SET007 Axioms: SET007+561.ax
%------------------------------------------------------------------------------
% File : SET007+561 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Representation Theorem for Free Continuous Lattices
% 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 : waybel22 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 5 unt; 0 def)
% Number of atoms : 327 ( 25 equ)
% Maximal formula atoms : 23 ( 10 avg)
% Number of connectives : 324 ( 29 ~; 0 |; 236 &)
% ( 7 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 40 ( 39 usr; 0 prp; 1-3 aty)
% Number of functors : 25 ( 25 usr; 0 con; 1-5 aty)
% Number of variables : 92 ( 83 !; 9 ?)
% 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_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
& v1_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v2_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v3_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v4_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v1_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v2_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v3_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v24_waybel_0(k2_yellow_1(k9_waybel_0(A)))
& v25_waybel_0(k2_yellow_1(k9_waybel_0(A)))
& v1_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v2_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v3_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v3_waybel_3(k2_yellow_1(k9_waybel_0(A))) ) ) ).
fof(cc1_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(k9_waybel_0(A))))
=> ~ v1_xboole_0(B) ) ) ).
fof(fc2_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_relat_1(k2_waybel22(A,B,C))
& v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(fc3_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_relat_1(k2_waybel22(A,B,C))
& v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& v17_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v19_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v21_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(t1_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_waybel_0(B,k2_yellow_1(k9_waybel_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_yellow_1(k9_waybel_0(A))))) )
=> k1_yellow_0(k2_yellow_1(k9_waybel_0(A)),B) = k3_tarski(B) ) ) ).
fof(t2_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& l1_orders_2(C) )
=> ! [D] :
( m1_waybel16(D,A,B)
=> ! [E] :
( m1_waybel16(E,B,C)
=> m1_waybel16(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ).
fof(t3_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> v17_waybel_0(k7_grcat_1(A),A,A) ) ).
fof(t4_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> v22_waybel_0(k7_grcat_1(A),A,A) ) ).
fof(t5_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> m1_waybel16(k7_grcat_1(A),A,A) ) ).
fof(t6_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
& v4_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& v7_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& v4_waybel_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& m1_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A))) ) ) ).
fof(d1_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
<=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& v3_waybel_3(C)
& l1_orders_2(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,u1_struct_0(C))
& m2_relset_1(D,B,u1_struct_0(C)) )
=> ? [E] :
( m1_waybel16(E,A,C)
& k7_relat_1(E,B) = D
& ! [F] :
( m1_waybel16(F,A,C)
=> ( k7_relat_1(F,B) = D
=> F = E ) ) ) ) ) ) ) ).
fof(t7_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t8_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
=> ! [C] :
( m1_waybel16(C,A,A)
=> ( k7_relat_1(C,B) = k1_pralg_3(B)
=> C = k7_grcat_1(A) ) ) ) ) ).
fof(t9_waybel22,axiom,
! [A] : r1_tarski(k1_waybel22(A),k9_waybel_0(k3_yellow_1(A))) ).
fof(t10_waybel22,axiom,
! [A] : k1_card_1(k1_waybel22(A)) = k1_card_1(A) ).
fof(t12_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> v5_orders_3(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(t13_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,k2_waybel22(A,B,C),k4_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))))) = k4_yellow_0(B) ) ) ).
fof(t14_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> k7_relat_1(k2_waybel22(A,B,C),k1_waybel22(A)) = C ) ) ).
fof(t15_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ! [D] :
( m1_waybel16(D,k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
=> ( k7_relat_1(D,k1_waybel22(A)) = C
=> D = k2_waybel22(A,B,C) ) ) ) ) ).
fof(t16_waybel22,axiom,
! [A] : r1_waybel22(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),k1_waybel22(A)) ).
fof(t17_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C,D] :
( ( r1_waybel22(A,C)
& r1_waybel22(B,D)
& k1_card_1(C) = k1_card_1(D) )
=> r5_waybel_1(A,B) ) ) ) ).
fof(t18_waybel22,axiom,
! [A,B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( r1_waybel22(B,C)
& k1_card_1(C) = k1_card_1(A) )
=> r5_waybel_1(B,k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))) ) ) ).
fof(dt_k1_waybel22,axiom,
! [A] : m1_subset_1(k1_waybel22(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_yellow_1(A))))) ).
fof(dt_k2_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& m2_relset_1(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) ) ) ).
fof(d2_waybel22,axiom,
! [A] : k1_waybel22(A) = a_1_0_waybel22(A) ).
fof(t11_waybel22,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_waybel_0(B,k3_yellow_1(A))
& v13_waybel_0(B,k3_yellow_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_yellow_1(A)))) )
=> B = k1_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),a_2_0_waybel22(A,B)) ) ).
fof(d3_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) )
=> ( D = k2_waybel22(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))))
=> k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,D,E) = k1_yellow_0(B,a_4_0_waybel22(A,B,C,E)) ) ) ) ) ) ).
fof(fraenkel_a_1_0_waybel22,axiom,
! [A,B] :
( r2_hidden(A,a_1_0_waybel22(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k3_yellow_1(B)))
& A = k7_waybel_0(k3_yellow_1(B),C)
& ? [D] :
( m1_subset_1(D,B)
& C = k1_tarski(D) ) ) ) ).
fof(fraenkel_a_2_0_waybel22,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v2_waybel_0(C,k3_yellow_1(B))
& v13_waybel_0(C,k3_yellow_1(B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_yellow_1(B)))) )
=> ( r2_hidden(A,a_2_0_waybel22(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = k2_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(B))),a_2_1_waybel22(B,D))
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_2_1_waybel22,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_1_waybel22(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k3_yellow_1(B)))
& A = k7_waybel_0(k3_yellow_1(B),D)
& ? [E] :
( m1_subset_1(E,B)
& D = k1_tarski(E)
& r2_hidden(E,C) ) ) ) ) ).
fof(fraenkel_a_4_0_waybel22,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& v3_waybel_3(C)
& l1_orders_2(C)
& v1_funct_1(D)
& v1_funct_2(D,k1_waybel22(B),u1_struct_0(C))
& m2_relset_1(D,k1_waybel22(B),u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(B))))) )
=> ( r2_hidden(A,a_4_0_waybel22(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& A = k2_yellow_0(C,a_4_1_waybel22(B,C,D,F))
& r2_hidden(F,E) ) ) ) ).
fof(fraenkel_a_4_1_waybel22,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& v3_waybel_3(C)
& l1_orders_2(C)
& v1_funct_1(D)
& v1_funct_2(D,k1_waybel22(B),u1_struct_0(C))
& m2_relset_1(D,k1_waybel22(B),u1_struct_0(C))
& m1_subset_1(E,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_4_1_waybel22(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k3_yellow_1(B)))
& A = k1_funct_1(D,k7_waybel_0(k3_yellow_1(B),F))
& ? [G] :
( m1_subset_1(G,B)
& F = k1_tarski(G)
& r2_hidden(G,E) ) ) ) ) ).
%------------------------------------------------------------------------------