SET007 Axioms: SET007+470.ax
%------------------------------------------------------------------------------
% File : SET007+470 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Examples of Category Structures
% 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 : msinst_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 18 ( 2 unt; 0 def)
% Number of atoms : 200 ( 22 equ)
% Maximal formula atoms : 26 ( 11 avg)
% Number of connectives : 236 ( 54 ~; 0 |; 107 &)
% ( 6 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 13 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-4 aty)
% Number of functors : 25 ( 25 usr; 2 con; 0-6 aty)
% Number of variables : 76 ( 71 !; 5 ?)
% 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_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k1_msinst_1(A))
& v2_altcat_1(k1_msinst_1(A))
& v6_altcat_1(k1_msinst_1(A))
& v11_altcat_1(k1_msinst_1(A))
& v12_altcat_1(k1_msinst_1(A)) ) ) ).
fof(rc1_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A)
& v1_msualg_6(B,A) ) ) ).
fof(fc2_msinst_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_msinst_1(A,B)) ) ).
fof(fc3_msinst_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& ~ v1_xboole_0(B) )
=> ( ~ v3_struct_0(k4_msinst_1(A,B))
& v2_altcat_1(k4_msinst_1(A,B))
& v6_altcat_1(k4_msinst_1(A,B))
& v11_altcat_1(k4_msinst_1(A,B))
& v12_altcat_1(k4_msinst_1(A,B)) ) ) ).
fof(d1_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_altcat_1(B)
& l2_altcat_1(B) )
=> ( B = k1_msinst_1(A)
<=> ( u1_struct_0(B) = k5_msalimit(A)
& ! [C] :
( m3_msalimit(C,k5_msalimit(A))
=> ! [D] :
( m3_msalimit(D,k5_msalimit(A))
=> k1_binop_1(u1_altcat_1(B),C,D) = k6_msalimit(C,D) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( ( r2_hidden(C,k5_msalimit(A))
& r2_hidden(D,k5_msalimit(A))
& r2_hidden(E,k5_msalimit(A)) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ! [H] :
( ( v1_relat_1(H)
& v1_funct_1(H) )
=> ! [I] :
( ( v1_relat_1(I)
& v1_funct_1(I) )
=> ( ( r2_hidden(k4_tarski(F,G),k1_binop_1(u1_altcat_1(B),C,D))
& r2_hidden(k4_tarski(H,I),k1_binop_1(u1_altcat_1(B),D,E)) )
=> k1_binop_1(k4_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B),C,D,E),k4_tarski(H,I),k4_tarski(F,G)) = k4_tarski(k5_relat_1(F,H),k5_relat_1(G,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v11_altcat_1(B)
& v12_altcat_1(B)
& l2_altcat_1(B) )
=> ( B = k1_msinst_1(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) ) ) ) ) ) ).
fof(d2_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( C = k2_msinst_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v4_msualg_1(E,A)
& v1_msualg_6(E,A)
& l3_msualg_1(E,A)
& D = E
& ! [F] :
( m1_subset_1(F,k2_relat_1(u4_msualg_1(A,E)))
=> r1_tarski(F,B) ) ) ) ) ) ) ).
fof(t2_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( l3_msualg_1(C,B)
=> ( r2_hidden(C,k2_msinst_1(B,A))
=> ( r2_hidden(u4_msualg_1(B,C),k1_funct_2(u1_struct_0(B),k1_zfmisc_1(A)))
& r2_hidden(u5_msualg_1(B,C),k1_funct_2(u1_msualg_1(B),k4_partfun1(k4_partfun1(k5_numbers,A),A))) ) ) ) ) ) ).
fof(t3_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_msualg_1(A))
=> ! [C] :
( l3_msualg_1(C,A)
=> ! [D] :
( l3_msualg_1(D,A)
=> ~ ( r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
& k3_msualg_1(A,B,C) != k1_xboole_0
& k3_msualg_1(A,B,D) = k1_xboole_0 ) ) ) ) ) ).
fof(t4_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_msualg_1(A))
=> ! [C] :
( ( v1_msualg_6(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v1_msualg_6(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( ( v1_msualg_6(E,A)
& l3_msualg_1(E,A) )
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
=> ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,E))
=> ! [H] :
( m1_subset_1(H,k3_msualg_1(A,B,C))
=> ~ ( k3_msualg_1(A,B,C) != k1_xboole_0
& r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
& r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,E))
& ! [I] :
( m3_pboole(I,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,E))
=> ~ ( I = k13_pboole(F,G)
& k5_msualg_3(A,C,E,B,I,H) = k5_msualg_3(A,D,E,B,G,k5_msualg_3(A,C,D,B,F,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t5_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v1_msualg_6(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_msualg_6(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v1_msualg_6(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
=> ~ ( r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
& r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
& r1_msualg_3(A,B,C,E)
& r1_msualg_3(A,C,D,F)
& ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D))
=> ~ ( G = k13_pboole(E,F)
& r1_msualg_3(A,B,D,G) ) ) ) ) ) ) ) ) ) ).
fof(d3_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D] :
( ( r2_hidden(C,k2_msinst_1(A,B))
& r2_hidden(D,k2_msinst_1(A,B)) )
=> ! [E] :
( E = k3_msinst_1(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ? [G] :
( v4_msualg_1(G,A)
& v1_msualg_6(G,A)
& l3_msualg_1(G,A)
& ? [H] :
( v4_msualg_1(H,A)
& v1_msualg_6(H,A)
& l3_msualg_1(H,A)
& ? [I] :
( m3_pboole(I,u1_struct_0(A),u4_msualg_1(A,G),u4_msualg_1(A,H))
& G = C
& H = D
& I = F
& r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,G),u4_msualg_1(A,H))
& r1_msualg_3(A,G,H,I) ) ) ) ) ) ) ) ) ).
fof(d4_msinst_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_altcat_1(C)
& l2_altcat_1(C) )
=> ( C = k4_msinst_1(A,B)
<=> ( u1_struct_0(C) = k2_msinst_1(A,B)
& ! [D] :
( m1_subset_1(D,k2_msinst_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_msinst_1(A,B))
=> k1_binop_1(u1_altcat_1(C),D,E) = k3_msinst_1(A,B,D,E) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G)
& v1_funcop_1(G) )
=> ! [H] :
( ( v1_relat_1(H)
& v1_funct_1(H)
& v1_funcop_1(H) )
=> ( ( r2_hidden(G,k1_binop_1(u1_altcat_1(C),D,E))
& r2_hidden(H,k1_binop_1(u1_altcat_1(C),E,F)) )
=> k1_binop_1(k4_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C),D,E,F),H,G) = k13_pboole(G,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_altcat_1(C)
& v11_altcat_1(C)
& v12_altcat_1(C)
& l2_altcat_1(C) )
=> ( C = k4_msinst_1(B,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( v4_msualg_1(D,B)
& v1_msualg_6(D,B)
& l3_msualg_1(D,B) ) ) ) ) ) ) ).
fof(dt_k1_msinst_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k1_msinst_1(A))
& v6_altcat_1(k1_msinst_1(A))
& l2_altcat_1(k1_msinst_1(A)) ) ) ).
fof(dt_k2_msinst_1,axiom,
$true ).
fof(dt_k3_msinst_1,axiom,
$true ).
fof(dt_k4_msinst_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& ~ v1_xboole_0(B) )
=> ( ~ v3_struct_0(k4_msinst_1(A,B))
& v6_altcat_1(k4_msinst_1(A,B))
& l2_altcat_1(k4_msinst_1(A,B)) ) ) ).
%------------------------------------------------------------------------------