TPTP Problem File: SEU351+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU351+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t34_lattice3
% 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-t34_lattice3 [Urb07]
% Status : Theorem
% Rating : 0.69 v8.2.0, 0.78 v8.1.0, 0.67 v7.5.0, 0.72 v7.4.0, 0.60 v7.3.0, 0.62 v7.1.0, 0.57 v7.0.0, 0.67 v6.4.0, 0.62 v6.3.0, 0.58 v6.2.0, 0.76 v6.1.0, 0.80 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.82 v5.3.0, 0.85 v5.2.0, 0.70 v5.1.0, 0.71 v5.0.0, 0.75 v4.1.0, 0.78 v4.0.0, 0.79 v3.7.0, 0.80 v3.5.0, 0.79 v3.3.0
% Syntax : Number of formulae : 37 ( 12 unt; 0 def)
% Number of atoms : 137 ( 8 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 124 ( 24 ~; 1 |; 54 &)
% ( 8 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 1 con; 0-2 aty)
% Number of variables : 65 ( 55 !; 10 ?)
% 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_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& lattice(A) )
=> ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A) ) ) ) ).
fof(cc2_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A) )
=> ( ~ empty_carrier(A)
& lattice(A) ) ) ) ).
fof(d16_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( latt_set_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,C)
=> below(A,B,D) ) ) ) ) ) ).
fof(d17_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( latt_element_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,C)
=> below(A,D,B) ) ) ) ) ) ).
fof(d21_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( C = join_of_latt_set(A,B)
<=> ( latt_element_smaller(A,C,B)
& ! [D] :
( element(D,the_carrier(A))
=> ( latt_element_smaller(A,D,B)
=> below(A,C,D) ) ) ) ) ) ) ) ).
fof(d22_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] : meet_of_latt_set(A,B) = join_of_latt_set(A,a_2_2_lattice3(A,B)) ) ).
fof(dt_k15_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> element(join_of_latt_set(A,B),the_carrier(A)) ) ).
fof(dt_k16_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> element(meet_of_latt_set(A,B),the_carrier(A)) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
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_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
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_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fraenkel_a_2_2_lattice3,axiom,
! [A,B,C] :
( ( ~ empty_carrier(B)
& latt_str(B) )
=> ( in(A,a_2_2_lattice3(B,C))
<=> ? [D] :
( element(D,the_carrier(B))
& A = D
& latt_set_smaller(B,D,C) ) ) ) ).
fof(fraenkel_a_2_3_lattice3,axiom,
! [A,B,C] :
( ( ~ empty_carrier(B)
& lattice(B)
& complete_latt_str(B)
& latt_str(B) )
=> ( in(A,a_2_3_lattice3(B,C))
<=> ? [D] :
( element(D,the_carrier(B))
& A = D
& latt_set_smaller(B,D,C) ) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(redefinition_r3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& join_absorbing(A)
& latt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> ( below_refl(A,B,C)
<=> below(A,B,C) ) ) ).
fof(reflexivity_r3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& join_absorbing(A)
& latt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> below_refl(A,B,B) ) ).
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(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) ).
fof(t34_lattice3,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( B = meet_of_latt_set(A,C)
<=> ( latt_set_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( latt_set_smaller(A,D,C)
=> below_refl(A,D,B) ) ) ) ) ) ) ).
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) ) ).
%------------------------------------------------------------------------------