TPTP Problem File: SEU402+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU402+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_funct_1__e2_11_1__funct_1
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-s1_funct_1__e2_11_1__funct_1 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.3.0
% Syntax : Number of formulae : 97 ( 21 unt; 0 def)
% Number of atoms : 440 ( 48 equ)
% Maximal formula atoms : 39 ( 4 avg)
% Number of connectives : 409 ( 66 ~; 1 |; 245 &)
% ( 14 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 1 con; 0-4 aty)
% Number of variables : 225 ( 173 !; 52 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_funct_1__e6_39_3__yellow19__1,conjecture,
! [A,B,C] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ( ! [D,E,F] :
( ( in(D,C)
& ? [G] :
( E = G
& ! [H] :
( in(H,G)
<=> ( in(H,the_carrier(B))
& ? [I] :
( netstr_induced_subset(I,A,B)
& ? [J] :
( element(J,the_carrier(B))
& D = topstr_closure(A,I)
& H = J
& I = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,J)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,J))) ) ) ) ) )
& in(D,C)
& ? [K] :
( F = K
& ! [L] :
( in(L,K)
<=> ( in(L,the_carrier(B))
& ? [M] :
( netstr_induced_subset(M,A,B)
& ? [N] :
( element(N,the_carrier(B))
& D = topstr_closure(A,M)
& L = N
& M = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,N)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,N))) ) ) ) ) ) )
=> E = F )
=> ? [D] :
( relation(D)
& function(D)
& ! [E,F] :
( in(ordered_pair(E,F),D)
<=> ( in(E,C)
& in(E,C)
& ? [O] :
( F = O
& ! [P] :
( in(P,O)
<=> ( in(P,the_carrier(B))
& ? [Q] :
( netstr_induced_subset(Q,A,B)
& ? [R] :
( element(R,the_carrier(B))
& E = topstr_closure(A,Q)
& P = R
& Q = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,R)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,R))) ) ) ) ) ) ) ) ) ) ) ).
fof(rc4_tops_1,axiom,
! [A] :
( top_str(A)
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& empty(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A) ) ) ).
fof(rc5_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& empty(B)
& open_subset(B,A)
& closed_subset(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A)
& nowhere_dense(B,A) ) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc1_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A) ) ) ).
fof(rc2_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A)
& closed_subset(B,A) ) ) ).
fof(rc3_tops_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B)
& open_subset(B,A)
& closed_subset(B,A) ) ) ).
fof(cc4_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( nowhere_dense(B,A)
=> boundary_set(B,A) ) ) ) ).
fof(cc5_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( closed_subset(B,A)
& boundary_set(B,A) )
=> ( boundary_set(B,A)
& nowhere_dense(B,A) ) ) ) ) ).
fof(cc6_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( open_subset(B,A)
& nowhere_dense(B,A) )
=> ( empty(B)
& open_subset(B,A)
& closed_subset(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& boundary_set(B,A)
& nowhere_dense(B,A) ) ) ) ) ).
fof(rc2_waybel_7,axiom,
! [A] :
? [B] :
( element(B,powerset(powerset(A)))
& ~ empty(B)
& finite(B) ) ).
fof(rc3_waybel_7,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( element(B,powerset(powerset(the_carrier(A))))
& ~ empty(B)
& finite(B) ) ) ).
fof(free_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ! [E,F,G,H] :
( net_str_of(A,B,C,D) = net_str_of(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(dt_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( strict_net_str(net_str_of(A,B,C,D),A)
& net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& closed_subset(B,A) ) ) ).
fof(rc7_pre_topc,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B)
& closed_subset(B,A) ) ) ).
fof(cc1_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> ( open_subset(B,A)
& closed_subset(B,A) ) ) ) ) ).
fof(cc2_tops_1,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> boundary_set(B,A) ) ) ) ).
fof(cc3_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( empty(B)
=> nowhere_dense(B,A) ) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(fc6_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& ~ empty(B)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( ~ empty_carrier(net_str_of(A,B,C,D))
& strict_net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(A)) ) ) ).
fof(abstractness_v6_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( strict_net_str(B,A)
=> B = net_str_of(A,the_carrier(B),the_InternalRel(B),the_mapping(A,B)) ) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k5_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( strict_net_str(netstr_restr_to_element(A,B,C),A)
& net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_m2_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( subnet(C,A,B)
=> ( ~ empty_carrier(C)
& transitive_relstr(C)
& directed_relstr(C)
& net_str(C,A) ) ) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc2_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> closed_subset(topstr_closure(A,B),A) ) ).
fof(rc4_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( net_str(B,A)
& strict_net_str(B,A) ) ) ).
fof(fc22_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( ~ empty_carrier(netstr_restr_to_element(A,B,C))
& strict_net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(fc26_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( ~ empty_carrier(netstr_restr_to_element(A,B,C))
& transitive_relstr(netstr_restr_to_element(A,B,C))
& strict_net_str(netstr_restr_to_element(A,B,C),A)
& directed_relstr(netstr_restr_to_element(A,B,C)) ) ) ).
fof(rc1_waybel_9,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ? [C] :
( subnet(C,A,B)
& ~ empty_carrier(C)
& transitive_relstr(C)
& strict_net_str(C,A)
& directed_relstr(C) ) ) ).
fof(fc15_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ( ~ empty(the_mapping(A,B))
& relation(the_mapping(A,B))
& function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(redefinition_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_rng_as_subset(A,B,C) = relation_rng(C) ) ).
fof(redefinition_k6_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> subnetstr_of_element(A,B,C) = netstr_restr_to_element(A,B,C) ) ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_rng_as_subset(A,B,C),powerset(B)) ) ).
fof(dt_k6_pre_topc,axiom,
! [A,B] :
( ( top_str(A)
& element(B,powerset(the_carrier(A))) )
=> element(topstr_closure(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k6_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( strict_net_str(subnetstr_of_element(A,B,C),A)
& subnet(subnetstr_of_element(A,B,C),A,B) ) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( top_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( net_str(B,A)
=> rel_str(B) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m1_yellow19,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( netstr_induced_subset(C,A,B)
=> element(C,powerset(the_carrier(A))) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u1_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A))
& relation_of2_as_subset(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(s1_tarski__e6_39_3__yellow19__2,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ( ! [D,E,F] :
( ( in(D,C)
& ? [G] :
( E = G
& ! [H] :
( in(H,G)
<=> ( in(H,the_carrier(B))
& ? [I] :
( netstr_induced_subset(I,A,B)
& ? [J] :
( element(J,the_carrier(B))
& D = topstr_closure(A,I)
& H = J
& I = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,J)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,J))) ) ) ) ) )
& in(D,C)
& ? [K] :
( F = K
& ! [L] :
( in(L,K)
<=> ( in(L,the_carrier(B))
& ? [M] :
( netstr_induced_subset(M,A,B)
& ? [N] :
( element(N,the_carrier(B))
& D = topstr_closure(A,M)
& L = N
& M = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,N)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,N))) ) ) ) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,C)
& in(F,C)
& ? [O] :
( E = O
& ! [P] :
( in(P,O)
<=> ( in(P,the_carrier(B))
& ? [Q] :
( netstr_induced_subset(Q,A,B)
& ? [R] :
( element(R,the_carrier(B))
& F = topstr_closure(A,Q)
& P = R
& Q = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,R)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,R))) ) ) ) ) ) ) ) ) ) ).
fof(s1_xboole_0__e6_39_3__yellow19__2,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [D] :
? [E] :
! [F] :
( in(F,E)
<=> ( in(F,cartesian_product2(C,D))
& ? [G,H] :
( ordered_pair(G,H) = F
& in(G,C)
& ? [I] :
( H = I
& ! [J] :
( in(J,I)
<=> ( in(J,the_carrier(B))
& ? [K] :
( netstr_induced_subset(K,A,B)
& ? [L] :
( element(L,the_carrier(B))
& G = topstr_closure(A,K)
& J = L
& K = relation_rng_as_subset(the_carrier(subnetstr_of_element(A,B,L)),the_carrier(A),the_mapping(A,subnetstr_of_element(A,B,L))) ) ) ) ) ) ) ) ) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(d1_funct_1,axiom,
! [A] :
( function(A)
<=> ! [B,C,D] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(B,D),A) )
=> C = D ) ) ).
fof(d1_relat_1,axiom,
! [A] :
( relation(A)
<=> ! [B] :
~ ( in(B,A)
& ! [C,D] : B != ordered_pair(C,D) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(t106_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(A,C)
& in(B,D) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t33_zfmisc_1,axiom,
! [A,B,C,D] :
( ordered_pair(A,B) = ordered_pair(C,D)
=> ( A = C
& B = D ) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
%------------------------------------------------------------------------------