SET007 Axioms: SET007+511.ax
%------------------------------------------------------------------------------
% File : SET007+511 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lattice of Substitutions
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : substlat [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 7 unt; 0 def)
% Number of atoms : 196 ( 34 equ)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 155 ( 9 ~; 0 |; 65 &)
% ( 5 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 26 ( 25 usr; 0 prp; 1-3 aty)
% Number of functors : 21 ( 21 usr; 1 con; 0-4 aty)
% Number of variables : 188 ( 182 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_substlat,axiom,
! [A,B] : ~ v1_xboole_0(k1_substlat(A,B)) ).
fof(rc1_substlat,axiom,
! [A,B] :
? [C] :
( m1_subset_1(C,k1_substlat(A,B))
& ~ v1_xboole_0(C) ) ).
fof(cc1_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_substlat(A,B))
=> v1_finset_1(C) ) ).
fof(cc2_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_substlat(A,B))
=> ( v1_finset_1(C)
& v1_fraenkel(C) ) ) ).
fof(rc2_substlat,axiom,
! [A,B] :
? [C] :
( m1_subset_1(C,k4_partfun1(A,B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C) ) ).
fof(fc2_substlat,axiom,
! [A,B] :
( ~ v3_struct_0(k5_substlat(A,B))
& v3_lattices(k5_substlat(A,B)) ) ).
fof(fc3_substlat,axiom,
! [A,B] :
( ~ v3_struct_0(k5_substlat(A,B))
& v3_lattices(k5_substlat(A,B))
& v4_lattices(k5_substlat(A,B))
& v5_lattices(k5_substlat(A,B))
& v6_lattices(k5_substlat(A,B))
& v7_lattices(k5_substlat(A,B))
& v8_lattices(k5_substlat(A,B))
& v9_lattices(k5_substlat(A,B))
& v10_lattices(k5_substlat(A,B)) ) ).
fof(fc4_substlat,axiom,
! [A,B] :
( ~ v3_struct_0(k5_substlat(A,B))
& v3_lattices(k5_substlat(A,B))
& v4_lattices(k5_substlat(A,B))
& v5_lattices(k5_substlat(A,B))
& v6_lattices(k5_substlat(A,B))
& v7_lattices(k5_substlat(A,B))
& v8_lattices(k5_substlat(A,B))
& v9_lattices(k5_substlat(A,B))
& v10_lattices(k5_substlat(A,B))
& v11_lattices(k5_substlat(A,B))
& v12_lattices(k5_substlat(A,B))
& v13_lattices(k5_substlat(A,B))
& v14_lattices(k5_substlat(A,B))
& v15_lattices(k5_substlat(A,B)) ) ).
fof(t1_substlat,axiom,
! [A,B] : r2_hidden(k1_xboole_0,k1_substlat(A,B)) ).
fof(t2_substlat,axiom,
! [A,B] : r2_hidden(k1_tarski(k1_xboole_0),k1_substlat(A,B)) ).
fof(t3_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k4_substlat(A,B,C,D) = k4_substlat(A,B,D,C) ) ) ).
fof(t4_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ( C = k1_tarski(k1_xboole_0)
=> k4_substlat(A,B,D,C) = D ) ) ) ).
fof(t5_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D,E] :
( ( r2_hidden(C,k1_substlat(A,B))
& r2_hidden(D,C)
& r2_hidden(E,C)
& r1_tarski(D,E) )
=> D = E ) ) ).
fof(t6_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( r2_hidden(D,k3_substlat(A,B,C))
=> ( r2_hidden(D,C)
& ! [E] :
( ( r2_hidden(E,C)
& r1_tarski(E,D) )
=> E = D ) ) ) ) ).
fof(t7_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( v1_finset_1(D)
=> ( ( r2_hidden(D,C)
& ! [E] :
( v1_finset_1(E)
=> ( ( r2_hidden(E,C)
& r1_tarski(E,D) )
=> E = D ) ) )
=> r2_hidden(D,k3_substlat(A,B,C)) ) ) ) ).
fof(t8_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> r1_tarski(k3_substlat(A,B,C),C) ) ).
fof(t9_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ( C = k1_xboole_0
=> k3_substlat(A,B,C) = k1_xboole_0 ) ) ).
fof(t10_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( v1_finset_1(D)
=> ~ ( r2_hidden(D,C)
& ! [E] :
~ ( r1_tarski(E,D)
& r2_hidden(E,k3_substlat(A,B,C)) ) ) ) ) ).
fof(t11_substlat,axiom,
! [A,B,C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> k3_substlat(A,B,C) = C ) ).
fof(t12_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> r1_tarski(k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),C,D)),k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k3_substlat(A,B,C),D)) ) ) ).
fof(t13_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k3_substlat(A,B,C),D)) = k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),C,D)) ) ) ).
fof(t14_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
=> ( r1_tarski(C,D)
=> r1_tarski(k4_substlat(A,B,C,E),k4_substlat(A,B,D,E)) ) ) ) ) ).
fof(t15_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
~ ( r2_hidden(E,k4_substlat(A,B,C,D))
& ! [F,G] :
~ ( r2_hidden(F,C)
& r2_hidden(G,D)
& E = k2_xboole_0(F,G) ) ) ) ) ).
fof(t16_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
( m1_subset_1(E,k4_partfun1(A,B))
=> ! [F] :
( m1_subset_1(F,k4_partfun1(A,B))
=> ( ( r2_hidden(E,C)
& r2_hidden(F,D)
& r1_partfun1(E,F) )
=> r2_hidden(k2_xboole_0(E,F),k4_substlat(A,B,C,D)) ) ) ) ) ) ).
fof(t17_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> r1_tarski(k3_substlat(A,B,k4_substlat(A,B,C,D)),k4_substlat(A,B,k3_substlat(A,B,C),D)) ) ) ).
fof(t18_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
=> ( r1_tarski(C,D)
=> r1_tarski(k4_substlat(A,B,E,C),k4_substlat(A,B,E,D)) ) ) ) ) ).
fof(t19_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k3_substlat(A,B,k4_substlat(A,B,k3_substlat(A,B,C),D)) = k3_substlat(A,B,k4_substlat(A,B,C,D)) ) ) ).
fof(t20_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k3_substlat(A,B,k4_substlat(A,B,C,k3_substlat(A,B,D))) = k3_substlat(A,B,k4_substlat(A,B,C,D)) ) ) ).
fof(t21_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
=> k4_substlat(A,B,C,k4_substlat(A,B,D,E)) = k4_substlat(A,B,k4_substlat(A,B,C,D),E) ) ) ) ).
fof(t22_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
=> k4_substlat(A,B,C,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),D,E)) = k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k4_substlat(A,B,C,D),k4_substlat(A,B,C,E)) ) ) ) ).
fof(t23_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> r1_tarski(C,k4_substlat(A,B,C,C)) ) ).
fof(t24_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> k3_substlat(A,B,k4_substlat(A,B,C,C)) = k3_substlat(A,B,C) ) ).
fof(t25_substlat,axiom,
! [A,B,C] :
( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> k3_substlat(A,B,k4_substlat(A,B,C,C)) = C ) ).
fof(d4_substlat,axiom,
! [A,B,C] :
( ( v3_lattices(C)
& l3_lattices(C) )
=> ( C = k5_substlat(A,B)
<=> ( u1_struct_0(C) = k1_substlat(A,B)
& ! [D] :
( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ! [E] :
( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
=> ( k1_binop_1(u2_lattices(C),D,E) = k3_substlat(A,B,k2_substlat(A,B,D,E))
& k1_binop_1(u1_lattices(C),D,E) = k3_substlat(A,B,k4_substlat(A,B,D,E)) ) ) ) ) ) ) ).
fof(t26_substlat,axiom,
! [A,B] : k5_lattices(k5_substlat(A,B)) = k1_xboole_0 ).
fof(t27_substlat,axiom,
! [A,B] : k6_lattices(k5_substlat(A,B)) = k1_tarski(k1_xboole_0) ).
fof(dt_k1_substlat,axiom,
! [A,B] : m1_subset_1(k1_substlat(A,B),k1_zfmisc_1(k5_finsub_1(k4_partfun1(A,B)))) ).
fof(dt_k2_substlat,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_substlat(A,B))
& m1_subset_1(D,k1_substlat(A,B)) )
=> m1_subset_1(k2_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) ).
fof(commutativity_k2_substlat,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_substlat(A,B))
& m1_subset_1(D,k1_substlat(A,B)) )
=> k2_substlat(A,B,C,D) = k2_substlat(A,B,D,C) ) ).
fof(idempotence_k2_substlat,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_substlat(A,B))
& m1_subset_1(D,k1_substlat(A,B)) )
=> k2_substlat(A,B,C,C) = C ) ).
fof(redefinition_k2_substlat,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_substlat(A,B))
& m1_subset_1(D,k1_substlat(A,B)) )
=> k2_substlat(A,B,C,D) = k2_xboole_0(C,D) ) ).
fof(dt_k3_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> m2_subset_1(k3_substlat(A,B,C),k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B)) ) ).
fof(dt_k4_substlat,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
& m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
=> m1_subset_1(k4_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) ).
fof(dt_k5_substlat,axiom,
! [A,B] :
( v3_lattices(k5_substlat(A,B))
& l3_lattices(k5_substlat(A,B)) ) ).
fof(d1_substlat,axiom,
! [A,B] : k1_substlat(A,B) = a_2_0_substlat(A,B) ).
fof(d2_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> k3_substlat(A,B,C) = a_3_0_substlat(A,B,C) ) ).
fof(d3_substlat,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> k4_substlat(A,B,C,D) = a_4_0_substlat(A,B,C,D) ) ) ).
fof(fraenkel_a_2_0_substlat,axiom,
! [A,B,C] :
( r2_hidden(A,a_2_0_substlat(B,C))
<=> ? [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
& A = D
& ! [E] :
( r2_hidden(E,D)
=> v1_finset_1(E) )
& ! [E] :
( m1_subset_1(E,k4_partfun1(B,C))
=> ! [F] :
( m1_subset_1(F,k4_partfun1(B,C))
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D)
& r1_tarski(E,F) )
=> E = F ) ) ) ) ) ).
fof(fraenkel_a_3_0_substlat,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
=> ( r2_hidden(A,a_3_0_substlat(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k4_partfun1(B,C))
& A = E
& v1_finset_1(E)
& ! [F] :
( m1_subset_1(F,k4_partfun1(B,C))
=> ( ( r2_hidden(F,D)
& r1_tarski(F,E) )
<=> F = E ) ) ) ) ) ).
fof(fraenkel_a_4_0_substlat,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
& m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C))) )
=> ( r2_hidden(A,a_4_0_substlat(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,k4_partfun1(B,C))
& m1_subset_1(G,k4_partfun1(B,C))
& A = k2_xboole_0(F,G)
& r2_hidden(F,D)
& r2_hidden(G,E)
& r1_partfun1(F,G) ) ) ) ).
%------------------------------------------------------------------------------