SET007 Axioms: SET007+737.ax
%------------------------------------------------------------------------------
% File : SET007+737 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homomorphisms of Order Sorted Algebras
% 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 : osalg_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 1 unt; 0 def)
% Number of atoms : 379 ( 4 equ)
% Maximal formula atoms : 21 ( 13 avg)
% Number of connectives : 401 ( 51 ~; 0 |; 233 &)
% ( 7 <=>; 110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 17 ( 17 usr; 0 con; 1-6 aty)
% Number of variables : 105 ( 103 !; 2 ?)
% 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_osalg_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A)
& v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v1_relat_1(k2_msualg_3(u1_struct_0(A),B))
& v1_funct_1(k2_msualg_3(u1_struct_0(A),B))
& v1_funcop_1(k2_msualg_3(u1_struct_0(A),B))
& v1_osalg_3(k2_msualg_3(u1_struct_0(A),B),A) ) ) ).
fof(rc1_osalg_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v12_osalg_1(B,A)
& v13_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msualg_2(C,A,B)
& v12_osalg_1(C,A)
& v13_osalg_1(C,A) ) ) ).
fof(cc1_osalg_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v12_osalg_1(B,A)
& v13_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_msualg_2(C,A,B)
=> ( v12_osalg_1(C,A)
=> ( v12_osalg_1(C,A)
& v13_osalg_1(C,A) ) ) ) ) ).
fof(fc2_osalg_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v4_msualg_1(g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)),A)
& v12_osalg_1(g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)),A) ) ) ).
fof(d1_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funcop_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v1_osalg_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_orders_2(A,C,D)
=> ! [E] :
( r2_hidden(E,k1_relat_1(k1_funct_1(B,C)))
=> ( r2_hidden(E,k1_relat_1(k1_funct_1(B,D)))
& k1_funct_1(k1_funct_1(B,C),E) = k1_funct_1(k1_funct_1(B,D),E) ) ) ) ) ) ) ) ) ).
fof(t1_osalg_3,axiom,
$true ).
fof(t2_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funcop_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v1_osalg_3(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_orders_2(A,C,D)
=> ( r1_tarski(k1_relat_1(k1_funct_1(B,C)),k1_relat_1(k1_funct_1(B,D)))
& r1_tarski(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) ) ).
fof(t3_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( ( v2_relat_1(C)
& v11_osalg_1(C,A)
& m1_pboole(C,u1_struct_0(A)) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),B,C)
=> ( v1_osalg_3(D,A)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r3_orders_2(A,E,F)
=> ! [G] :
( r2_hidden(G,k1_funct_1(B,E))
=> k1_funct_1(k1_msualg_3(u1_struct_0(A),B,C,D,E),G) = k1_funct_1(k1_msualg_3(u1_struct_0(A),B,C,D,F),G) ) ) ) ) ) ) ) ) ) ).
fof(t4_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funcop_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ( v1_osalg_3(B,A)
=> ! [C] :
( m1_subset_1(C,k13_finseq_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k13_finseq_1(u1_struct_0(A)))
=> ( r2_osalg_1(A,C,D)
=> r1_tarski(k1_funct_1(k6_pboole(u1_struct_0(A),B),C),k1_funct_1(k6_pboole(u1_struct_0(A),B),D)) ) ) ) ) ) ) ).
fof(t5_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> v1_osalg_3(k2_msualg_3(u1_struct_0(A),B),A) ) ) ).
fof(t6_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( ( v2_relat_1(C)
& v11_osalg_1(C,A)
& m1_pboole(C,u1_struct_0(A)) )
=> ! [D] :
( ( v2_relat_1(D)
& v11_osalg_1(D,A)
& m1_pboole(D,u1_struct_0(A)) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),B,C)
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),C,D)
=> ( ( v1_osalg_3(E,A)
& v1_osalg_3(F,A) )
=> v1_osalg_3(k3_msualg_3(u1_struct_0(A),B,C,D,E,F),A) ) ) ) ) ) ) ) ).
fof(t7_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( ( v11_osalg_1(C,A)
& m1_pboole(C,u1_struct_0(A)) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),B,C)
=> ( ( v1_msualg_3(D)
& v2_msualg_3(D,u1_struct_0(A),B,C)
& v1_osalg_3(D,A) )
=> v1_osalg_3(k4_msualg_3(u1_struct_0(A),B,C,D),A) ) ) ) ) ) ).
fof(t8_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v11_osalg_1(B,A)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,u1_struct_0(A)) )
=> ( v1_osalg_3(C,A)
=> ( v11_osalg_1(k14_pboole(u1_struct_0(A),B,C),A)
& m1_pboole(k14_pboole(u1_struct_0(A),B,C),u1_struct_0(A)) ) ) ) ) ) ).
fof(d2_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r1_osalg_3(A,B,C)
<=> ? [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
& r4_msualg_3(A,B,C,D)
& v1_osalg_3(D,A) ) ) ) ) ) ).
fof(t9_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> r1_osalg_3(A,B,B) ) ) ).
fof(t10_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r1_osalg_3(A,B,C)
=> r1_osalg_3(A,C,B) ) ) ) ) ).
fof(t11_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v5_msualg_1(D,A)
& v12_osalg_1(D,A)
& l3_msualg_1(D,A) )
=> ( ( r3_osalg_3(A,B,C)
& r3_osalg_3(A,C,D) )
=> r3_osalg_3(A,B,D) ) ) ) ) ) ).
fof(t12_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( ( v1_osalg_3(D,A)
& r1_msualg_3(A,B,C,D) )
=> v12_osalg_1(k7_msualg_3(A,B,C,D),A) ) ) ) ) ) ).
fof(t13_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( v1_osalg_3(D,A)
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(A))
=> ! [F] :
( m1_subset_1(F,u1_msualg_1(A))
=> ( r6_osalg_1(A,E,F)
=> ! [G] :
( m1_subset_1(G,k3_msualg_1(A,E,B))
=> ! [H] :
( m1_subset_1(H,k3_msualg_1(A,F,B))
=> ( G = H
=> k6_msualg_3(A,B,C,E,D,G) = k6_msualg_3(A,B,C,F,D,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& v13_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( ( v1_osalg_3(D,A)
& r1_msualg_3(A,B,C,D) )
=> ( v12_osalg_1(k7_msualg_3(A,B,C,D),A)
& v12_osalg_1(k7_msualg_3(A,B,C,D),A)
& v13_osalg_1(k7_msualg_3(A,B,C,D),A)
& l3_msualg_1(k7_msualg_3(A,B,C,D),A) ) ) ) ) ) ) ).
fof(t15_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v12_osalg_1(B,A)
& v13_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v12_osalg_1(C,A)
& m1_msualg_2(C,A,B) )
=> v13_osalg_1(C,A) ) ) ) ).
fof(t16_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ~ ( r1_msualg_3(A,B,C,D)
& v1_osalg_3(D,A)
& ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k7_msualg_3(A,B,C,D)))
=> ~ ( r6_pboole(u1_struct_0(A),D,E)
& v1_osalg_3(E,A)
& r2_msualg_3(A,B,k7_msualg_3(A,B,C,D),E) ) ) ) ) ) ) ) ).
fof(t17_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ~ ( r1_msualg_3(A,B,C,D)
& v1_osalg_3(D,A)
& ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k7_msualg_3(A,B,C,D)))
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,k7_msualg_3(A,B,C,D)),u4_msualg_1(A,C))
=> ~ ( r2_msualg_3(A,B,k7_msualg_3(A,B,C,D),E)
& r3_msualg_3(A,k7_msualg_3(A,B,C,D),C,F)
& r6_pboole(u1_struct_0(A),D,k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,k7_msualg_3(A,B,C,D)),u4_msualg_1(A,C),E,F))
& v1_osalg_3(E,A)
& v1_osalg_3(F,A) ) ) ) ) ) ) ) ) ).
fof(t18_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v13_osalg_1(B,A)
<=> v13_osalg_1(g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)),A) ) ) ) ).
fof(t19_osalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r3_osalg_3(A,B,C)
=> ( v13_osalg_1(B,A)
<=> v13_osalg_1(C,A) ) ) ) ) ) ).
fof(reflexivity_r2_osalg_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> r2_osalg_3(A,B,B) ) ).
fof(redefinition_r2_osalg_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r2_osalg_3(A,B,C)
<=> r1_osalg_3(A,B,C) ) ) ).
fof(symmetry_r3_osalg_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A)
& v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r3_osalg_3(A,B,C)
=> r3_osalg_3(A,C,B) ) ) ).
fof(redefinition_r3_osalg_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& v4_osalg_1(A)
& v5_osalg_1(A)
& l3_osalg_1(A)
& v5_msualg_1(B,A)
& v12_osalg_1(B,A)
& l3_msualg_1(B,A)
& v5_msualg_1(C,A)
& v12_osalg_1(C,A)
& l3_msualg_1(C,A) )
=> ( r3_osalg_3(A,B,C)
<=> r1_osalg_3(A,B,C) ) ) ).
%------------------------------------------------------------------------------