TPTP Problem File: LAT285+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LAT285+3 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Representation Theorem for Boolean Algebras T14
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Wal93] Walijewski (1993), Representation Theorem for Boolean
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t14_lopclset [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 12370 (2672 unt; 0 def)
% Number of atoms : 74569 (8763 equ)
% Maximal formula atoms : 52 ( 6 avg)
% Number of connectives : 70669 (8470 ~; 425 |;35181 &)
% (2046 <=>;24547 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 692 ( 690 usr; 2 prp; 0-6 aty)
% Number of functors : 1781 (1781 usr; 496 con; 0-10 aty)
% Number of variables : 30568 (29155 !;1413 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Chainy small version: includes all preceding MML articles that
% are included in any Bushy version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+5.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+15.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+19.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+22.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+32.ax').
include('Axioms/SET007/SET007+33.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+50.ax').
include('Axioms/SET007/SET007+51.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+59.ax').
include('Axioms/SET007/SET007+60.ax').
include('Axioms/SET007/SET007+61.ax').
include('Axioms/SET007/SET007+64.ax').
include('Axioms/SET007/SET007+66.ax').
include('Axioms/SET007/SET007+67.ax').
include('Axioms/SET007/SET007+68.ax').
include('Axioms/SET007/SET007+71.ax').
include('Axioms/SET007/SET007+75.ax').
include('Axioms/SET007/SET007+76.ax').
include('Axioms/SET007/SET007+77.ax').
include('Axioms/SET007/SET007+79.ax').
include('Axioms/SET007/SET007+80.ax').
include('Axioms/SET007/SET007+86.ax').
include('Axioms/SET007/SET007+91.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+125.ax').
include('Axioms/SET007/SET007+126.ax').
include('Axioms/SET007/SET007+148.ax').
include('Axioms/SET007/SET007+159.ax').
include('Axioms/SET007/SET007+165.ax').
include('Axioms/SET007/SET007+170.ax').
include('Axioms/SET007/SET007+182.ax').
include('Axioms/SET007/SET007+186.ax').
include('Axioms/SET007/SET007+188.ax').
include('Axioms/SET007/SET007+190.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+202.ax').
include('Axioms/SET007/SET007+205.ax').
include('Axioms/SET007/SET007+206.ax').
include('Axioms/SET007/SET007+207.ax').
include('Axioms/SET007/SET007+209.ax').
include('Axioms/SET007/SET007+210.ax').
include('Axioms/SET007/SET007+211.ax').
include('Axioms/SET007/SET007+212.ax').
include('Axioms/SET007/SET007+213.ax').
include('Axioms/SET007/SET007+217.ax').
include('Axioms/SET007/SET007+218.ax').
include('Axioms/SET007/SET007+223.ax').
include('Axioms/SET007/SET007+224.ax').
include('Axioms/SET007/SET007+225.ax').
include('Axioms/SET007/SET007+227.ax').
include('Axioms/SET007/SET007+237.ax').
include('Axioms/SET007/SET007+241.ax').
include('Axioms/SET007/SET007+242.ax').
include('Axioms/SET007/SET007+246.ax').
include('Axioms/SET007/SET007+247.ax').
include('Axioms/SET007/SET007+248.ax').
include('Axioms/SET007/SET007+252.ax').
include('Axioms/SET007/SET007+253.ax').
include('Axioms/SET007/SET007+255.ax').
include('Axioms/SET007/SET007+256.ax').
include('Axioms/SET007/SET007+276.ax').
include('Axioms/SET007/SET007+278.ax').
include('Axioms/SET007/SET007+279.ax').
include('Axioms/SET007/SET007+280.ax').
include('Axioms/SET007/SET007+281.ax').
include('Axioms/SET007/SET007+293.ax').
include('Axioms/SET007/SET007+295.ax').
include('Axioms/SET007/SET007+297.ax').
include('Axioms/SET007/SET007+298.ax').
include('Axioms/SET007/SET007+299.ax').
include('Axioms/SET007/SET007+301.ax').
include('Axioms/SET007/SET007+308.ax').
include('Axioms/SET007/SET007+309.ax').
include('Axioms/SET007/SET007+311.ax').
include('Axioms/SET007/SET007+312.ax').
include('Axioms/SET007/SET007+317.ax').
include('Axioms/SET007/SET007+321.ax').
include('Axioms/SET007/SET007+322.ax').
include('Axioms/SET007/SET007+327.ax').
include('Axioms/SET007/SET007+335.ax').
include('Axioms/SET007/SET007+338.ax').
include('Axioms/SET007/SET007+339.ax').
include('Axioms/SET007/SET007+354.ax').
include('Axioms/SET007/SET007+363.ax').
include('Axioms/SET007/SET007+365.ax').
include('Axioms/SET007/SET007+370.ax').
include('Axioms/SET007/SET007+375.ax').
%------------------------------------------------------------------------------
fof(fraenkel_a_1_0_lopclset,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ( r2_hidden(A,a_1_0_lopclset(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v3_pre_topc(C,B)
& v4_pre_topc(C,B) ) ) ) ).
fof(dt_k1_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> m1_subset_1(k1_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k2_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> m2_subset_1(k2_lopclset(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ).
fof(commutativity_k2_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k2_lopclset(A,B,C) = k2_lopclset(A,C,B) ) ).
fof(idempotence_k2_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k2_lopclset(A,B,B) = B ) ).
fof(redefinition_k2_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k2_lopclset(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k3_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> m2_subset_1(k3_lopclset(A,B,C),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ).
fof(commutativity_k3_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k3_lopclset(A,B,C) = k3_lopclset(A,C,B) ) ).
fof(idempotence_k3_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k3_lopclset(A,B,B) = B ) ).
fof(redefinition_k3_lopclset,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_lopclset(A))
& m1_subset_1(C,k1_lopclset(A)) )
=> k3_lopclset(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k4_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k4_lopclset(A))
& v1_funct_2(k4_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
& m2_relset_1(k4_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) ) ) ).
fof(dt_k5_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k5_lopclset(A))
& v1_funct_2(k5_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
& m2_relset_1(k5_lopclset(A),k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) ) ) ).
fof(dt_k6_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k6_lopclset(A))
& v10_lattices(k6_lopclset(A))
& l3_lattices(k6_lopclset(A)) ) ) ).
fof(dt_k7_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> m1_subset_1(k7_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k8_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( v1_relat_1(k8_lopclset(A))
& v1_funct_1(k8_lopclset(A)) ) ) ).
fof(dt_k9_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( v1_funct_1(k9_lopclset(A))
& v1_funct_2(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)))
& m2_relset_1(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A))) ) ) ).
fof(redefinition_k9_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> k9_lopclset(A) = k8_lopclset(A) ) ).
fof(dt_k10_lopclset,axiom,
$true ).
fof(dt_k11_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( v1_pre_topc(k11_lopclset(A))
& v2_pre_topc(k11_lopclset(A))
& l1_pre_topc(k11_lopclset(A)) ) ) ).
fof(dt_k12_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(k12_lopclset(A))
& v10_lattices(k12_lopclset(A))
& l3_lattices(k12_lopclset(A)) ) ) ).
fof(dt_k13_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> m1_lattice4(k13_lopclset(A),A,k12_lopclset(A)) ) ).
fof(redefinition_k13_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& ~ v3_realset2(A)
& l3_lattices(A) )
=> k13_lopclset(A) = k8_lopclset(A) ) ).
fof(d1_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k1_lopclset(A) = a_1_0_lopclset(A) ) ).
fof(fc1_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k1_lopclset(A)) ) ).
fof(t1_lopclset,axiom,
$true ).
fof(t2_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k1_lopclset(A))
=> v3_pre_topc(B,A) ) ) ) ).
fof(t3_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,k1_lopclset(A))
=> v4_pre_topc(B,A) ) ) ) ).
fof(t4_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
& v4_pre_topc(B,A) )
=> r2_hidden(B,k1_lopclset(A)) ) ) ) ).
fof(d2_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
& m2_relset_1(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) )
=> ( B = k4_lopclset(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> k2_binop_1(k1_lopclset(A),k1_lopclset(A),k1_lopclset(A),B,C,D) = k2_lopclset(A,C,D) ) ) ) ) ) ).
fof(d3_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A))
& m2_relset_1(B,k2_zfmisc_1(k1_lopclset(A),k1_lopclset(A)),k1_lopclset(A)) )
=> ( B = k5_lopclset(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> k2_binop_1(k1_lopclset(A),k1_lopclset(A),k1_lopclset(A),B,C,D) = k3_lopclset(A,C,D) ) ) ) ) ) ).
fof(t5_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ( ( B = D
& C = E )
=> k1_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)),B,C) = k2_lopclset(A,D,E) ) ) ) ) ) ) ).
fof(t6_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> ( ( B = D
& C = E )
=> k2_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)),B,C) = k3_lopclset(A,D,E) ) ) ) ) ) ) ).
fof(t7_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> m2_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ).
fof(t8_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> m2_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ).
fof(t9_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A))
=> m2_subset_1(k3_subset_1(u1_struct_0(A),B),k1_zfmisc_1(u1_struct_0(A)),k1_lopclset(A)) ) ) ).
fof(t10_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)))
& v10_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)))
& l3_lattices(g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A))) ) ) ).
fof(d4_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k6_lopclset(A) = g3_lattices(k1_lopclset(A),k4_lopclset(A),k5_lopclset(A)) ) ).
fof(t11_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k6_lopclset(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_lopclset(A)))
=> k3_lattices(k6_lopclset(A),B,C) = k2_xboole_0(B,C) ) ) ) ).
fof(t12_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k6_lopclset(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_lopclset(A)))
=> k4_lattices(k6_lopclset(A),B,C) = k3_xboole_0(B,C) ) ) ) ).
fof(t13_lopclset,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> u1_struct_0(k6_lopclset(A)) = k1_lopclset(A) ) ).
fof(t14_lopclset,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v17_lattices(k6_lopclset(A)) ) ).
%------------------------------------------------------------------------------