SET007 Axioms: SET007+363.ax
%------------------------------------------------------------------------------
% File : SET007+363 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Notation of Universal Algebra
% 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 : unialg_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 3 unt; 0 def)
% Number of atoms : 181 ( 19 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 173 ( 27 ~; 0 |; 78 &)
% ( 9 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 3 con; 0-2 aty)
% Number of variables : 67 ( 61 !; 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(rc1_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_relset_1(B,k13_finseq_1(A),A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_unialg_1(B,A)
& v2_unialg_1(B,A) ) ) ).
fof(rc2_unialg_1,axiom,
? [A] :
( l1_unialg_1(A)
& v3_unialg_1(A) ) ).
fof(rc3_unialg_1,axiom,
? [A] :
( l1_unialg_1(A)
& ~ v3_struct_0(A)
& v3_unialg_1(A) ) ).
fof(fc1_unialg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k4_partfun1(k13_finseq_1(A),A)) )
=> ( ~ v3_struct_0(g1_unialg_1(A,B))
& v3_unialg_1(g1_unialg_1(A,B)) ) ) ).
fof(rc4_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v4_unialg_1(B,A)
& v5_unialg_1(B,A) ) ) ).
fof(rc5_unialg_1,axiom,
? [A] :
( l1_unialg_1(A)
& ~ v3_struct_0(A)
& v3_unialg_1(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A) ) ).
fof(fc2_unialg_1,axiom,
! [A] :
( ( v6_unialg_1(A)
& l1_unialg_1(A) )
=> ( v1_relat_1(u1_unialg_1(A))
& v1_funct_1(u1_unialg_1(A))
& v1_finseq_1(u1_unialg_1(A))
& v4_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ).
fof(fc3_unialg_1,axiom,
! [A] :
( ( v7_unialg_1(A)
& l1_unialg_1(A) )
=> ( v1_relat_1(u1_unialg_1(A))
& v1_funct_1(u1_unialg_1(A))
& v1_finseq_1(u1_unialg_1(A))
& v5_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ).
fof(fc4_unialg_1,axiom,
! [A] :
( ( v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( ~ v1_xboole_0(u1_unialg_1(A))
& v1_relat_1(u1_unialg_1(A))
& v2_relat_1(u1_unialg_1(A))
& v1_funct_1(u1_unialg_1(A))
& v1_finseq_1(u1_unialg_1(A)) ) ) ).
fof(d1_unialg_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k13_finseq_1(A),A) )
=> ( v1_unialg_1(B,A)
<=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( ( r2_hidden(C,k1_relat_1(B))
& r2_hidden(D,k1_relat_1(B)) )
=> k3_finseq_1(C) = k3_finseq_1(D) ) ) ) ) ) ).
fof(d2_unialg_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k13_finseq_1(A),A) )
=> ( v2_unialg_1(B,A)
<=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& r2_hidden(C,k1_relat_1(B)) )
=> r2_hidden(D,k1_relat_1(B)) ) ) ) ) ) ).
fof(t1_unialg_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k13_finseq_1(A),A) )
=> ( ~ ( ~ v1_xboole_0(B)
& k1_relat_1(B) = k1_xboole_0 )
& ~ ( k1_relat_1(B) != k1_xboole_0
& v1_xboole_0(B) ) ) ) ).
fof(t2_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ v1_xboole_0(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B))
& v1_funct_1(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B))
& v1_unialg_1(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B),A)
& v2_unialg_1(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B),A)
& m2_relset_1(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B),k13_finseq_1(A),A) ) ) ) ).
fof(t3_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> m1_subset_1(k2_funcop_1(k1_tarski(k6_finseq_1(A)),B),k4_partfun1(k13_finseq_1(A),A)) ) ) ).
fof(d3_unialg_1,axiom,
$true ).
fof(d4_unialg_1,axiom,
! [A,B] :
( m2_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
=> ( v4_unialg_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k13_finseq_1(A),A) )
=> ( ( r2_hidden(C,k4_finseq_1(B))
& D = k1_funct_1(B,C) )
=> v1_unialg_1(D,A) ) ) ) ) ) ).
fof(d5_unialg_1,axiom,
! [A,B] :
( m2_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
=> ( v5_unialg_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k13_finseq_1(A),A) )
=> ( ( r2_hidden(C,k4_finseq_1(B))
& D = k1_funct_1(B,C) )
=> v2_unialg_1(D,A) ) ) ) ) ) ).
fof(d6_unialg_1,axiom,
$true ).
fof(d7_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> ( v6_unialg_1(A)
<=> v4_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ).
fof(d8_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> ( v7_unialg_1(A)
<=> v5_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ).
fof(d9_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> ( v8_unialg_1(A)
<=> ( u1_unialg_1(A) != k1_xboole_0
& v2_relat_1(u1_unialg_1(A)) ) ) ) ).
fof(t4_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,k4_partfun1(k13_finseq_1(A),A))
=> ( C = k2_funcop_1(k1_tarski(k6_finseq_1(A)),B)
=> ( v4_unialg_1(k1_unialg_1(A,C),A)
& v5_unialg_1(k1_unialg_1(A,C),A)
& v2_relat_1(k1_unialg_1(A,C)) ) ) ) ) ) ).
fof(d10_unialg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(B)
& v1_unialg_1(B,A)
& m2_relset_1(B,k13_finseq_1(A),A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k2_unialg_1(A,B)
<=> ! [D] :
( m2_finseq_1(D,A)
=> ( r2_hidden(D,k1_relat_1(B))
=> C = k3_finseq_1(D) ) ) ) ) ) ) ).
fof(t5_unialg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k4_finseq_1(u1_unialg_1(A)))
=> ( v1_funct_1(k1_funct_1(u1_unialg_1(A),B))
& m2_relset_1(k1_funct_1(u1_unialg_1(A),B),k13_finseq_1(u1_struct_0(A)),u1_struct_0(A)) ) ) ) ) ).
fof(d11_unialg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ( B = k3_unialg_1(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(u1_unialg_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_funct_1(D)
& v1_unialg_1(D,u1_struct_0(A))
& m2_relset_1(D,k13_finseq_1(u1_struct_0(A)),u1_struct_0(A)) )
=> ( D = k1_funct_1(u1_unialg_1(A),C)
=> k1_funct_1(B,C) = k2_unialg_1(u1_struct_0(A),D) ) ) ) ) ) ) ) ) ).
fof(dt_l1_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_unialg_1,axiom,
? [A] : l1_unialg_1(A) ).
fof(abstractness_v3_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> ( v3_unialg_1(A)
=> A = g1_unialg_1(u1_struct_0(A),u1_unialg_1(A)) ) ) ).
fof(dt_k1_unialg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k4_partfun1(k13_finseq_1(A),A)) )
=> m2_finseq_1(k1_unialg_1(A,B),k4_partfun1(k13_finseq_1(A),A)) ) ).
fof(redefinition_k1_unialg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k4_partfun1(k13_finseq_1(A),A)) )
=> k1_unialg_1(A,B) = k5_finseq_1(B) ) ).
fof(dt_k2_unialg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(B)
& v1_unialg_1(B,A)
& m1_relset_1(B,k13_finseq_1(A),A) )
=> m2_subset_1(k2_unialg_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k3_unialg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> m2_finseq_1(k3_unialg_1(A),k5_numbers) ) ).
fof(dt_u1_unialg_1,axiom,
! [A] :
( l1_unialg_1(A)
=> m2_finseq_1(u1_unialg_1(A),k4_partfun1(k13_finseq_1(u1_struct_0(A)),u1_struct_0(A))) ) ).
fof(dt_g1_unialg_1,axiom,
! [A,B] :
( m1_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
=> ( v3_unialg_1(g1_unialg_1(A,B))
& l1_unialg_1(g1_unialg_1(A,B)) ) ) ).
fof(free_g1_unialg_1,axiom,
! [A,B] :
( m1_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
=> ! [C,D] :
( g1_unialg_1(A,B) = g1_unialg_1(C,D)
=> ( A = C
& B = D ) ) ) ).
%------------------------------------------------------------------------------