SET007 Axioms: SET007+562.ax
%------------------------------------------------------------------------------
% File : SET007+562 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Oriented Chains
% 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 : graph_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 4 unt; 0 def)
% Number of atoms : 376 ( 39 equ)
% Maximal formula atoms : 23 ( 8 avg)
% Number of connectives : 342 ( 8 ~; 3 |; 185 &)
% ( 7 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-4 aty)
% Number of functors : 26 ( 26 usr; 4 con; 0-4 aty)
% Number of variables : 138 ( 128 !; 10 ?)
% 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_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v8_graph_1(B,A)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ).
fof(rc2_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v8_graph_1(B,A)
& v1_graph_4(B,A) ) ) ).
fof(rc3_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v8_graph_1(B,A)
& v3_graph_2(B,A) ) ) ).
fof(cc1_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ( ( v1_xboole_0(B)
& v8_graph_1(B,A) )
=> ( v2_funct_1(B)
& v8_graph_1(B,A) ) ) ) ) ).
fof(d1_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( r1_graph_4(A,B,C,D)
<=> ( k1_funct_1(u3_graph_1(A),D) = B
& k1_funct_1(u4_graph_1(A),D) = C ) ) ) ) ) ).
fof(t1_graph_4,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ( r1_graph_4(B,C,D,A)
=> r2_graph_1(B,C,D,A) ) ) ) ) ).
fof(d2_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m1_subset_1(C,u1_graph_1(A))
=> ( r2_graph_4(A,B,C)
<=> ? [D] :
( r2_hidden(D,u2_graph_1(A))
& r1_graph_4(A,B,C,D) ) ) ) ) ) ).
fof(t2_graph_4,axiom,
! [A,B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_graph_1(B))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(B))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(B))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(B))
=> ( ( r1_graph_4(B,C,D,A)
& r1_graph_4(B,E,F,A) )
=> ( C = E
& D = F ) ) ) ) ) ) ) ).
fof(t3_graph_4,axiom,
$true ).
fof(t4_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( k1_graph_4(A,k1_xboole_0) = k1_xboole_0
& k2_graph_4(A,k1_xboole_0) = k1_xboole_0 ) ) ).
fof(d5_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r3_graph_4(A,B,C)
<=> ( k3_finseq_1(B) = k1_nat_1(k3_finseq_1(C),np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(C)) )
=> r1_graph_4(A,k4_finseq_4(k5_numbers,u1_graph_1(A),B,D),k4_finseq_4(k5_numbers,u1_graph_1(A),B,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) ) ) ) ) ).
fof(t5_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( r3_graph_4(A,B,C)
=> r1_graph_2(A,B,C) ) ) ) ) ).
fof(t6_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( r3_graph_4(A,B,C)
=> r1_tarski(k1_graph_4(A,k2_relat_1(C)),k2_relat_1(B)) ) ) ) ) ).
fof(t7_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( r3_graph_4(A,B,C)
=> r1_tarski(k2_graph_4(A,k2_relat_1(C)),k2_relat_1(B)) ) ) ) ) ).
fof(t8_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( r3_graph_4(A,B,C)
=> ( C = k1_xboole_0
| r1_tarski(k2_relat_1(B),k2_xboole_0(k1_graph_4(A,k2_relat_1(C)),k2_graph_4(A,k2_relat_1(C)))) ) ) ) ) ) ).
fof(t9_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> r3_graph_4(A,k13_binarith(u1_graph_1(A),B),k1_xboole_0) ) ) ).
fof(t10_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r3_graph_4(A,C,B) ) ) ) ).
fof(t11_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ( ( r3_graph_4(A,B,D)
& r3_graph_4(A,C,D) )
=> ( D = k1_xboole_0
| B = C ) ) ) ) ) ) ).
fof(d6_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ( B != k1_xboole_0
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ( C = k3_graph_4(A,B)
<=> r3_graph_4(A,C,B) ) ) ) ) ) ).
fof(t12_graph_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(B))
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(B))
=> ! [E] :
( ( v8_graph_1(E,B)
& m2_graph_1(E,B) )
=> ! [F] :
( ( v8_graph_1(F,B)
& m2_graph_1(F,B) )
=> ( ( r3_graph_4(B,C,E)
& F = k7_relat_1(E,k2_finseq_1(A))
& D = k7_relat_1(C,k2_finseq_1(k1_nat_1(A,np__1))) )
=> r3_graph_4(B,D,F) ) ) ) ) ) ) ) ).
fof(t13_graph_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v2_graph_1(D)
& l1_graph_1(D) )
=> ! [E] :
( ( v8_graph_1(E,D)
& m2_graph_1(E,D) )
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,C)
& r1_xreal_0(C,k3_finseq_1(E))
& A = k2_graph_2(u2_graph_1(D),E,B,C) )
=> ( v8_graph_1(A,D)
& m2_graph_1(A,D) ) ) ) ) ) ) ) ).
fof(t14_graph_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(C))
=> ! [E] :
( m2_finseq_1(E,u1_graph_1(C))
=> ! [F] :
( ( v8_graph_1(F,C)
& m2_graph_1(F,C) )
=> ! [G] :
( ( v8_graph_1(G,C)
& m2_graph_1(G,C) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,B)
& r1_xreal_0(B,k3_finseq_1(F))
& G = k2_graph_2(u2_graph_1(C),F,A,B)
& r3_graph_4(C,D,F)
& E = k2_graph_2(u1_graph_1(C),D,A,k1_nat_1(B,np__1)) )
=> r3_graph_4(C,E,G) ) ) ) ) ) ) ) ) ).
fof(t15_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( ( v8_graph_1(D,A)
& m2_graph_1(D,A) )
=> ! [E] :
( ( v8_graph_1(E,A)
& m2_graph_1(E,A) )
=> ( ( r3_graph_4(A,B,D)
& r3_graph_4(A,C,E)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1) )
=> ( v8_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A)
& m2_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A) ) ) ) ) ) ) ) ).
fof(t16_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(A))
=> ! [E] :
( ( v8_graph_1(E,A)
& m2_graph_1(E,A) )
=> ! [F] :
( ( v8_graph_1(F,A)
& m2_graph_1(F,A) )
=> ! [G] :
( ( v8_graph_1(G,A)
& m2_graph_1(G,A) )
=> ( ( r3_graph_4(A,B,E)
& r3_graph_4(A,C,F)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1)
& G = k8_finseq_1(u2_graph_1(A),E,F)
& D = k4_graph_2(u1_graph_1(A),B,C) )
=> r3_graph_4(A,D,G) ) ) ) ) ) ) ) ) ).
fof(d7_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& m2_graph_1(B,A) )
=> ( v1_graph_4(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r3_graph_4(A,C,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(E,k3_finseq_1(C))
& k1_funct_1(C,D) = k1_funct_1(C,E) )
=> ( r1_xreal_0(E,D)
| ( D = np__1
& E = k3_finseq_1(C) ) ) ) ) ) ) ) ) ) ).
fof(t17_graph_4,axiom,
$true ).
fof(t18_graph_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v8_graph_1(C,B)
& m2_graph_1(C,B) )
=> ( v8_graph_1(k7_relat_1(C,k2_finseq_1(A)),B)
& m2_graph_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ) ) ).
fof(t19_graph_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v8_graph_1(C,B)
& v3_graph_2(C,B)
& m2_graph_1(C,B) )
=> ( v8_graph_1(k7_relat_1(C,k2_finseq_1(A)),B)
& v3_graph_2(k7_relat_1(C,k2_finseq_1(A)),B)
& m2_graph_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ) ) ).
fof(t20_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ( C = B
=> v1_graph_4(C,A) ) ) ) ) ).
fof(t21_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) ) ) ) ).
fof(t22_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ~ ( ~ v1_graph_4(C,A)
& r3_graph_4(A,B,C)
& ! [D] :
( m1_graph_2(D,u2_graph_1(A),C)
=> ! [E] :
( m1_graph_2(E,u1_graph_1(A),B)
=> ! [F] :
( ( v8_graph_1(F,A)
& m2_graph_1(F,A) )
=> ! [G] :
( m2_finseq_1(G,u1_graph_1(A))
=> ~ ( ~ r1_xreal_0(k3_finseq_1(C),k3_finseq_1(F))
& r3_graph_4(A,G,F)
& ~ r1_xreal_0(k3_finseq_1(B),k3_finseq_1(G))
& k1_funct_1(B,np__1) = k1_funct_1(G,np__1)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(G,k3_finseq_1(G))
& k15_finseq_1(D) = F
& k15_finseq_1(E) = G ) ) ) ) ) ) ) ) ) ).
fof(t23_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m2_graph_1(C,A) )
=> ~ ( r3_graph_4(A,B,C)
& ! [D] :
( m1_graph_2(D,u2_graph_1(A),C)
=> ! [E] :
( m1_graph_2(E,u1_graph_1(A),B)
=> ! [F] :
( ( v8_graph_1(F,A)
& v3_graph_2(F,A)
& m2_graph_1(F,A) )
=> ! [G] :
( m2_finseq_1(G,u1_graph_1(A))
=> ~ ( k15_finseq_1(D) = F
& k15_finseq_1(E) = G
& r3_graph_4(A,G,F)
& k1_funct_1(B,np__1) = k1_funct_1(G,np__1)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(G,k3_finseq_1(G)) ) ) ) ) ) ) ) ) ) ).
fof(t24_graph_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( v2_funct_1(A)
& v8_graph_1(A,C)
& m2_graph_1(A,C) )
=> ( v2_funct_1(k7_relat_1(A,k2_finseq_1(B)))
& v8_graph_1(k7_relat_1(A,k2_finseq_1(B)),C)
& m2_graph_1(k7_relat_1(A,k2_finseq_1(B)),C) ) ) ) ) ) ).
fof(t25_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ( v2_funct_1(B)
& v8_graph_1(B,A)
& m2_graph_1(B,A) ) ) ) ).
fof(t26_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) )
=> ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) ) )
& ( ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ( v8_graph_1(B,A)
& v1_graph_4(B,A)
& m2_graph_1(B,A) ) )
& ( ( v8_graph_1(B,A)
& v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ( v2_funct_1(B)
& v8_graph_1(B,A)
& m2_graph_1(B,A) ) ) ) ) ) ).
fof(dt_k1_graph_4,axiom,
$true ).
fof(dt_k2_graph_4,axiom,
$true ).
fof(dt_k3_graph_4,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> m2_finseq_1(k3_graph_4(A,B),u1_graph_1(A)) ) ).
fof(d3_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] : k1_graph_4(A,B) = a_2_0_graph_4(A,B) ) ).
fof(d4_graph_4,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] : k2_graph_4(A,B) = a_2_1_graph_4(A,B) ) ).
fof(fraenkel_a_2_0_graph_4,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r2_hidden(A,a_2_0_graph_4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_graph_1(B))
& A = D
& ? [E] :
( m1_subset_1(E,u2_graph_1(B))
& r2_hidden(E,C)
& D = k1_funct_1(u3_graph_1(B),E) ) ) ) ) ).
fof(fraenkel_a_2_1_graph_4,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r2_hidden(A,a_2_1_graph_4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_graph_1(B))
& A = D
& ? [E] :
( m1_subset_1(E,u2_graph_1(B))
& r2_hidden(E,C)
& D = k1_funct_1(u4_graph_1(B),E) ) ) ) ) ).
%------------------------------------------------------------------------------