SET007 Axioms: SET007+462.ax
%------------------------------------------------------------------------------
% File : SET007+462 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Correspondence Between Signatures and Graphs. Part II
% Version : [Urb08] axioms.
% English : Correspondence Between Monotonic Many Sorted Signatures and
% Well-Founded Graphs. Part II
% 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 : msscyc_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 15 ( 1 unt; 0 def)
% Number of atoms : 101 ( 16 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 103 ( 17 ~; 0 |; 50 &)
% ( 6 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 26 ( 26 usr; 3 con; 0-4 aty)
% Number of variables : 39 ( 32 !; 7 ?)
% 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(d1_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> ! [B] :
( B = k1_msscyc_2(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D,E] :
( C = k4_tarski(D,E)
& r2_hidden(D,u1_msualg_1(A))
& r2_hidden(E,u1_struct_0(A))
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ? [G] :
( m1_subset_1(G,k13_finseq_1(u1_struct_0(A)))
& k1_funct_1(u2_msualg_1(A),D) = G
& r2_hidden(F,k4_finseq_1(G))
& k1_funct_1(G,F) = E ) ) ) ) ) ) ).
fof(t1_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> r1_tarski(k1_msscyc_2(A),k2_zfmisc_1(u1_msualg_1(A),u1_struct_0(A))) ) ).
fof(d2_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_msscyc_2(A),u1_struct_0(A))
& m2_relset_1(B,k1_msscyc_2(A),u1_struct_0(A)) )
=> ( B = k2_msscyc_2(A)
<=> ! [C] :
( r2_hidden(C,k1_msscyc_2(A))
=> k1_funct_1(B,C) = k2_mcart_1(C) ) ) ) ) ).
fof(d3_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_msscyc_2(A),u1_struct_0(A))
& m2_relset_1(B,k1_msscyc_2(A),u1_struct_0(A)) )
=> ( B = k3_msscyc_2(A)
<=> ! [C] :
( r2_hidden(C,k1_msscyc_2(A))
=> k1_funct_1(B,C) = k1_funct_1(u3_msualg_1(A),k1_mcart_1(C)) ) ) ) ) ).
fof(d4_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> k4_msscyc_2(A) = g1_graph_1(u1_struct_0(A),k1_msscyc_2(A),k2_msscyc_2(A),k3_msscyc_2(A)) ) ).
fof(t2_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( ~ ( ? [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
& k9_msafree2(A,B,C,E) = D )
& ! [E] :
( ( v8_graph_1(E,k4_msscyc_2(A))
& m1_graph_1(E,k4_msscyc_2(A)) )
=> ~ ( k3_finseq_1(E) = D
& k1_funct_1(k7_graph_2(k4_msscyc_2(A),E),k1_nat_1(k3_finseq_1(E),np__1)) = C ) ) )
& ~ ( ? [E] :
( v8_graph_1(E,k4_msscyc_2(A))
& m1_graph_1(E,k4_msscyc_2(A))
& k3_finseq_1(E) = D
& k1_funct_1(k7_graph_2(k4_msscyc_2(A),E),k1_nat_1(k3_finseq_1(E),np__1)) = C )
& ! [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> k9_msafree2(A,B,C,E) != D ) ) ) ) ) ) ) ) ).
fof(t3_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( v5_msafree2(A)
<=> v4_msscyc_1(k4_msscyc_2(A)) ) ) ).
fof(t4_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( v5_msafree2(A)
=> v4_msscyc_1(k4_msscyc_2(A)) ) ) ).
fof(t6_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( ( v5_msscyc_1(A)
& v4_msscyc_1(k4_msscyc_2(A)) )
=> v5_msafree2(A) ) ) ).
fof(dt_k1_msscyc_2,axiom,
$true ).
fof(dt_k2_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> ( v1_funct_1(k2_msscyc_2(A))
& v1_funct_2(k2_msscyc_2(A),k1_msscyc_2(A),u1_struct_0(A))
& m2_relset_1(k2_msscyc_2(A),k1_msscyc_2(A),u1_struct_0(A)) ) ) ).
fof(dt_k3_msscyc_2,axiom,
! [A] :
( l1_msualg_1(A)
=> ( v1_funct_1(k3_msscyc_2(A))
& v1_funct_2(k3_msscyc_2(A),k1_msscyc_2(A),u1_struct_0(A))
& m2_relset_1(k3_msscyc_2(A),k1_msscyc_2(A),u1_struct_0(A)) ) ) ).
fof(dt_k4_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ( v2_graph_1(k4_msscyc_2(A))
& l1_graph_1(k4_msscyc_2(A)) ) ) ).
fof(t5_msscyc_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_pre_circ(B,u1_struct_0(A))
& m1_pboole(B,u1_struct_0(A)) )
=> ( v5_msscyc_1(A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> v1_finset_1(a_4_0_msscyc_2(A,B,C,D)) ) ) ) ) ) ).
fof(fraenkel_a_4_0_msscyc_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& v1_pre_circ(C,u1_struct_0(B))
& m1_pboole(C,u1_struct_0(B))
& m2_subset_1(D,k1_numbers,k5_numbers)
& m1_subset_1(E,u1_struct_0(B)) )
=> ( r2_hidden(A,a_4_0_msscyc_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_funct_1(u4_msualg_1(B,k11_msafree(B,C)),E))
& A = F
& r1_xreal_0(k9_msafree2(B,C,E,F),D) ) ) ) ).
%------------------------------------------------------------------------------