TPTP Problem File: SEU304+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU304+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t23_lattices
% 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-t23_lattices [Urb07]
% Status : Theorem
% Rating : 0.28 v8.2.0, 0.25 v7.5.0, 0.28 v7.4.0, 0.13 v7.3.0, 0.28 v7.2.0, 0.24 v7.1.0, 0.17 v7.0.0, 0.20 v6.4.0, 0.27 v6.3.0, 0.25 v6.2.0, 0.28 v6.1.0, 0.37 v6.0.0, 0.35 v5.5.0, 0.37 v5.4.0, 0.39 v5.3.0, 0.44 v5.2.0, 0.25 v5.1.0, 0.24 v5.0.0, 0.33 v4.1.0, 0.39 v4.0.0, 0.42 v3.7.0, 0.40 v3.5.0, 0.37 v3.4.0, 0.32 v3.3.0
% Syntax : Number of formulae : 50 ( 16 unt; 0 def)
% Number of atoms : 149 ( 9 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 124 ( 25 ~; 1 |; 57 &)
% ( 4 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 1 con; 0-6 aty)
% Number of variables : 93 ( 84 !; 9 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_finsub_1,axiom,
! [A] :
( preboolean(A)
=> ( cup_closed(A)
& diff_closed(A) ) ) ).
fof(cc2_finsub_1,axiom,
! [A] :
( ( cup_closed(A)
& diff_closed(A) )
=> preboolean(A) ) ).
fof(commutativity_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> meet_commut(A,B,C) = meet_commut(A,C,B) ) ).
fof(d1_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> join(A,B,C) = apply_binary_as_element(the_carrier(A),the_carrier(A),the_carrier(A),the_L_join(A),B,C) ) ) ) ).
fof(d2_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> meet(A,B,C) = apply_binary_as_element(the_carrier(A),the_carrier(A),the_carrier(A),the_L_meet(A),B,C) ) ) ) ).
fof(d3_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( below(A,B,C)
<=> join(A,B,C) = C ) ) ) ) ).
fof(d8_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ( meet_absorbing(A)
<=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> join(A,meet(A,B,C),C) = C ) ) ) ) ).
fof(dt_k1_binop_1,axiom,
$true ).
fof(dt_k1_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(join(A,B,C),the_carrier(A)) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ empty(A)
& ~ empty(B)
& function(D)
& quasi_total(D,cartesian_product2(A,B),C)
& relation_of2(D,cartesian_product2(A,B),C)
& element(E,A)
& element(F,B) )
=> element(apply_binary_as_element(A,B,C,D,E,F),C) ) ).
fof(dt_k2_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(meet(A,B,C),the_carrier(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(meet_commut(A,B,C),the_carrier(A)) ) ).
fof(dt_l1_lattices,axiom,
! [A] :
( meet_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( join_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( meet_semilatt_str(A)
& join_semilatt_str(A) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_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_u1_lattices,axiom,
! [A] :
( meet_semilatt_str(A)
=> ( function(the_L_meet(A))
& quasi_total(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& relation_of2_as_subset(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_lattices,axiom,
! [A] :
( join_semilatt_str(A)
=> ( function(the_L_join(A))
& quasi_total(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& relation_of2_as_subset(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(existence_l1_lattices,axiom,
? [A] : meet_semilatt_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_l2_lattices,axiom,
? [A] : join_semilatt_str(A) ).
fof(existence_l3_lattices,axiom,
? [A] : latt_str(A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(fc1_finsub_1,axiom,
! [A] :
( ~ empty(powerset(A))
& cup_closed(powerset(A))
& diff_closed(powerset(A))
& preboolean(powerset(A)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(redefinition_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ empty(A)
& ~ empty(B)
& function(D)
& quasi_total(D,cartesian_product2(A,B),C)
& relation_of2(D,cartesian_product2(A,B),C)
& element(E,A)
& element(F,B) )
=> apply_binary_as_element(A,B,C,D,E,F) = apply_binary(D,E,F) ) ).
fof(redefinition_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> meet_commut(A,B,C) = meet(A,B,C) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t23_lattices,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> below(A,meet_commut(A,B,C),B) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
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) ) ).
%------------------------------------------------------------------------------