a,V14a0,V15a1),V16a2) = f2525(A_27a,V14a0,V15a1,V16a2) ) ) ).
fof(lameq_f2527,axiom,
! [A_27a,V14a0] :
( mem(V14a0,ty_2Enum_2Enum)
=> ! [V15a1] : ap(f2527(A_27a,V14a0),V15a1) = f2526(A_27a,V14a0,V15a1) ) ).
fof(lameq_f2528,axiom,
! [A_27a,V14a0] : ap(f2528(A_27a),V14a0) = f2527(A_27a,V14a0) ).
fof(lameq_f2529,axiom,
! [A_27a,V11a1] :
( mem(V11a1,ty_2Enum_2Enum)
=> ! [V10a0] :
( mem(V10a0,ty_2Enum_2Enum)
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V12a2] :
( mem(V12a2,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V2_27ptree_27] :
( mem(V2_27ptree_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool))
=> ! [V13a3] : ap(f2529(A_27a,V11a1,V10a0,V3a0_27,V12a2,V2_27ptree_27),V13a3) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),V3a0_27),ap(ap(ap(ap(f2528(A_27a),V10a0),V11a1),V12a2),V13a3))),ap(ap(c_2Ebool_2E_2F_5C,ap(V2_27ptree_27,V12a2)),ap(V2_27ptree_27,V13a3))) ) ) ) ) ) ).
fof(lameq_f2530,axiom,
! [A_27a,V2_27ptree_27] :
( mem(V2_27ptree_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool))
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V10a0] :
( mem(V10a0,ty_2Enum_2Enum)
=> ! [V11a1] :
( mem(V11a1,ty_2Enum_2Enum)
=> ! [V12a2] : ap(f2530(A_27a,V2_27ptree_27,V3a0_27,V10a0,V11a1),V12a2) = ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),f2529(A_27a,V11a1,V10a0,V3a0_27,V12a2,V2_27ptree_27)) ) ) ) ) ).
fof(lameq_f2531,axiom,
! [A_27a,V10a0] :
( mem(V10a0,ty_2Enum_2Enum)
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V2_27ptree_27] :
( mem(V2_27ptree_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool))
=> ! [V11a1] : ap(f2531(A_27a,V10a0,V3a0_27,V2_27ptree_27),V11a1) = ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),f2530(A_27a,V2_27ptree_27,V3a0_27,V10a0,V11a1)) ) ) ) ).
fof(lameq_f2532,axiom,
! [A_27a,V2_27ptree_27] :
( mem(V2_27ptree_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool))
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V10a0] : ap(f2532(A_27a,V2_27ptree_27,V3a0_27),V10a0) = ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f2531(A_27a,V10a0,V3a0_27,V2_27ptree_27)) ) ) ).
fof(lameq_f2533,axiom,
! [A_27a,V2_27ptree_27] :
( mem(V2_27ptree_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool))
=> ! [V3a0_27] : ap(f2533(A_27a,V2_27ptree_27),V3a0_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),V3a0_27),ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),c_2Enum_2E0),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)),c_2Ebool_2EARB(ty_2Enum_2Enum)),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Enum_2Enum),c_2Ebool_2EARB(A_27a)),c_2Ebool_2EARB(ty_2Enum_2Enum)))),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))))),ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f2524(A_27a,V3a0_27))),ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f2532(A_27a,V2_27ptree_27,V3a0_27))))),ap(V2_27ptree_27,V3a0_27)) ) ).
fof(lameq_f2534,axiom,
! [A_27a,V1a0_27] :
( mem(V1a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))))
=> ! [V2_27ptree_27] : ap(f2534(A_27a,V1a0_27),V2_27ptree_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))),f2533(A_27a,V2_27ptree_27))),ap(V2_27ptree_27,V1a0_27)) ) ).
fof(lameq_f2535,axiom,
! [A_27a,V1a0_27] : ap(f2535(A_27a),V1a0_27) = ap(c_2Ebool_2E_21(arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),bool)),f2534(A_27a,V1a0_27)) ).
fof(ax_thm_2Epatricia_2Eptree__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Epatricia_2Eptree(A_27a),ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum)))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(A_27a,ty_2Enum_2Enum))),ty_2Epatricia_2Eptree(A_27a)),f2535(A_27a)),V0rep)) ) ) ).
fof(ax_thm_2Epatricia_2Eptree__case__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0v] :
( mem(V0v,A_27b)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V2f1] :
( mem(V2f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),c_2Epatricia_2EEmpty(A_27a)),V0v),V1f),V2f1) = V0v ) ) )
& ! [V3a0] :
( mem(V3a0,ty_2Enum_2Enum)
=> ! [V4a1] :
( mem(V4a1,A_27a)
=> ! [V5v] :
( mem(V5v,A_27b)
=> ! [V6f] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V7f1] :
( mem(V7f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),ap(ap(c_2Epatricia_2ELeaf(A_27a),V3a0),V4a1)),V5v),V6f),V7f1) = ap(ap(V6f,V3a0),V4a1) ) ) ) ) )
& ! [V8a0] :
( mem(V8a0,ty_2Enum_2Enum)
=> ! [V9a1] :
( mem(V9a1,ty_2Enum_2Enum)
=> ! [V10a2] :
( mem(V10a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V11a3] :
( mem(V11a3,ty_2Epatricia_2Eptree(A_27a))
=> ! [V12v] :
( mem(V12v,A_27b)
=> ! [V13f] :
( mem(V13f,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V14f1] :
( mem(V14f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V8a0),V9a1),V10a2),V11a3)),V12v),V13f),V14f1) = ap(ap(ap(ap(V14f1,V8a0),V9a1),V10a2),V11a3) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2Eptree__size__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ap(ap(c_2Epatricia_2Eptree__size(A_27a),V0f),c_2Epatricia_2EEmpty(A_27a)) = c_2Enum_2E0 )
& ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2a0] :
( mem(V2a0,ty_2Enum_2Enum)
=> ! [V3a1] :
( mem(V3a1,A_27a)
=> ap(ap(c_2Epatricia_2Eptree__size(A_27a),V1f),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2a0),V3a1)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2Earithmetic_2E_2B,V2a0),ap(V1f,V3a1))) ) ) )
& ! [V4f] :
( mem(V4f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V5a0] :
( mem(V5a0,ty_2Enum_2Enum)
=> ! [V6a1] :
( mem(V6a1,ty_2Enum_2Enum)
=> ! [V7a2] :
( mem(V7a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8a3] :
( mem(V8a3,ty_2Epatricia_2Eptree(A_27a))
=> ap(ap(c_2Epatricia_2Eptree__size(A_27a),V4f),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5a0),V6a1),V7a2),V8a3)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2Earithmetic_2E_2B,V5a0),ap(ap(c_2Earithmetic_2E_2B,V6a1),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Epatricia_2Eptree__size(A_27a),V4f),V7a2)),ap(ap(c_2Epatricia_2Eptree__size(A_27a),V4f),V8a3))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Edatatype__ptree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0ptree] :
( mem(V0ptree,arr(ty_2Epatricia_2Eptree(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,ty_2Epatricia_2Eptree(A_27a))),arr(arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))))),bool))))
=> p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(ap(V0ptree,c_2Epatricia_2EEmpty(A_27a)),c_2Epatricia_2ELeaf(A_27a)),c_2Epatricia_2EBranch(A_27a)))) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a0] :
( mem(V0a0,ty_2Enum_2Enum)
=> ! [V1a1] :
( mem(V1a1,A_27a)
=> ! [V2a0_27] :
( mem(V2a0_27,ty_2Enum_2Enum)
=> ! [V3a1_27] :
( mem(V3a1_27,A_27a)
=> ( ap(ap(c_2Epatricia_2ELeaf(A_27a),V0a0),V1a1) = ap(ap(c_2Epatricia_2ELeaf(A_27a),V2a0_27),V3a1_27)
<=> ( V0a0 = V2a0_27
& V1a1 = V3a1_27 ) ) ) ) ) )
& ! [V4a0] :
( mem(V4a0,ty_2Enum_2Enum)
=> ! [V5a1] :
( mem(V5a1,ty_2Enum_2Enum)
=> ! [V6a2] :
( mem(V6a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V7a3] :
( mem(V7a3,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8a0_27] :
( mem(V8a0_27,ty_2Enum_2Enum)
=> ! [V9a1_27] :
( mem(V9a1_27,ty_2Enum_2Enum)
=> ! [V10a2_27] :
( mem(V10a2_27,ty_2Epatricia_2Eptree(A_27a))
=> ! [V11a3_27] :
( mem(V11a3_27,ty_2Epatricia_2Eptree(A_27a))
=> ( ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V4a0),V5a1),V6a2),V7a3) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V8a0_27),V9a1_27),V10a2_27),V11a3_27)
<=> ( V4a0 = V8a0_27
& V5a1 = V9a1_27
& V6a2 = V10a2_27
& V7a3 = V11a3_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__distinct,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a1] :
( mem(V0a1,A_27a)
=> ! [V1a0] :
( mem(V1a0,ty_2Enum_2Enum)
=> c_2Epatricia_2EEmpty(A_27a) != ap(ap(c_2Epatricia_2ELeaf(A_27a),V1a0),V0a1) ) )
& ! [V2a3] :
( mem(V2a3,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3a2] :
( mem(V3a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4a1] :
( mem(V4a1,ty_2Enum_2Enum)
=> ! [V5a0] :
( mem(V5a0,ty_2Enum_2Enum)
=> c_2Epatricia_2EEmpty(A_27a) != ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5a0),V4a1),V3a2),V2a3) ) ) ) )
& ! [V6a3] :
( mem(V6a3,ty_2Epatricia_2Eptree(A_27a))
=> ! [V7a2] :
( mem(V7a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8a1_27] :
( mem(V8a1_27,ty_2Enum_2Enum)
=> ! [V9a1] :
( mem(V9a1,A_27a)
=> ! [V10a0_27] :
( mem(V10a0_27,ty_2Enum_2Enum)
=> ! [V11a0] :
( mem(V11a0,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2ELeaf(A_27a),V11a0),V9a1) != ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V10a0_27),V8a1_27),V7a2),V6a3) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__nchotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0pp] :
( mem(V0pp,ty_2Epatricia_2Eptree(A_27a))
=> ( V0pp = c_2Epatricia_2EEmpty(A_27a)
| ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& ? [V2a] :
( mem(V2a,A_27a)
& V0pp = ap(ap(c_2Epatricia_2ELeaf(A_27a),V1n),V2a) ) )
| ? [V3n0] :
( mem(V3n0,ty_2Enum_2Enum)
& ? [V4n] :
( mem(V4n,ty_2Enum_2Enum)
& ? [V5p] :
( mem(V5p,ty_2Epatricia_2Eptree(A_27a))
& ? [V6p0] :
( mem(V6p0,ty_2Epatricia_2Eptree(A_27a))
& V0pp = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3n0),V4n),V5p),V6p0) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f0] :
( mem(V0f0,A_27b)
=> ! [V1f1] :
( mem(V1f1,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V2f2] :
( mem(V2f2,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),arr(A_27b,arr(A_27b,A_27b)))))))
=> ? [V3fn] :
( mem(V3fn,arr(ty_2Epatricia_2Eptree(A_27a),A_27b))
& ap(V3fn,c_2Epatricia_2EEmpty(A_27a)) = V0f0
& ! [V4a0] :
( mem(V4a0,ty_2Enum_2Enum)
=> ! [V5a1] :
( mem(V5a1,A_27a)
=> ap(V3fn,ap(ap(c_2Epatricia_2ELeaf(A_27a),V4a0),V5a1)) = ap(ap(V1f1,V4a0),V5a1) ) )
& ! [V6a0] :
( mem(V6a0,ty_2Enum_2Enum)
=> ! [V7a1] :
( mem(V7a1,ty_2Enum_2Enum)
=> ! [V8a2] :
( mem(V8a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V9a3] :
( mem(V9a3,ty_2Epatricia_2Eptree(A_27a))
=> ap(V3fn,ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V6a0),V7a1),V8a2),V9a3)) = ap(ap(ap(ap(ap(ap(V2f2,V6a0),V7a1),V8a2),V9a3),ap(V3fn,V8a2)),ap(V3fn,V9a3)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epatricia_2Eptree(A_27a),bool))
=> ( ( p(ap(V0P,c_2Epatricia_2EEmpty(A_27a)))
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2a] :
( mem(V2a,A_27a)
=> p(ap(V0P,ap(ap(c_2Epatricia_2ELeaf(A_27a),V1n),V2a))) ) )
& ! [V3p] :
( mem(V3p,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4p0] :
( mem(V4p0,ty_2Epatricia_2Eptree(A_27a))
=> ( ( p(ap(V0P,V3p))
& p(ap(V0P,V4p0)) )
=> ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ! [V6n0] :
( mem(V6n0,ty_2Enum_2Enum)
=> p(ap(V0P,ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V6n0),V5n),V3p),V4p0))) ) ) ) ) ) )
=> ! [V7p] :
( mem(V7p,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,V7p)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__case__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0v_27] :
( mem(V0v_27,A_27b)
=> ! [V1f_27] :
( mem(V1f_27,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V2f1_27] :
( mem(V2f1_27,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ! [V3M] :
( mem(V3M,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4M_27] :
( mem(V4M_27,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5v] :
( mem(V5v,A_27b)
=> ! [V6f] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V7f1] :
( mem(V7f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ( ( V3M = V4M_27
& ( V4M_27 = c_2Epatricia_2EEmpty(A_27a)
=> V5v = V0v_27 )
& ! [V8a0] :
( mem(V8a0,ty_2Enum_2Enum)
=> ! [V9a1] :
( mem(V9a1,A_27a)
=> ( V4M_27 = ap(ap(c_2Epatricia_2ELeaf(A_27a),V8a0),V9a1)
=> ap(ap(V6f,V8a0),V9a1) = ap(ap(V1f_27,V8a0),V9a1) ) ) )
& ! [V10a0] :
( mem(V10a0,ty_2Enum_2Enum)
=> ! [V11a1] :
( mem(V11a1,ty_2Enum_2Enum)
=> ! [V12a2] :
( mem(V12a2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V13a3] :
( mem(V13a3,ty_2Epatricia_2Eptree(A_27a))
=> ( V4M_27 = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V10a0),V11a1),V12a2),V13a3)
=> ap(ap(ap(ap(V7f1,V10a0),V11a1),V12a2),V13a3) = ap(ap(ap(ap(V2f1_27,V10a0),V11a1),V12a2),V13a3) ) ) ) ) ) )
=> ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),V3M),V5v),V6f),V7f1) = ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),V4M_27),V0v_27),V1f_27),V2f1_27) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2Eptree__case__eq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1v] :
( mem(V1v,A_27b)
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,A_27b)))
=> ! [V3f1] :
( mem(V3f1,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),A_27b)))))
=> ! [V4v_27] :
( mem(V4v_27,A_27b)
=> ( ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,A_27b),V0x),V1v),V2f),V3f1) = V4v_27
<=> ( ( V0x = c_2Epatricia_2EEmpty(A_27a)
& V1v = V4v_27 )
| ? [V5n] :
( mem(V5n,ty_2Enum_2Enum)
& ? [V6a] :
( mem(V6a,A_27a)
& V0x = ap(ap(c_2Epatricia_2ELeaf(A_27a),V5n),V6a)
& ap(ap(V2f,V5n),V6a) = V4v_27 ) )
| ? [V7n0] :
( mem(V7n0,ty_2Enum_2Enum)
& ? [V8n] :
( mem(V8n,ty_2Enum_2Enum)
& ? [V9p] :
( mem(V9p,ty_2Epatricia_2Eptree(A_27a))
& ? [V10p0] :
( mem(V10p0,ty_2Epatricia_2Eptree(A_27a))
& V0x = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7n0),V8n),V9p),V10p0)
& ap(ap(ap(ap(V3f1,V7n0),V8n),V9p),V10p0) = V4v_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCHING__BIT__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)))
=> ( ! [V1p0] :
( mem(V1p0,ty_2Enum_2Enum)
=> ! [V2p1] :
( mem(V2p1,ty_2Enum_2Enum)
=> ( ( ~ ( ( p(ap(c_2Earithmetic_2EODD,V1p0))
<=> p(ap(c_2Earithmetic_2EEVEN,V2p1)) )
| V1p0 = V2p1 )
=> p(ap(ap(V0P,ap(c_2Earithmetic_2EDIV2,V1p0)),ap(c_2Earithmetic_2EDIV2,V2p1))) )
=> p(ap(ap(V0P,V1p0),V2p1)) ) ) )
=> ! [V3v] :
( mem(V3v,ty_2Enum_2Enum)
=> ! [V4v1] :
( mem(V4v1,ty_2Enum_2Enum)
=> p(ap(ap(V0P,V3v),V4v1)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCHING__BIT__def,axiom,
! [V0p1] :
( mem(V0p1,ty_2Enum_2Enum)
=> ! [V1p0] :
( mem(V1p0,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EBRANCHING__BIT,V1p0),V0p1) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(bool),ap(c_2Earithmetic_2EODD,V1p0)),ap(c_2Earithmetic_2EEVEN,V0p1))),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1p0),V0p1))),c_2Enum_2E0),ap(c_2Enum_2ESUC,ap(ap(c_2Epatricia_2EBRANCHING__BIT,ap(c_2Earithmetic_2EDIV2,V1p0)),ap(c_2Earithmetic_2EDIV2,V0p1)))) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Enum_2Enum,bool)))
=> ( ( ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> p(ap(ap(V0P,c_2Epatricia_2EEmpty(A_27a)),V1k)) )
& ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ! [V4k] :
( mem(V4k,ty_2Enum_2Enum)
=> p(ap(ap(V0P,ap(ap(c_2Epatricia_2ELeaf(A_27a),V2j),V3d)),V4k)) ) ) )
& ! [V5p] :
( mem(V5p,ty_2Enum_2Enum)
=> ! [V6m] :
( mem(V6m,ty_2Enum_2Enum)
=> ! [V7l] :
( mem(V7l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8r] :
( mem(V8r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V9k] :
( mem(V9k,ty_2Enum_2Enum)
=> ( p(ap(ap(V0P,ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Ebit_2EBIT,V6m),V9k)),V7l),V8r)),V9k))
=> p(ap(ap(V0P,ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5p),V6m),V7l),V8r)),V9k)) ) ) ) ) ) ) )
=> ! [V10v] :
( mem(V10v,ty_2Epatricia_2Eptree(A_27a))
=> ! [V11v1] :
( mem(V11v1,ty_2Enum_2Enum)
=> p(ap(ap(V0P,V10v),V11v1)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),c_2Epatricia_2EEmpty(A_27a)),V0k) = c_2Eoption_2ENONE(A_27a) )
& ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2j),V3d)),V1k) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1k),V2j)),ap(c_2Eoption_2ESOME(A_27a),V3d)),c_2Eoption_2ENONE(A_27a)) ) ) )
& ! [V4r] :
( mem(V4r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5p] :
( mem(V5p,ty_2Enum_2Enum)
=> ! [V6m] :
( mem(V6m,ty_2Enum_2Enum)
=> ! [V7l] :
( mem(V7l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8k] :
( mem(V8k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5p),V6m),V7l),V4r)),V8k) = ap(ap(c_2Epatricia_2EPEEK(A_27a),ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Ebit_2EBIT,V6m),V8k)),V7l),V4r)),V8k) ) ) ) ) ) ) ) ).
fof(lameq_f2536,axiom,
! [A_27a,V3t1] :
( mem(V3t1,ty_2Epatricia_2Eptree(A_27a))
=> ! [V0p0] :
( mem(V0p0,ty_2Enum_2Enum)
=> ! [V1t0] :
( mem(V1t0,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4m] : ap(f2536(A_27a,V3t1,V0p0,V1t0),V4m) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Ebit_2EBIT,V4m),V0p0)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),ap(ap(c_2Ebit_2EMOD__2EXP,V4m),V0p0)),V4m),V1t0),V3t1)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),ap(ap(c_2Ebit_2EMOD__2EXP,V4m),V0p0)),V4m),V3t1),V1t0)) ) ) ) ).
fof(ax_thm_2Epatricia_2EJOIN__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p0] :
( mem(V0p0,ty_2Enum_2Enum)
=> ! [V1t0] :
( mem(V1t0,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2p1] :
( mem(V2p1,ty_2Enum_2Enum)
=> ! [V3t1] :
( mem(V3t1,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2EJOIN(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)))),V0p0),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a))),V1t0),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)),V2p1),V3t1)))) = ap(ap(c_2Ebool_2ELET(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)),f2536(A_27a,V3t1,V0p0,V1t0)),ap(ap(c_2Epatricia_2EBRANCHING__BIT,V0p0),V2p1)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a),bool)))
=> ( ( ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2e] :
( mem(V2e,A_27a)
=> p(ap(ap(V0P,c_2Epatricia_2EEmpty(A_27a)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2e))) ) )
& ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ! [V4d] :
( mem(V4d,A_27a)
=> ! [V5k] :
( mem(V5k,ty_2Enum_2Enum)
=> ! [V6e] :
( mem(V6e,A_27a)
=> p(ap(ap(V0P,ap(ap(c_2Epatricia_2ELeaf(A_27a),V3j),V4d)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V5k),V6e))) ) ) ) )
& ! [V7p] :
( mem(V7p,ty_2Enum_2Enum)
=> ! [V8m] :
( mem(V8m,ty_2Enum_2Enum)
=> ! [V9l] :
( mem(V9l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V10r] :
( mem(V10r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V11k] :
( mem(V11k,ty_2Enum_2Enum)
=> ! [V12e] :
( mem(V12e,A_27a)
=> ( ( ( ( p(ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V8m),V11k),V7p))
& ~ p(ap(ap(c_2Ebit_2EBIT,V8m),V11k)) )
=> p(ap(ap(V0P,V10r),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V11k),V12e))) )
& ( ( p(ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V8m),V11k),V7p))
& p(ap(ap(c_2Ebit_2EBIT,V8m),V11k)) )
=> p(ap(ap(V0P,V9l),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V11k),V12e))) ) )
=> p(ap(ap(V0P,ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7p),V8m),V9l),V10r)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V11k),V12e))) ) ) ) ) ) ) ) )
=> ! [V13v] :
( mem(V13v,ty_2Epatricia_2Eptree(A_27a))
=> ! [V14v1] :
( mem(V14v1,ty_2Enum_2Enum)
=> ! [V15v2] :
( mem(V15v2,A_27a)
=> p(ap(ap(V0P,V13v),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V14v1),V15v2))) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ! [V1e] :
( mem(V1e,A_27a)
=> ap(ap(c_2Epatricia_2EADD(A_27a),c_2Epatricia_2EEmpty(A_27a)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V0k),V1e)) = ap(ap(c_2Epatricia_2ELeaf(A_27a),V0k),V1e) ) )
& ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ! [V4e] :
( mem(V4e,A_27a)
=> ! [V5d] :
( mem(V5d,A_27a)
=> ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V3j),V5d)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V2k),V4e)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V3j),V2k)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2k),V4e)),ap(c_2Epatricia_2EJOIN(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)))),V2k),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a))),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2k),V4e)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)),V3j),ap(ap(c_2Epatricia_2ELeaf(A_27a),V3j),V5d)))))) ) ) ) )
& ! [V6r] :
( mem(V6r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V7p] :
( mem(V7p,ty_2Enum_2Enum)
=> ! [V8m] :
( mem(V8m,ty_2Enum_2Enum)
=> ! [V9l] :
( mem(V9l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V10k] :
( mem(V10k,ty_2Enum_2Enum)
=> ! [V11e] :
( mem(V11e,A_27a)
=> ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7p),V8m),V9l),V6r)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V10k),V11e)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V8m),V10k),V7p)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Ebit_2EBIT,V8m),V10k)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7p),V8m),ap(ap(c_2Epatricia_2EADD(A_27a),V9l),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V10k),V11e))),V6r)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7p),V8m),V9l),ap(ap(c_2Epatricia_2EADD(A_27a),V6r),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V10k),V11e))))),ap(c_2Epatricia_2EJOIN(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)))),V10k),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a))),ap(ap(c_2Epatricia_2ELeaf(A_27a),V10k),V11e)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(A_27a)),V7p),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V7p),V8m),V9l),V6r)))))) ) ) ) ) ) ) ) ) ).
fof(lameq_f2537,axiom,
! [A_27a,V0R] : ap(f2537(A_27a),V0R) = ap(c_2Erelation_2EWF(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))))),V0R) ).
fof(lameq_f2538,axiom,
! [A_27a,V11v30] :
( mem(V11v30,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V12v31] : ap(f2538(A_27a,V11v30,V5m,V3p,V9v18,V10v19),V12v31) = ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3p),V5m),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v18),V10v19)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V11v30),V12v31))) ) ) ) ) ) ).
fof(lameq_f2539,axiom,
! [A_27a,V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V11v30] : ap(f2539(A_27a,V5m,V3p,V9v18,V10v19),V11v30) = f2538(A_27a,V11v30,V5m,V3p,V9v18,V10v19) ) ) ) ) ).
fof(lameq_f2540,axiom,
! [A_27a,V14v33] :
( mem(V14v33,ty_2Enum_2Enum)
=> ! [V13v32] :
( mem(V13v32,ty_2Enum_2Enum)
=> ! [V15v34] :
( mem(V15v34,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V16v35] : ap(f2540(A_27a,V14v33,V13v32,V15v34,V5m,V3p,V9v18,V10v19),V16v35) = ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3p),V5m),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v18),V10v19)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V13v32),V14v33),V15v34),V16v35))) ) ) ) ) ) ) ) ).
fof(lameq_f2541,axiom,
! [A_27a,V14v33] :
( mem(V14v33,ty_2Enum_2Enum)
=> ! [V13v32] :
( mem(V13v32,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V15v34] : ap(f2541(A_27a,V14v33,V13v32,V5m,V3p,V9v18,V10v19),V15v34) = f2540(A_27a,V14v33,V13v32,V15v34,V5m,V3p,V9v18,V10v19) ) ) ) ) ) ) ).
fof(lameq_f2542,axiom,
! [A_27a,V13v32] :
( mem(V13v32,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V14v33] : ap(f2542(A_27a,V13v32,V5m,V3p,V9v18,V10v19),V14v33) = f2541(A_27a,V14v33,V13v32,V5m,V3p,V9v18,V10v19) ) ) ) ) ) ).
fof(lameq_f2543,axiom,
! [A_27a,V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] :
( mem(V10v19,A_27a)
=> ! [V13v32] : ap(f2543(A_27a,V5m,V3p,V9v18,V10v19),V13v32) = f2542(A_27a,V13v32,V5m,V3p,V9v18,V10v19) ) ) ) ) ).
fof(lameq_f2544,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] :
( mem(V9v18,ty_2Enum_2Enum)
=> ! [V10v19] : ap(f2544(A_27a,V8t,V5m,V3p,V9v18),V10v19) = ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,ty_2Epatricia_2Eptree(A_27a)),V8t),ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v18),V10v19))),f2539(A_27a,V5m,V3p,V9v18,V10v19)),f2543(A_27a,V5m,V3p,V9v18,V10v19)) ) ) ) ) ).
fof(lameq_f2545,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V9v18] : ap(f2545(A_27a,V8t,V5m,V3p),V9v18) = f2544(A_27a,V8t,V5m,V3p,V9v18) ) ) ) ).
fof(lameq_f2546,axiom,
! [A_27a,V21v42] :
( mem(V21v42,ty_2Enum_2Enum)
=> ! [V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V22v43] : ap(f2546(A_27a,V21v42,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V22v43) = ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3p),V5m),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v20),V18v21),V19v22),V20v23)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V21v42),V22v43))) ) ) ) ) ) ) ) ).
fof(lameq_f2547,axiom,
! [A_27a,V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V21v42] : ap(f2547(A_27a,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V21v42) = f2546(A_27a,V21v42,V20v23,V18v21,V17v20,V19v22,V3p,V5m) ) ) ) ) ) ) ).
fof(lameq_f2548,axiom,
! [A_27a,V24v45] :
( mem(V24v45,ty_2Enum_2Enum)
=> ! [V23v44] :
( mem(V23v44,ty_2Enum_2Enum)
=> ! [V25v46] :
( mem(V25v46,ty_2Epatricia_2Eptree(A_27a))
=> ! [V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V26v47] : ap(f2548(A_27a,V24v45,V23v44,V25v46,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V26v47) = ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3p),V5m),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v20),V18v21),V19v22),V20v23)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V23v44),V24v45),V25v46),V26v47))) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f2549,axiom,
! [A_27a,V24v45] :
( mem(V24v45,ty_2Enum_2Enum)
=> ! [V23v44] :
( mem(V23v44,ty_2Enum_2Enum)
=> ! [V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V25v46] : ap(f2549(A_27a,V24v45,V23v44,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V25v46) = f2548(A_27a,V24v45,V23v44,V25v46,V20v23,V18v21,V17v20,V19v22,V3p,V5m) ) ) ) ) ) ) ) ) ).
fof(lameq_f2550,axiom,
! [A_27a,V23v44] :
( mem(V23v44,ty_2Enum_2Enum)
=> ! [V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V24v45] : ap(f2550(A_27a,V23v44,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V24v45) = f2549(A_27a,V24v45,V23v44,V20v23,V18v21,V17v20,V19v22,V3p,V5m) ) ) ) ) ) ) ) ).
fof(lameq_f2551,axiom,
! [A_27a,V20v23] :
( mem(V20v23,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V23v44] : ap(f2551(A_27a,V20v23,V18v21,V17v20,V19v22,V3p,V5m),V23v44) = f2550(A_27a,V23v44,V20v23,V18v21,V17v20,V19v22,V3p,V5m) ) ) ) ) ) ) ).
fof(lameq_f2552,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V19v22] :
( mem(V19v22,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V20v23] : ap(f2552(A_27a,V8t,V18v21,V17v20,V19v22,V3p,V5m),V20v23) = ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,ty_2Epatricia_2Eptree(A_27a)),V8t),ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v20),V18v21),V19v22),V20v23))),f2547(A_27a,V20v23,V18v21,V17v20,V19v22,V3p,V5m)),f2551(A_27a,V20v23,V18v21,V17v20,V19v22,V3p,V5m)) ) ) ) ) ) ) ).
fof(lameq_f2553,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V18v21] :
( mem(V18v21,ty_2Enum_2Enum)
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V19v22] : ap(f2553(A_27a,V8t,V18v21,V17v20,V3p,V5m),V19v22) = f2552(A_27a,V8t,V18v21,V17v20,V19v22,V3p,V5m) ) ) ) ) ) ).
fof(lameq_f2554,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V17v20] :
( mem(V17v20,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V18v21] : ap(f2554(A_27a,V8t,V17v20,V3p,V5m),V18v21) = f2553(A_27a,V8t,V18v21,V17v20,V3p,V5m) ) ) ) ) ).
fof(lameq_f2555,axiom,
! [A_27a,V8t] :
( mem(V8t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V17v20] : ap(f2555(A_27a,V8t,V3p,V5m),V17v20) = f2554(A_27a,V8t,V17v20,V3p,V5m) ) ) ) ).
fof(lameq_f2556,axiom,
! [A_27a,V7v4] :
( mem(V7v4,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V8t] : ap(f2556(A_27a,V7v4,V3p,V5m),V8t) = ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,ty_2Epatricia_2Eptree(A_27a)),V7v4),ap(c_2Ecombin_2EI(ty_2Epatricia_2Eptree(A_27a)),V8t)),f2545(A_27a,V8t,V5m,V3p)),f2555(A_27a,V8t,V3p,V5m)) ) ) ) ).
fof(lameq_f2557,axiom,
! [A_27a,V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V7v4] : ap(f2557(A_27a,V3p,V5m),V7v4) = f2556(A_27a,V7v4,V3p,V5m) ) ) ).
fof(lameq_f2558,axiom,
! [A_27a,V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V6v3] : ap(f2558(A_27a,V5m,V3p),V6v3) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V6v3),f2557(A_27a,V3p,V5m)) ) ) ).
fof(lameq_f2559,axiom,
! [A_27a,V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V5m] : ap(f2559(A_27a,V3p),V5m) = f2558(A_27a,V5m,V3p) ) ).
fof(lameq_f2560,axiom,
! [A_27a,V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V4v1] : ap(f2560(A_27a,V3p),V4v1) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epatricia_2Eptree(A_27a),ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V4v1),f2559(A_27a,V3p)) ) ).
fof(lameq_f2561,axiom,
! [A_27a,V3p] : ap(f2561(A_27a),V3p) = f2560(A_27a,V3p) ).
fof(lameq_f2562,axiom,
! [A_27a,V2a] : ap(f2562(A_27a),V2a) = ap(ap(c_2Epair_2Epair__CASE(ty_2Epatricia_2Eptree(A_27a),ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V2a),f2561(A_27a)) ).
fof(ax_thm_2Epatricia_2EBRANCH__primitive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epatricia_2EBRANCH(A_27a) = ap(ap(c_2Erelation_2EWFREC(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),ty_2Epatricia_2Eptree(A_27a)),ap(c_2Emin_2E_40(arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),bool))),f2537(A_27a))),k(arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),ty_2Epatricia_2Eptree(A_27a)),f2562(A_27a))) ) ).
fof(conj_thm_2Epatricia_2EBRANCH__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),bool))
=> ( ( ! [V1p] :
( mem(V1p,ty_2Enum_2Enum)
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3t] :
( mem(V3t,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V1p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V2m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EEmpty(A_27a)),V3t))))) ) ) )
& ! [V4p] :
( mem(V4p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6v6] :
( mem(V6v6,ty_2Enum_2Enum)
=> ! [V7v7] :
( mem(V7v7,A_27a)
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V4p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V5m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V6v6),V7v7)),c_2Epatricia_2EEmpty(A_27a)))))) ) ) ) )
& ! [V8p] :
( mem(V8p,ty_2Enum_2Enum)
=> ! [V9m] :
( mem(V9m,ty_2Enum_2Enum)
=> ! [V10v8] :
( mem(V10v8,ty_2Enum_2Enum)
=> ! [V11v9] :
( mem(V11v9,ty_2Enum_2Enum)
=> ! [V12v10] :
( mem(V12v10,ty_2Epatricia_2Eptree(A_27a))
=> ! [V13v11] :
( mem(V13v11,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V8p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V9m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V10v8),V11v9),V12v10),V13v11)),c_2Epatricia_2EEmpty(A_27a)))))) ) ) ) ) ) )
& ! [V14p] :
( mem(V14p,ty_2Enum_2Enum)
=> ! [V15m] :
( mem(V15m,ty_2Enum_2Enum)
=> ! [V16v12] :
( mem(V16v12,ty_2Enum_2Enum)
=> ! [V17v13] :
( mem(V17v13,A_27a)
=> ! [V18v24] :
( mem(V18v24,ty_2Enum_2Enum)
=> ! [V19v25] :
( mem(V19v25,A_27a)
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V14p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V15m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V16v12),V17v13)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V18v24),V19v25)))))) ) ) ) ) ) )
& ! [V20p] :
( mem(V20p,ty_2Enum_2Enum)
=> ! [V21m] :
( mem(V21m,ty_2Enum_2Enum)
=> ! [V22v12] :
( mem(V22v12,ty_2Enum_2Enum)
=> ! [V23v13] :
( mem(V23v13,A_27a)
=> ! [V24v26] :
( mem(V24v26,ty_2Enum_2Enum)
=> ! [V25v27] :
( mem(V25v27,ty_2Enum_2Enum)
=> ! [V26v28] :
( mem(V26v28,ty_2Epatricia_2Eptree(A_27a))
=> ! [V27v29] :
( mem(V27v29,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V20p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V21m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V22v12),V23v13)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V24v26),V25v27),V26v28),V27v29)))))) ) ) ) ) ) ) ) )
& ! [V28p] :
( mem(V28p,ty_2Enum_2Enum)
=> ! [V29m] :
( mem(V29m,ty_2Enum_2Enum)
=> ! [V30v14] :
( mem(V30v14,ty_2Enum_2Enum)
=> ! [V31v15] :
( mem(V31v15,ty_2Enum_2Enum)
=> ! [V32v16] :
( mem(V32v16,ty_2Epatricia_2Eptree(A_27a))
=> ! [V33v17] :
( mem(V33v17,ty_2Epatricia_2Eptree(A_27a))
=> ! [V34v36] :
( mem(V34v36,ty_2Enum_2Enum)
=> ! [V35v37] :
( mem(V35v37,A_27a)
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V28p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V29m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V30v14),V31v15),V32v16),V33v17)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V34v36),V35v37)))))) ) ) ) ) ) ) ) )
& ! [V36p] :
( mem(V36p,ty_2Enum_2Enum)
=> ! [V37m] :
( mem(V37m,ty_2Enum_2Enum)
=> ! [V38v14] :
( mem(V38v14,ty_2Enum_2Enum)
=> ! [V39v15] :
( mem(V39v15,ty_2Enum_2Enum)
=> ! [V40v16] :
( mem(V40v16,ty_2Epatricia_2Eptree(A_27a))
=> ! [V41v17] :
( mem(V41v17,ty_2Epatricia_2Eptree(A_27a))
=> ! [V42v38] :
( mem(V42v38,ty_2Enum_2Enum)
=> ! [V43v39] :
( mem(V43v39,ty_2Enum_2Enum)
=> ! [V44v40] :
( mem(V44v40,ty_2Epatricia_2Eptree(A_27a))
=> ! [V45v41] :
( mem(V45v41,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V36p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V37m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V38v14),V39v15),V40v16),V41v17)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V42v38),V43v39),V44v40),V45v41)))))) ) ) ) ) ) ) ) ) ) ) )
=> ! [V46v] :
( mem(V46v,ty_2Enum_2Enum)
=> ! [V47v1] :
( mem(V47v1,ty_2Enum_2Enum)
=> ! [V48v2] :
( mem(V48v2,ty_2Epatricia_2Eptree(A_27a))
=> ! [V49v3] :
( mem(V49v3,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V46v),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V47v1),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V48v2),V49v3))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCH__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2t] :
( mem(V2t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3v6] :
( mem(V3v6,ty_2Enum_2Enum)
=> ! [V4v7] :
( mem(V4v7,A_27a)
=> ! [V5v8] :
( mem(V5v8,ty_2Enum_2Enum)
=> ! [V6v9] :
( mem(V6v9,ty_2Enum_2Enum)
=> ! [V7v10] :
( mem(V7v10,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8v11] :
( mem(V8v11,ty_2Epatricia_2Eptree(A_27a))
=> ! [V9v12] :
( mem(V9v12,ty_2Enum_2Enum)
=> ! [V10v13] :
( mem(V10v13,A_27a)
=> ! [V11v24] :
( mem(V11v24,ty_2Enum_2Enum)
=> ! [V12v25] :
( mem(V12v25,A_27a)
=> ! [V13v26] :
( mem(V13v26,ty_2Enum_2Enum)
=> ! [V14v27] :
( mem(V14v27,ty_2Enum_2Enum)
=> ! [V15v28] :
( mem(V15v28,ty_2Epatricia_2Eptree(A_27a))
=> ! [V16v29] :
( mem(V16v29,ty_2Epatricia_2Eptree(A_27a))
=> ! [V17v14] :
( mem(V17v14,ty_2Enum_2Enum)
=> ! [V18v15] :
( mem(V18v15,ty_2Enum_2Enum)
=> ! [V19v16] :
( mem(V19v16,ty_2Epatricia_2Eptree(A_27a))
=> ! [V20v17] :
( mem(V20v17,ty_2Epatricia_2Eptree(A_27a))
=> ! [V21v36] :
( mem(V21v36,ty_2Enum_2Enum)
=> ! [V22v37] :
( mem(V22v37,A_27a)
=> ! [V23v38] :
( mem(V23v38,ty_2Enum_2Enum)
=> ! [V24v39] :
( mem(V24v39,ty_2Enum_2Enum)
=> ! [V25v40] :
( mem(V25v40,ty_2Epatricia_2Eptree(A_27a))
=> ! [V26v41] :
( mem(V26v41,ty_2Epatricia_2Eptree(A_27a))
=> ( ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EEmpty(A_27a)),V2t)))) = V2t
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V3v6),V4v7)),c_2Epatricia_2EEmpty(A_27a))))) = ap(ap(c_2Epatricia_2ELeaf(A_27a),V3v6),V4v7)
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5v8),V6v9),V7v10),V8v11)),c_2Epatricia_2EEmpty(A_27a))))) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5v8),V6v9),V7v10),V8v11)
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v12),V10v13)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V11v24),V12v25))))) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v12),V10v13)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V11v24),V12v25))
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v12),V10v13)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V13v26),V14v27),V15v28),V16v29))))) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),ap(ap(c_2Epatricia_2ELeaf(A_27a),V9v12),V10v13)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V13v26),V14v27),V15v28),V16v29))
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v14),V18v15),V19v16),V20v17)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V21v36),V22v37))))) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v14),V18v15),V19v16),V20v17)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V21v36),V22v37))
& ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v14),V18v15),V19v16),V20v17)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V23v38),V24v39),V25v40),V26v41))))) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V17v14),V18v15),V19v16),V20v17)),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V23v38),V24v39),V25v40),V26v41)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EREMOVE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),c_2Epatricia_2EEmpty(A_27a)),V0k) = c_2Epatricia_2EEmpty(A_27a) )
& ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ! [V3k] :
( mem(V3k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V1j),V2d)),V3k) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1j),V3k)),c_2Epatricia_2EEmpty(A_27a)),ap(ap(c_2Epatricia_2ELeaf(A_27a),V1j),V2d)) ) ) )
& ! [V4p] :
( mem(V4p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6l] :
( mem(V6l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V7r] :
( mem(V7r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8k] :
( mem(V8k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V4p),V5m),V6l),V7r)),V8k) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V5m),V8k),V4p)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Ebit_2EBIT,V5m),V8k)),ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V4p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V5m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V6l),V8k)),V7r))))),ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V4p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V5m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V6l),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V7r),V8k))))))),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V4p),V5m),V6l),V7r)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2ETRAVERSE__AUX__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,ty_2Elist_2Elist(ty_2Enum_2Enum))
=> ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),c_2Epatricia_2EEmpty(A_27a)),V0a) = V0a )
& ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ! [V3a] :
( mem(V3a,ty_2Elist_2Elist(ty_2Enum_2Enum))
=> ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V1k),V2d)),V3a) = ap(ap(c_2Elist_2ECONS(ty_2Enum_2Enum),V1k),V3a) ) ) )
& ! [V4p] :
( mem(V4p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6l] :
( mem(V6l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V7r] :
( mem(V7r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8a] :
( mem(V8a,ty_2Elist_2Elist(ty_2Enum_2Enum))
=> ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V4p),V5m),V6l),V7r)),V8a) = ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),V6l),ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),V7r),V8a)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2ETRAVERSE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ap(c_2Epatricia_2ETRAVERSE(A_27a),c_2Epatricia_2EEmpty(A_27a)) = c_2Elist_2ENIL(ty_2Enum_2Enum)
& ! [V0j] :
( mem(V0j,ty_2Enum_2Enum)
=> ! [V1d] :
( mem(V1d,A_27a)
=> ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V0j),V1d)) = ap(ap(c_2Elist_2ECONS(ty_2Enum_2Enum),V0j),c_2Elist_2ENIL(ty_2Enum_2Enum)) ) )
& ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4l] :
( mem(V4l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5r] :
( mem(V5r,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V2p),V3m),V4l),V5r)) = ap(ap(c_2Elist_2EAPPEND(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V4l)),ap(c_2Epatricia_2ETRAVERSE(A_27a),V5r)) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EKEYS__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2EKEYS(A_27a),V0t) = ap(ap(c_2Esorting_2EQSORT(ty_2Enum_2Enum),c_2Eprim__rec_2E_3C),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)) ) ) ).
fof(ax_thm_2Epatricia_2ETRANSFORM__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27b,A_27a))
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),V0f),c_2Epatricia_2EEmpty(A_27b)) = c_2Epatricia_2EEmpty(A_27a) )
& ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27b)
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),V1f),ap(ap(c_2Epatricia_2ELeaf(A_27b),V2j),V3d)) = ap(ap(c_2Epatricia_2ELeaf(A_27a),V2j),ap(V1f,V3d)) ) ) )
& ! [V4f] :
( mem(V4f,arr(A_27b,A_27a))
=> ! [V5p] :
( mem(V5p,ty_2Enum_2Enum)
=> ! [V6m] :
( mem(V6m,ty_2Enum_2Enum)
=> ! [V7l] :
( mem(V7l,ty_2Epatricia_2Eptree(A_27b))
=> ! [V8r] :
( mem(V8r,ty_2Epatricia_2Eptree(A_27b))
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),V4f),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27b),V5p),V6m),V7l),V8r)) = ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5p),V6m),ap(ap(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),V4f),V7l)),ap(ap(c_2Epatricia_2ETRANSFORM(A_27a,A_27b),V4f),V8r)) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EEVERY__LEAF__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),c_2Epatricia_2EEmpty(A_27a)))
<=> $true ) )
& ! [V1P] :
( mem(V1P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V1P),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2j),V3d)))
<=> p(ap(ap(V1P,V2j),V3d)) ) ) ) )
& ! [V4P] :
( mem(V4P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5p] :
( mem(V5p,ty_2Enum_2Enum)
=> ! [V6m] :
( mem(V6m,ty_2Enum_2Enum)
=> ! [V7l] :
( mem(V7l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8r] :
( mem(V8r,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V4P),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5p),V6m),V7l),V8r)))
<=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V4P),V7l))
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V4P),V8r)) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EEXISTS__LEAF__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Epatricia_2EEXISTS__LEAF(A_27a),V0P),c_2Epatricia_2EEmpty(A_27a)))
<=> $false ) )
& ! [V1P] :
( mem(V1P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ( p(ap(ap(c_2Epatricia_2EEXISTS__LEAF(A_27a),V1P),ap(ap(c_2Epatricia_2ELeaf(A_27a),V2j),V3d)))
<=> p(ap(ap(V1P,V2j),V3d)) ) ) ) )
& ! [V4P] :
( mem(V4P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5p] :
( mem(V5p,ty_2Enum_2Enum)
=> ! [V6m] :
( mem(V6m,ty_2Enum_2Enum)
=> ! [V7l] :
( mem(V7l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V8r] :
( mem(V8r,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(ap(c_2Epatricia_2EEXISTS__LEAF(A_27a),V4P),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V5p),V6m),V7l),V8r)))
<=> ( p(ap(ap(c_2Epatricia_2EEXISTS__LEAF(A_27a),V4P),V7l))
| p(ap(ap(c_2Epatricia_2EEXISTS__LEAF(A_27a),V4P),V8r)) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2ESIZE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2ESIZE(A_27a),V0t) = ap(c_2Elist_2ELENGTH(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)) ) ) ).
fof(ax_thm_2Epatricia_2EDEPTH__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ap(c_2Epatricia_2EDEPTH(A_27a),c_2Epatricia_2EEmpty(A_27a)) = c_2Enum_2E0
& ! [V0j] :
( mem(V0j,ty_2Enum_2Enum)
=> ! [V1d] :
( mem(V1d,A_27a)
=> ap(c_2Epatricia_2EDEPTH(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V0j),V1d)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) )
& ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4l] :
( mem(V4l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5r] :
( mem(V5r,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2EDEPTH(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V2p),V3m),V4l),V5r)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2Earithmetic_2EMAX,ap(c_2Epatricia_2EDEPTH(A_27a),V4l)),ap(c_2Epatricia_2EDEPTH(A_27a),V5r))) ) ) ) ) ) ) ).
fof(lameq_f2563,axiom,
! [A_27a,V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V6k] : ap(f2563(A_27a,V2p,V3m),V6k) = k(A_27a,ap(ap(c_2Ebool_2E_2F_5C,ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V3m),V6k),V2p)),ap(ap(c_2Ebit_2EBIT,V3m),V6k))) ) ) ).
fof(lameq_f2564,axiom,
! [A_27a,V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V8k] : ap(f2564(A_27a,V2p,V3m),V8k) = k(A_27a,ap(ap(c_2Ebool_2E_2F_5C,ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V3m),V8k),V2p)),ap(c_2Ebool_2E_7E,ap(ap(c_2Ebit_2EBIT,V3m),V8k)))) ) ) ).
fof(ax_thm_2Epatricia_2EIS__PTREE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),c_2Epatricia_2EEmpty(A_27a)))
<=> $true )
& ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ! [V1d] :
( mem(V1d,A_27a)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V0k),V1d)))
<=> $true ) ) )
& ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4l] :
( mem(V4l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5r] :
( mem(V5r,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V2p),V3m),V4l),V5r)))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2p),ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3m)))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V4l))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V5r))
& V4l != c_2Epatricia_2EEmpty(A_27a)
& V5r != c_2Epatricia_2EEmpty(A_27a)
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),f2563(A_27a,V2p,V3m)),V4l))
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),f2564(A_27a,V2p,V3m)),V5r)) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EIN__PTREE__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V0n),V1t))
<=> p(ap(c_2Eoption_2EIS__SOME(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPEEK(ty_2Eone_2Eone),V1t),V0n))) ) ) ) ).
fof(ax_thm_2Epatricia_2EINSERT__PTREE__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ap(ap(c_2Epatricia_2EINSERT__PTREE,V0n),V1t) = ap(ap(c_2Epatricia_2EADD(ty_2Eone_2Eone),V1t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Eone_2Eone),V0n),c_2Eone_2Eone)) ) ) ).
fof(ax_thm_2Epatricia_2EPTREE__OF__NUMSET__def,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s) = ap(ap(ap(c_2Elist_2EFOLDL(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(ty_2Eone_2Eone)),ap(c_2Ecombin_2EC(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(ty_2Eone_2Eone),ty_2Epatricia_2Eptree(ty_2Eone_2Eone)),c_2Epatricia_2EINSERT__PTREE)),V0t),ap(c_2Elist_2ESET__TO__LIST(ty_2Enum_2Enum),V1s)) ) ) ).
fof(ax_thm_2Epatricia_2ENUMSET__OF__PTREE__def,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t) = ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t)) ) ).
fof(ax_thm_2Epatricia_2EUNION__PTREE__def,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),V1t2) = ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t1),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V1t2)) ) ) ).
fof(lameq_f2565,axiom,
! [A_27a,V0R] : ap(f2565(A_27a),V0R) = ap(c_2Erelation_2EWF(ty_2Epatricia_2Eptree(A_27a)),V0R) ).
fof(lameq_f2566,axiom,
! [A_27a,V2a] : ap(f2566(A_27a),V2a) = ap(ap(ap(ap(c_2Epatricia_2Eptree__CASE(A_27a,bool),V2a),ap(c_2Ecombin_2EI(bool),c_2Ebool_2ET)),k(ty_2Enum_2Enum,k(A_27a,ap(c_2Ecombin_2EI(bool),c_2Ebool_2EF)))),k(ty_2Enum_2Enum,k(ty_2Enum_2Enum,k(ty_2Epatricia_2Eptree(A_27a),k(ty_2Epatricia_2Eptree(A_27a),ap(c_2Ecombin_2EI(bool),c_2Ebool_2EF)))))) ).
fof(ax_thm_2Epatricia_2EIS__EMPTY__primitive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epatricia_2EIS__EMPTY(A_27a) = ap(ap(c_2Erelation_2EWFREC(ty_2Epatricia_2Eptree(A_27a),bool),ap(c_2Emin_2E_40(arr(ty_2Epatricia_2Eptree(A_27a),arr(ty_2Epatricia_2Eptree(A_27a),bool))),f2565(A_27a))),k(arr(ty_2Epatricia_2Eptree(A_27a),bool),f2566(A_27a))) ) ).
fof(conj_thm_2Epatricia_2EIS__EMPTY__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epatricia_2Eptree(A_27a),bool))
=> ( ( p(ap(V0P,c_2Epatricia_2EEmpty(A_27a)))
& ! [V1v] :
( mem(V1v,ty_2Enum_2Enum)
=> ! [V2v1] :
( mem(V2v1,A_27a)
=> p(ap(V0P,ap(ap(c_2Epatricia_2ELeaf(A_27a),V1v),V2v1))) ) )
& ! [V3v2] :
( mem(V3v2,ty_2Enum_2Enum)
=> ! [V4v3] :
( mem(V4v3,ty_2Enum_2Enum)
=> ! [V5v4] :
( mem(V5v4,ty_2Epatricia_2Eptree(A_27a))
=> ! [V6v5] :
( mem(V6v5,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V3v2),V4v3),V5v4),V6v5))) ) ) ) ) )
=> ! [V7v] :
( mem(V7v,ty_2Epatricia_2Eptree(A_27a))
=> p(ap(V0P,V7v)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIS__EMPTY__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0v] :
( mem(V0v,ty_2Enum_2Enum)
=> ! [V1v1] :
( mem(V1v1,A_27a)
=> ! [V2v2] :
( mem(V2v2,ty_2Enum_2Enum)
=> ! [V3v3] :
( mem(V3v3,ty_2Enum_2Enum)
=> ! [V4v4] :
( mem(V4v4,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5v5] :
( mem(V5v5,ty_2Epatricia_2Eptree(A_27a))
=> ( ( p(ap(c_2Epatricia_2EIS__EMPTY(A_27a),c_2Epatricia_2EEmpty(A_27a)))
<=> $true )
& ( p(ap(c_2Epatricia_2EIS__EMPTY(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V0v),V1v1)))
<=> $false )
& ( p(ap(c_2Epatricia_2EIS__EMPTY(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V2v2),V3v3),V4v4),V5v5)))
<=> $false ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epatricia_2EFIND__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EFIND(A_27a),V0t),V1k) = ap(c_2Eoption_2ETHE(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V0t),V1k)) ) ) ) ).
fof(ax_thm_2Epatricia_2EADD__LIST__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epatricia_2EADD__LIST(A_27a) = ap(c_2Elist_2EFOLDL(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a),ty_2Epatricia_2Eptree(A_27a)),c_2Epatricia_2EADD(A_27a)) ) ).
fof(conj_thm_2Epatricia_2EBRANCHING__BIT,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ( V0a != V1b
=> ~ ( p(ap(ap(c_2Ebit_2EBIT,ap(ap(c_2Epatricia_2EBRANCHING__BIT,V0a),V1b)),V0a))
<=> p(ap(ap(c_2Ebit_2EBIT,ap(ap(c_2Epatricia_2EBRANCHING__BIT,V0a),V1b)),V1b)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCHING__BIT__ZERO,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ( ap(ap(c_2Epatricia_2EBRANCHING__BIT,V0a),V1b) = c_2Enum_2E0
<=> ( ( p(ap(c_2Earithmetic_2EODD,V0a))
<=> p(ap(c_2Earithmetic_2EEVEN,V1b)) )
| V0a = V1b ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCHING__BIT__SYM,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EBRANCHING__BIT,V0a),V1b) = ap(ap(c_2Epatricia_2EBRANCHING__BIT,V1b),V0a) ) ) ).
fof(conj_thm_2Epatricia_2EEVERY__LEAF__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ( ( p(ap(ap(V0P,V2k),V3d))
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V1t)) )
=> p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),ap(ap(c_2Epatricia_2EADD(A_27a),V1t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V2k),V3d)))) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EMONO__EVERY__LEAF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1Q] :
( mem(V1Q,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2t] :
( mem(V2t,ty_2Epatricia_2Eptree(A_27a))
=> ( ( ! [V3k] :
( mem(V3k,ty_2Enum_2Enum)
=> ! [V4d] :
( mem(V4d,A_27a)
=> ( p(ap(ap(V0P,V3k),V4d))
=> p(ap(ap(V1Q,V3k),V4d)) ) ) )
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V2t)) )
=> p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V1Q),V2t)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ENOT__ADD__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d)) != c_2Epatricia_2EEmpty(A_27a) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EEMPTY__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),c_2Epatricia_2EEmpty(A_27a))) ) ).
fof(conj_thm_2Epatricia_2EADD__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1x] :
( mem(V1x,ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),V1x))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EEVERY__LEAF__BRANCH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1p] :
( mem(V1p,ty_2Enum_2Enum)
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3l] :
( mem(V3l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4r] :
( mem(V4r,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V1p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V2m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V3l),V4r))))))
<=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V3l))
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V4r)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EEVERY__LEAF__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V1t))
=> p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V1t),V2k))) ) ) ) ) ) ).
fof(lameq_f2567,axiom,
! [A_27a,V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V4k] : ap(f2567(A_27a,V0p,V1m),V4k) = k(A_27a,ap(ap(c_2Ebool_2E_2F_5C,ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V1m),V4k),V0p)),ap(ap(c_2Ebit_2EBIT,V1m),V4k))) ) ) ).
fof(lameq_f2568,axiom,
! [A_27a,V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V6k] : ap(f2568(A_27a,V0p,V1m),V6k) = k(A_27a,ap(ap(c_2Ebool_2E_2F_5C,ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V1m),V6k),V0p)),ap(c_2Ebool_2E_7E,ap(ap(c_2Ebit_2EBIT,V1m),V6k)))) ) ) ).
fof(conj_thm_2Epatricia_2EIS__PTREE__BRANCH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2l] :
( mem(V2l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3r] :
( mem(V3r,ty_2Epatricia_2Eptree(A_27a))
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V0p),ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1m)))
& ~ ( V2l = c_2Epatricia_2EEmpty(A_27a)
& V3r = c_2Epatricia_2EEmpty(A_27a) )
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),f2567(A_27a,V0p,V1m)),V2l))
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),f2568(A_27a,V0p,V1m)),V3r))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V2l))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V3r)) )
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V2l),V3r)))))) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EREMOVE__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__NONE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( ( ! [V3d] :
( mem(V3d,A_27a)
=> ~ p(ap(ap(V0P,V2k),V3d)) )
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V1t)) )
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),V1t),V2k) = c_2Eoption_2ENONE(A_27a) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))),V3j) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1k),V3j)),ap(c_2Eoption_2ESOME(A_27a),V2d)),ap(ap(c_2Epatricia_2EPEEK(A_27a),V0t),V3j)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EBRANCH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2l] :
( mem(V2l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3r] :
( mem(V3r,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V0p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V1m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V2l),V3r)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Epatricia_2Eptree(A_27a)),V2l),c_2Epatricia_2EEmpty(A_27a))),V3r),ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Epatricia_2Eptree(A_27a)),V3r),c_2Epatricia_2EEmpty(A_27a))),V2l),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),V2l),V3r))) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k)),V2j) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1k),V2j)),c_2Eoption_2ENONE(A_27a)),ap(ap(c_2Epatricia_2EPEEK(A_27a),V0t),V2j)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EEVERY__LEAF__TRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1Q] :
( mem(V1Q,arr(ty_2Enum_2Enum,arr(A_27b,bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3t] :
( mem(V3t,ty_2Epatricia_2Eptree(A_27a))
=> ( ( ! [V4k] :
( mem(V4k,ty_2Enum_2Enum)
=> ! [V5d] :
( mem(V5d,A_27a)
=> ( p(ap(ap(V0P,V4k),V5d))
=> p(ap(ap(V1Q,V4k),ap(V2f,V5d))) ) ) )
& p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V3t)) )
=> p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27b),V1Q),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V2f),V3t))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ETRANSFORM__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ( ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t) = c_2Epatricia_2EEmpty(A_27b)
<=> V1t = c_2Epatricia_2EEmpty(A_27a) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ETRANSFORM__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V1t))
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27b),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t))) ) ) ) ) ) ).
fof(lameq_f2569,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V3x] : ap(f2569(A_27b,A_27a,V0f),V3x) = ap(c_2Eoption_2ESOME(A_27b),ap(V0f,V3x)) ) ).
fof(conj_thm_2Epatricia_2EPEEK__TRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EPEEK(A_27b),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t)),V2k) = ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Epatricia_2EPEEK(A_27a),V1t),V2k)),c_2Eoption_2ENONE(A_27b)),f2569(A_27b,A_27a,V0f)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__TRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),ap(ap(c_2Epatricia_2EADD(A_27a),V1t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V2k),V3d))) = ap(ap(c_2Epatricia_2EADD(A_27b),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27b),V2k),ap(V0f,V3d))) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ETRANSFORM__BRANCH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1p] :
( mem(V1p,ty_2Enum_2Enum)
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3l] :
( mem(V3l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4r] :
( mem(V4r,ty_2Epatricia_2Eptree(A_27a))
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),ap(c_2Epatricia_2EBRANCH(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)))),V1p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a))),V2m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27a),ty_2Epatricia_2Eptree(A_27a)),V3l),V4r))))) = ap(c_2Epatricia_2EBRANCH(A_27b),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27b),ty_2Epatricia_2Eptree(A_27b)))),V1p),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epatricia_2Eptree(A_27b),ty_2Epatricia_2Eptree(A_27b))),V2m),ap(ap(c_2Epair_2E_2C(ty_2Epatricia_2Eptree(A_27b),ty_2Epatricia_2Eptree(A_27b)),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V3l)),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V4r))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EREMOVE__TRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V1t),V2k)) = ap(ap(c_2Epatricia_2EREMOVE(A_27b),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t)),V2k) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EREMOVE__ADD__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))),V1k) = ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ! [V3e] :
( mem(V3e,A_27a)
=> ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V3e)) = ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V3e)) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EEVERY__LEAF__PEEK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Epatricia_2EEVERY__LEAF(A_27a),V0P),V1t))
& p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V1t),V2k))) )
=> p(ap(ap(V0P,V2k),ap(c_2Eoption_2ETHE(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V1t),V2k)))) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIS__PTREE__PEEK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ( ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ~ p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),c_2Epatricia_2EEmpty(A_27a)),V0k))) )
& ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3b] :
( mem(V3b,A_27b)
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27b),ap(ap(c_2Epatricia_2EPEEK(A_27b),ap(ap(c_2Epatricia_2ELeaf(A_27b),V2j),V3b)),V1k)))
<=> V2j = V1k ) ) ) )
& ! [V4p] :
( mem(V4p,ty_2Enum_2Enum)
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6l] :
( mem(V6l,ty_2Epatricia_2Eptree(A_27c))
=> ! [V7r] :
( mem(V7r,ty_2Epatricia_2Eptree(A_27c))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27c),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27c),V4p),V5m),V6l),V7r)))
=> ( ? [V8k] :
( mem(V8k,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebit_2EBIT,V5m),V8k))
& p(ap(c_2Eoption_2EIS__SOME(A_27c),ap(ap(c_2Epatricia_2EPEEK(A_27c),V6l),V8k))) )
& ? [V9k] :
( mem(V9k,ty_2Enum_2Enum)
& ~ p(ap(ap(c_2Ebit_2EBIT,V5m),V9k))
& p(ap(c_2Eoption_2EIS__SOME(A_27c),ap(ap(c_2Epatricia_2EPEEK(A_27c),V7r),V9k))) )
& ! [V10k] :
( mem(V10k,ty_2Enum_2Enum)
=> ! [V11n] :
( mem(V11n,ty_2Enum_2Enum)
=> ( ( ~ p(ap(ap(ap(c_2Ebit_2EMOD__2EXP__EQ,V5m),V10k),V4p))
| ( p(ap(ap(c_2Eprim__rec_2E_3C,V11n),V5m))
& ~ ( p(ap(ap(c_2Ebit_2EBIT,V11n),V4p))
<=> p(ap(ap(c_2Ebit_2EBIT,V11n),V10k)) ) ) )
=> ( ~ p(ap(c_2Eoption_2EIS__SOME(A_27c),ap(ap(c_2Epatricia_2EPEEK(A_27c),V6l),V10k)))
& ~ p(ap(c_2Eoption_2EIS__SOME(A_27c),ap(ap(c_2Epatricia_2EPEEK(A_27c),V7r),V10k))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(A_27a))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V1t2)) )
=> ( ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EPEEK(A_27a),V0t1),V2k) = ap(ap(c_2Epatricia_2EPEEK(A_27a),V1t2),V2k) )
<=> V0t1 = V1t2 ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EREMOVE__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k)),V1k) = ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EREMOVE__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(ap(c_2Epatricia_2EREMOVE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))),V3j) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Epatricia_2Eptree(A_27a)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1k),V3j)),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V3j)),ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V3j)),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__ADD__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,A_27a)
=> ! [V4e] :
( mem(V4e,A_27a)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
& V1k != V2j )
=> ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V3d))),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V2j),V4e)) = ap(ap(c_2Epatricia_2EADD(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V2j),V4e))),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V3d)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EFILTER__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1l] :
( mem(V1l,ty_2Elist_2Elist(A_27a))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),ap(c_2Elist_2ELENGTH(A_27a),V1l)))
=> ~ p(ap(V0P,ap(ap(c_2Elist_2EEL(A_27a),V2n),V1l))) ) )
<=> ap(ap(c_2Elist_2EFILTER(A_27a),V0P),V1l) = c_2Elist_2ENIL(A_27a) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ETRAVERSE__TRANSFORM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2ETRAVERSE(A_27b),ap(ap(c_2Epatricia_2ETRANSFORM(A_27b,A_27a),V0f),V1t)) = ap(c_2Epatricia_2ETRAVERSE(A_27a),V1t) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EMEM__TRAVERSE__PEEK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t))))
<=> p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V0t),V1k))) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIN__NUMSET__OF__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t)))
<=> p(ap(ap(c_2Epatricia_2EIN__PTREE,V1n),V0t)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__LIST__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1l] :
( mem(V1l,ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a)))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(c_2Epatricia_2EADD__LIST(A_27a),V0t),V1l))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EADD__LIST__TO__EMPTY__IS__PTREE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Enum_2Enum,A_27a)))
=> p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(c_2Epatricia_2EADD__LIST(A_27a),c_2Epatricia_2EEmpty(A_27a)),V0l))) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s))) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE__EMPTY,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s))) ) ).
fof(conj_thm_2Epatricia_2ENOT__KEY__LEFT__AND__RIGHT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2l] :
( mem(V2l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V3r] :
( mem(V3r,ty_2Epatricia_2Eptree(A_27a))
=> ! [V4k] :
( mem(V4k,ty_2Enum_2Enum)
=> ! [V5j] :
( mem(V5j,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V0p),V1m),V2l),V3r)))
& p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V2l),V4k)))
& p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Epatricia_2EPEEK(A_27a),V3r),V5j))) )
=> V4k != V5j ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EALL__DISTINCT__TRAVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> p(ap(c_2Elist_2EALL__DISTINCT(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t))) ) ) ) ).
fof(conj_thm_2Epatricia_2EMEM__ALL__DISTINCT__IMP__PERM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l1] :
( mem(V0l1,ty_2Elist_2Elist(A_27a))
=> ! [V1l2] :
( mem(V1l2,ty_2Elist_2Elist(A_27a))
=> ( ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),V0l1))
& p(ap(c_2Elist_2EALL__DISTINCT(A_27a),V1l2))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V0l1)))
<=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V1l2))) ) ) )
=> p(ap(ap(c_2Esorting_2EPERM(A_27a),V0l1),V1l2)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EMEM__TRAVERSE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t))))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EINSERT__PTREE__IS__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EINSERT__PTREE,V1x),V0t))) ) ) ) ).
fof(conj_thm_2Epatricia_2EFINITE__NUMSET__OF__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t))) ) ).
fof(conj_thm_2Epatricia_2EADD__INSERT,axiom,
! [V0v] :
( mem(V0v,ty_2Eone_2Eone)
=> ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Epatricia_2EADD(ty_2Eone_2Eone),V1t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Eone_2Eone),V2n),V0v)) = ap(ap(c_2Epatricia_2EINSERT__PTREE,V2n),V1t) ) ) ) ).
fof(conj_thm_2Epatricia_2EPEEK__INSERT__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2j] :
( mem(V2j,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ap(ap(c_2Epatricia_2EPEEK(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EINSERT__PTREE,V1k),V0t)),V2j) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(ty_2Eone_2Eone)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1k),V2j)),ap(c_2Eoption_2ESOME(ty_2Eone_2Eone),c_2Eone_2Eone)),ap(ap(c_2Epatricia_2EPEEK(ty_2Eone_2Eone),V0t),V2j)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EMEM__TRAVERSE__INSERT__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ! [V2h] :
( mem(V2h,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EINSERT__PTREE,V2h),V0t)))))
<=> ( V1x = V2h
| ( V1x != V2h
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPERM__INSERT__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> p(ap(ap(c_2Esorting_2EPERM(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),ap(ap(ap(c_2Elist_2EFOLDL(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(ty_2Eone_2Eone)),ap(c_2Ecombin_2EC(ty_2Enum_2Enum,ty_2Epatricia_2Eptree(ty_2Eone_2Eone),ty_2Epatricia_2Eptree(ty_2Eone_2Eone)),c_2Epatricia_2EINSERT__PTREE)),V0t),ap(c_2Elist_2ESET__TO__LIST(ty_2Enum_2Enum),V1s)))),ap(c_2Elist_2ESET__TO__LIST(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t)),V1s)))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__OF__NUMSET,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) )
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s)))
<=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),V0t))
| p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2n),V1s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__EMPTY,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ~ p(ap(ap(c_2Epatricia_2EIN__PTREE,V0n),c_2Epatricia_2EEmpty(ty_2Eone_2Eone))) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),V0s))
<=> p(ap(ap(c_2Epatricia_2EIN__PTREE,V1n),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__EXTENSION,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t2)) )
=> ( V0t1 = V1t2
<=> ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2x),V0t1))
<=> p(ap(ap(c_2Epatricia_2EIN__PTREE,V2x),V1t2)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__NUMSET__OF__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) )
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t)),V1s)) = ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s) ) ) ) ).
fof(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) )
=> ap(c_2Epatricia_2ENUMSET__OF__PTREE,ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s)) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t)),V1s) ) ) ) ).
fof(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__EMPTY,axiom,
ap(c_2Epatricia_2ENUMSET__OF__PTREE,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__EMPTY,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = V0t ) ).
fof(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ap(c_2Epatricia_2ENUMSET__OF__PTREE,ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s)) = V0s ) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__INSERT__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),ap(ap(c_2Epatricia_2EINSERT__PTREE,V1m),V0t)))
<=> ( V1m = V2n
| p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),V0t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__REMOVE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),ap(ap(c_2Epatricia_2EREMOVE(ty_2Eone_2Eone),V0t),V1m)))
<=> ( V2n != V1m
& p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),V0t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EIN__PTREE__UNION__PTREE,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t2)) )
=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),V1t2)))
<=> ( p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),V0t1))
| p(ap(ap(c_2Epatricia_2EIN__PTREE,V2n),V1t2)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EUNION__PTREE__IS__PTREE,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t2)) )
=> p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),V1t2))) ) ) ) ).
fof(conj_thm_2Epatricia_2EUNION__PTREE__COMM,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t2)) )
=> ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),V1t2) = ap(ap(c_2Epatricia_2EUNION__PTREE,V1t2),V0t1) ) ) ) ).
fof(conj_thm_2Epatricia_2EUNION__PTREE__COMM__EMPTY,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ap(ap(c_2Epatricia_2EUNION__PTREE,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0t) = ap(ap(c_2Epatricia_2EUNION__PTREE,V0t),c_2Epatricia_2EEmpty(ty_2Eone_2Eone)) ) ) ).
fof(conj_thm_2Epatricia_2EUNION__PTREE__EMPTY,axiom,
( ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ap(ap(c_2Epatricia_2EUNION__PTREE,V0t),c_2Epatricia_2EEmpty(ty_2Eone_2Eone)) = V0t )
& ! [V1t] :
( mem(V1t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t))
=> ap(ap(c_2Epatricia_2EUNION__PTREE,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V1t) = V1t ) ) ) ).
fof(conj_thm_2Epatricia_2EUNION__PTREE__ASSOC,axiom,
! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V2t3] :
( mem(V2t3,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V1t2))
& p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V2t3)) )
=> ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),ap(ap(c_2Epatricia_2EUNION__PTREE,V1t2),V2t3)) = ap(ap(c_2Epatricia_2EUNION__PTREE,ap(ap(c_2Epatricia_2EUNION__PTREE,V0t1),V1t2)),V2t3) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__UNION,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s1] :
( mem(V1s1,arr(ty_2Enum_2Enum,bool))
=> ! [V2s2] :
( mem(V2s2,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s1))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2s2)) )
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V1s1),V2s2)) = ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s1)),V2s2) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) )
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V2x),V1s)) = ap(ap(c_2Epatricia_2EINSERT__PTREE,V2x),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT__EMPTY,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V1x),V0s)) = ap(ap(c_2Epatricia_2EINSERT__PTREE,V1x),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s)) ) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__DELETE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V0s),V1x)) = ap(ap(c_2Epatricia_2EREMOVE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s)),V1x) ) ) ) ).
fof(conj_thm_2Epatricia_2ETRAVERSE__AUX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t) = ap(ap(c_2Epatricia_2ETRAVERSE__AUX(A_27a),V0t),c_2Elist_2ENIL(ty_2Enum_2Enum)) ) ) ).
fof(conj_thm_2Epatricia_2EPTREE__TRAVERSE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(A_27b))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27b),V1t2)) )
=> ( ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t1))))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27b),V1t2)))) ) )
<=> ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t1) = ap(c_2Epatricia_2ETRAVERSE(A_27b),V1t2) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EQSORT__MEM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l2] :
( mem(V0l2,ty_2Elist_2Elist(A_27a))
=> ! [V1l1] :
( mem(V1l1,ty_2Elist_2Elist(A_27a))
=> ! [V2R] :
( mem(V2R,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(c_2Esorting_2EQSORT(A_27a),V2R),V1l1) = ap(ap(c_2Esorting_2EQSORT(A_27a),V2R),V0l2)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V1l1)))
<=> p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Elist_2ELIST__TO__SET(A_27a),V0l2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EKEYS__PEEK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t1] :
( mem(V0t1,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1t2] :
( mem(V1t2,ty_2Epatricia_2Eptree(A_27b))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t1))
& p(ap(c_2Epatricia_2EIS__PTREE(A_27b),V1t2)) )
=> ( ap(c_2Epatricia_2EKEYS(A_27a),V0t1) = ap(c_2Epatricia_2EKEYS(A_27b),V1t2)
<=> ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t1) = ap(c_2Epatricia_2ETRAVERSE(A_27b),V1t2) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPERM__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
& ~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) )
=> p(ap(ap(c_2Esorting_2EPERM(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d)))),ap(ap(c_2Elist_2ECONS(ty_2Enum_2Enum),V1k),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPERM__NOT__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) )
=> ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))) = ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPERM__NOT__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
& ~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) )
=> ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k)) = ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t) ) ) ) ) ).
fof(lameq_f2570,axiom,
! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2x] : ap(f2570(V1k),V2x) = ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2x),V1k)) ) ).
fof(conj_thm_2Epatricia_2EPERM__DELETE__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t)))) )
=> p(ap(ap(c_2Esorting_2EPERM(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EREMOVE(ty_2Eone_2Eone),V0t),V1k))),ap(ap(c_2Elist_2EFILTER(ty_2Enum_2Enum),f2570(V1k)),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t)))) ) ) ) ).
fof(conj_thm_2Epatricia_2EFILTER__NONE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1l] :
( mem(V1l,ty_2Elist_2Elist(A_27a))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),ap(c_2Elist_2ELENGTH(A_27a),V1l)))
=> p(ap(V0P,ap(ap(c_2Elist_2EEL(A_27a),V2n),V1l))) ) )
=> ap(ap(c_2Elist_2EFILTER(A_27a),V0P),V1l) = V1l ) ) ) ) ).
fof(conj_thm_2Epatricia_2EPERM__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) )
=> p(ap(ap(c_2Esorting_2EPERM(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k))),ap(ap(c_2Elist_2EFILTER(ty_2Enum_2Enum),f2570(V1k)),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ESIZE__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,A_27a)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(c_2Epatricia_2ESIZE(A_27a),ap(ap(c_2Epatricia_2EADD(A_27a),V0t),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,A_27a),V1k),V2d))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))),ap(c_2Epatricia_2ESIZE(A_27a),V0t)),ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epatricia_2ESIZE(A_27a),V0t)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ESIZE__REMOVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(A_27a))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( p(ap(c_2Epatricia_2EIS__PTREE(A_27a),V0t))
=> ap(c_2Epatricia_2ESIZE(A_27a),ap(ap(c_2Epatricia_2EREMOVE(A_27a),V0t),V1k)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1k),ap(c_2Elist_2ELIST__TO__SET(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(A_27a),V0t)))),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epatricia_2ESIZE(A_27a),V0t)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Epatricia_2ESIZE(A_27a),V0t)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ESIZE,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ap(c_2Epatricia_2ESIZE(A_27a),c_2Epatricia_2EEmpty(A_27a)) = c_2Enum_2E0
& ! [V0k] :
( mem(V0k,ty_2Enum_2Enum)
=> ! [V1d] :
( mem(V1d,A_27a)
=> ap(c_2Epatricia_2ESIZE(A_27a),ap(ap(c_2Epatricia_2ELeaf(A_27a),V0k),V1d)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) )
& ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4l] :
( mem(V4l,ty_2Epatricia_2Eptree(A_27a))
=> ! [V5r] :
( mem(V5r,ty_2Epatricia_2Eptree(A_27a))
=> ap(c_2Epatricia_2ESIZE(A_27a),ap(ap(ap(ap(c_2Epatricia_2EBranch(A_27a),V2p),V3m),V4l),V5r)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epatricia_2ESIZE(A_27a),V4l)),ap(c_2Epatricia_2ESIZE(A_27a),V5r)) ) ) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
=> ( ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
& p(ap(c_2Elist_2EALL__DISTINCT(ty_2Enum_2Enum),ap(ap(c_2Elist_2EAPPEND(ty_2Enum_2Enum),ap(c_2Epatricia_2ETRAVERSE(ty_2Eone_2Eone),V0t)),ap(c_2Elist_2ESET__TO__LIST(ty_2Enum_2Enum),V1s)))) )
=> ap(c_2Epatricia_2ESIZE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,V0t),V1s)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epatricia_2ESIZE(ty_2Eone_2Eone),V0t)),ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),V1s)) ) ) ) ) ).
fof(conj_thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ap(c_2Epatricia_2ESIZE(ty_2Eone_2Eone),ap(ap(c_2Epatricia_2EPTREE__OF__NUMSET,c_2Epatricia_2EEmpty(ty_2Eone_2Eone)),V0s)) = ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),V0s) ) ) ).
fof(conj_thm_2Epatricia_2ECARD__LIST__TO__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,ty_2Elist_2Elist(A_27a))
=> ( p(ap(c_2Elist_2EALL__DISTINCT(A_27a),V0l))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Elist_2ELIST__TO__SET(A_27a),V0l)) = ap(c_2Elist_2ELENGTH(A_27a),V0l) ) ) ) ).
fof(conj_thm_2Epatricia_2ECARD__NUMSET__OF__PTREE,axiom,
! [V0t] :
( mem(V0t,ty_2Epatricia_2Eptree(ty_2Eone_2Eone))
=> ( p(ap(c_2Epatricia_2EIS__PTREE(ty_2Eone_2Eone),V0t))
=> ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(c_2Epatricia_2ENUMSET__OF__PTREE,V0t)) = ap(c_2Epatricia_2ESIZE(ty_2Eone_2Eone),V0t) ) ) ).
fof(conj_thm_2Epatricia_2EDELETE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( mem(V2s2,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s1),V2s2)),V0x) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s1),V0x)),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s2),V0x)) ) ) ) ) ).
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