SET007 Axioms: SET007+204.ax
%------------------------------------------------------------------------------
% File : SET007+204 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Graphs
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 97 ( 10 unt; 0 def)
% Number of atoms : 652 ( 69 equ)
% Maximal formula atoms : 26 ( 6 avg)
% Number of connectives : 571 ( 16 ~; 4 |; 329 &)
% ( 29 <=>; 193 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-4 aty)
% Number of functors : 27 ( 27 usr; 5 con; 0-4 aty)
% Number of variables : 215 ( 190 !; 25 ?)
% 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_1,axiom,
? [A] :
( l1_graph_1(A)
& v1_graph_1(A) ) ).
fof(rc2_graph_1,axiom,
? [A] :
( l1_graph_1(A)
& v1_graph_1(A)
& v2_graph_1(A) ) ).
fof(rc3_graph_1,axiom,
? [A] :
( l1_graph_1(A)
& v7_graph_1(A) ) ).
fof(rc4_graph_1,axiom,
? [A] :
( l1_graph_1(A)
& v2_graph_1(A)
& v3_graph_1(A)
& v4_graph_1(A)
& v5_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A) ) ).
fof(rc5_graph_1,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) ) ) ).
fof(rc6_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ).
fof(rc7_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& 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) ) ) ).
fof(rc8_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v10_graph_1(B,A) ) ) ).
fof(rc9_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& 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)
& v10_graph_1(B,A) ) ) ).
fof(rc10_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m3_graph_1(B,A)
& v1_graph_1(B)
& v2_graph_1(B) ) ) ).
fof(d1_graph_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( v2_graph_1(A)
<=> ~ v1_xboole_0(u1_graph_1(A)) ) ) ).
fof(d2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> ! [C] :
( ( v1_graph_1(C)
& v2_graph_1(C)
& l1_graph_1(C) )
=> ( C = k1_graph_1(A,B)
<=> ( u1_graph_1(C) = k2_xboole_0(u1_graph_1(A),u1_graph_1(B))
& u2_graph_1(C) = k2_xboole_0(u2_graph_1(A),u2_graph_1(B))
& ! [D] :
( r2_hidden(D,u2_graph_1(A))
=> ( k1_funct_1(u3_graph_1(C),D) = k1_funct_1(u3_graph_1(A),D)
& k1_funct_1(u4_graph_1(C),D) = k1_funct_1(u4_graph_1(A),D) ) )
& ! [D] :
( r2_hidden(D,u2_graph_1(B))
=> ( k1_funct_1(u3_graph_1(C),D) = k1_funct_1(u3_graph_1(B),D)
& k1_funct_1(u4_graph_1(C),D) = k1_funct_1(u4_graph_1(B),D) ) ) ) ) ) ) ) ) ).
fof(d3_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( r1_graph_1(A,B,C)
<=> ( r1_partfun1(u4_graph_1(B),u4_graph_1(C))
& r1_partfun1(u3_graph_1(B),u3_graph_1(C))
& g1_graph_1(u1_graph_1(A),u2_graph_1(A),u3_graph_1(A),u4_graph_1(A)) = k1_graph_1(B,C) ) ) ) ) ) ).
fof(d4_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v3_graph_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,u2_graph_1(A))
& r2_hidden(C,u2_graph_1(A))
& k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u3_graph_1(A),C)
& k1_funct_1(u4_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C) )
=> B = C ) ) ) ).
fof(d5_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v4_graph_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,u2_graph_1(A))
& r2_hidden(C,u2_graph_1(A)) )
=> ( ( ~ ( k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u3_graph_1(A),C)
& k1_funct_1(u4_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C) )
& ~ ( k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C)
& k1_funct_1(u3_graph_1(A),C) = k1_funct_1(u4_graph_1(A),B) ) )
| B = C ) ) ) ) ).
fof(d6_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v5_graph_1(A)
<=> ! [B] :
~ ( r2_hidden(B,u2_graph_1(A))
& k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u4_graph_1(A),B) ) ) ) ).
fof(d7_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v6_graph_1(A)
<=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ~ ( r1_xboole_0(u1_graph_1(B),u1_graph_1(C))
& r1_graph_1(A,B,C) ) ) ) ) ) ).
fof(d8_graph_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( v7_graph_1(A)
<=> ( v1_finset_1(u1_graph_1(A))
& v1_finset_1(u2_graph_1(A)) ) ) ) ).
fof(d9_graph_1,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] :
( r2_graph_1(A,B,C,D)
<=> ( ( k1_funct_1(u3_graph_1(A),D) = B
& k1_funct_1(u4_graph_1(A),D) = C )
| ( k1_funct_1(u3_graph_1(A),D) = C
& k1_funct_1(u4_graph_1(A),D) = B ) ) ) ) ) ) ).
fof(d10_graph_1,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))
=> ( r3_graph_1(A,B,C)
<=> ? [D] :
( r2_hidden(D,u2_graph_1(A))
& r2_graph_1(A,B,C,D) ) ) ) ) ) ).
fof(d11_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( m1_graph_1(B,A)
<=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> r2_hidden(k1_funct_1(B,C),u2_graph_1(A)) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& k3_finseq_1(C) = k1_nat_1(k3_finseq_1(B),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)) )
=> r2_hidden(k1_funct_1(C,D),u1_graph_1(A)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B))
& ! [E] :
( m1_subset_1(E,u1_graph_1(A))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(A))
=> ~ ( E = k1_funct_1(C,D)
& F = k1_funct_1(C,k1_nat_1(D,np__1))
& r2_graph_1(A,E,F,k1_funct_1(B,D)) ) ) ) ) ) ) ) ) ) ) ).
fof(d12_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ( v8_graph_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(B),C)
| k1_funct_1(u3_graph_1(A),k1_funct_1(B,k1_nat_1(C,np__1))) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) ) ).
fof(d13_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ( v2_funct_1(B)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& ~ r1_xreal_0(D,C)
& r1_xreal_0(D,k3_finseq_1(B))
& k1_funct_1(B,C) = k1_funct_1(B,D) ) ) ) ) ) ) ).
fof(d14_graph_1,axiom,
$true ).
fof(d15_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_funct_1(B)
& m2_graph_1(B,A) )
=> ( v10_graph_1(B,A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& k3_finseq_1(C) = k1_nat_1(k3_finseq_1(B),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)) )
=> r2_hidden(k1_funct_1(C,D),u1_graph_1(A)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B))
& ! [E] :
( m1_subset_1(E,u1_graph_1(A))
=> ! [F] :
( m1_subset_1(F,u1_graph_1(A))
=> ~ ( E = k1_funct_1(C,D)
& F = k1_funct_1(C,k1_nat_1(D,np__1))
& r2_graph_1(A,E,F,k1_funct_1(B,D)) ) ) ) ) )
& k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) ).
fof(d16_graph_1,axiom,
$true ).
fof(d17_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( m3_graph_1(B,A)
<=> ( r1_tarski(u1_graph_1(B),u1_graph_1(A))
& r1_tarski(u2_graph_1(B),u2_graph_1(A))
& ! [C] :
( r2_hidden(C,u2_graph_1(B))
=> ( k1_funct_1(u3_graph_1(B),C) = k1_funct_1(u3_graph_1(A),C)
& k1_funct_1(u4_graph_1(B),C) = k1_funct_1(u4_graph_1(A),C)
& r2_hidden(k1_funct_1(u3_graph_1(A),C),u1_graph_1(B))
& r2_hidden(k1_funct_1(u4_graph_1(A),C),u1_graph_1(B)) ) ) ) ) ) ) ).
fof(d18_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k2_graph_1(A)
<=> ? [C] :
( v1_finset_1(C)
& C = u1_graph_1(A)
& B = k4_card_1(C) ) ) ) ) ).
fof(d19_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k3_graph_1(A)
<=> ? [C] :
( v1_finset_1(C)
& C = u2_graph_1(A)
& B = k4_card_1(C) ) ) ) ) ).
fof(d20_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k4_graph_1(A,B)
<=> ? [D] :
( v1_finset_1(D)
& ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,u2_graph_1(A))
& k1_funct_1(u4_graph_1(A),E) = B ) )
& C = k4_card_1(D) ) ) ) ) ) ).
fof(d21_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k5_graph_1(A,B)
<=> ? [D] :
( v1_finset_1(D)
& ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,u2_graph_1(A))
& k1_funct_1(u3_graph_1(A),E) = B ) )
& C = k4_card_1(D) ) ) ) ) ) ).
fof(d22_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> k6_graph_1(A,B) = k1_nat_1(k4_graph_1(A,B),k5_graph_1(A,B)) ) ) ).
fof(d23_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
<=> m3_graph_1(A,B) ) ) ) ).
fof(d24_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( B = k7_graph_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_graph_1(C)
& m3_graph_1(C,A) ) ) ) ) ).
fof(t1_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( k1_relat_1(u3_graph_1(A)) = u2_graph_1(A)
& k1_relat_1(u4_graph_1(A)) = u2_graph_1(A)
& r1_tarski(k2_relat_1(u3_graph_1(A)),u1_graph_1(A))
& r1_tarski(k2_relat_1(u4_graph_1(A)),u1_graph_1(A)) ) ) ).
fof(t2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> r2_hidden(B,u1_graph_1(A)) ) ) ).
fof(t3_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( r2_hidden(B,u2_graph_1(A))
=> ( r2_hidden(k1_funct_1(u3_graph_1(A),B),u1_graph_1(A))
& r2_hidden(k1_funct_1(u4_graph_1(A),B),u1_graph_1(A)) ) ) ) ).
fof(t4_graph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( m2_graph_1(C,B)
=> m2_graph_1(k2_partfun1(k5_numbers,u2_graph_1(B),C,k2_finseq_1(A)),B) ) ) ) ).
fof(t5_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
=> ( r1_tarski(u3_graph_1(A),u3_graph_1(B))
& r1_tarski(u4_graph_1(A),u4_graph_1(B)) ) ) ) ) ).
fof(t6_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> ( u3_graph_1(k1_graph_1(A,B)) = k2_xboole_0(u3_graph_1(A),u3_graph_1(B))
& u4_graph_1(k1_graph_1(A,B)) = k2_xboole_0(u4_graph_1(A),u4_graph_1(B)) ) ) ) ) ).
fof(t7_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> A = k1_graph_1(A,A) ) ).
fof(t8_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> k1_graph_1(A,B) = k1_graph_1(B,A) ) ) ) ).
fof(t9_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B))
& r1_partfun1(u3_graph_1(A),u3_graph_1(C))
& r1_partfun1(u4_graph_1(A),u4_graph_1(C))
& r1_partfun1(u3_graph_1(B),u3_graph_1(C))
& r1_partfun1(u4_graph_1(B),u4_graph_1(C)) )
=> k1_graph_1(k1_graph_1(A,B),C) = k1_graph_1(A,k1_graph_1(B,C)) ) ) ) ) ).
fof(t10_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( r1_graph_1(A,B,C)
=> r1_graph_1(A,C,B) ) ) ) ) ).
fof(t11_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> r1_graph_1(A,A,A) ) ).
fof(t12_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ? [C] :
( v2_graph_1(C)
& l1_graph_1(C)
& r4_graph_1(A,C)
& r4_graph_1(B,C) )
=> k1_graph_1(A,B) = k1_graph_1(B,A) ) ) ) ).
fof(t13_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ? [D] :
( v2_graph_1(D)
& l1_graph_1(D)
& r4_graph_1(A,D)
& r4_graph_1(B,D)
& r4_graph_1(C,D) )
=> k1_graph_1(k1_graph_1(A,B),C) = k1_graph_1(A,k1_graph_1(B,C)) ) ) ) ) ).
fof(t14_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v2_funct_1(k1_xboole_0)
& v8_graph_1(k1_xboole_0,A)
& v10_graph_1(k1_xboole_0,A)
& m2_graph_1(k1_xboole_0,A) ) ) ).
fof(t15_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& m3_graph_1(B,A) )
=> ! [C] :
( ( v1_graph_1(C)
& m3_graph_1(C,A) )
=> ( ( u1_graph_1(B) = u1_graph_1(C)
& u2_graph_1(B) = u2_graph_1(C) )
=> B = C ) ) ) ) ).
fof(t16_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r4_graph_1(A,B)
& r4_graph_1(B,A) )
=> A = B ) ) ) ).
fof(t17_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( r4_graph_1(A,B)
& r4_graph_1(B,C) )
=> r4_graph_1(A,C) ) ) ) ) ).
fof(t18_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( r1_graph_1(A,B,C)
=> ( r4_graph_1(B,A)
& r4_graph_1(C,A) ) ) ) ) ) ).
fof(t19_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> ( r4_graph_1(A,k1_graph_1(A,B))
& r4_graph_1(B,k1_graph_1(A,B)) ) ) ) ) ).
fof(t20_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ? [C] :
( v2_graph_1(C)
& l1_graph_1(C)
& r4_graph_1(A,C)
& r4_graph_1(B,C) )
=> ( r4_graph_1(A,k1_graph_1(A,B))
& r4_graph_1(B,k1_graph_1(A,B)) ) ) ) ) ).
fof(t21_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ! [D] :
( ( v2_graph_1(D)
& l1_graph_1(D) )
=> ( ( r4_graph_1(A,B)
& r4_graph_1(C,B)
& r1_graph_1(D,A,C) )
=> r4_graph_1(D,B) ) ) ) ) ) ).
fof(t22_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( r4_graph_1(A,B)
& r4_graph_1(C,B) )
=> r4_graph_1(k1_graph_1(A,C),B) ) ) ) ) ).
fof(t23_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
=> ( k1_graph_1(A,B) = B
& k1_graph_1(B,A) = B ) ) ) ) ).
fof(t24_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> ( ( k1_graph_1(A,B) != B
& k1_graph_1(B,A) != B )
| r4_graph_1(A,B) ) ) ) ) ).
fof(t25_graph_1,axiom,
$true ).
fof(t26_graph_1,axiom,
$true ).
fof(t27_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& v3_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
=> v3_graph_1(A) ) ) ) ).
fof(t28_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& v4_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
=> v4_graph_1(A) ) ) ) ).
fof(t29_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& v5_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(A,B)
=> v5_graph_1(A) ) ) ) ).
fof(t30_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ( r2_hidden(B,k7_graph_1(A))
<=> r4_graph_1(B,A) ) ) ) ).
fof(t31_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> r2_hidden(A,k7_graph_1(A)) ) ).
fof(t32_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ( r4_graph_1(B,A)
<=> r1_tarski(k7_graph_1(B),k7_graph_1(A)) ) ) ) ).
fof(t33_graph_1,axiom,
$true ).
fof(t34_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> r1_tarski(k1_tarski(A),k7_graph_1(A)) ) ).
fof(t35_graph_1,axiom,
! [A] :
( ( v1_graph_1(A)
& v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_graph_1(B)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ~ ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B))
& r1_tarski(k7_graph_1(k1_graph_1(A,B)),k2_xboole_0(k7_graph_1(A),k7_graph_1(B)))
& ~ r4_graph_1(A,B)
& ~ r4_graph_1(B,A) ) ) ) ).
fof(t36_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
& r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
=> r1_tarski(k2_xboole_0(k7_graph_1(A),k7_graph_1(B)),k7_graph_1(k1_graph_1(A,B))) ) ) ) ).
fof(t37_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v2_graph_1(C)
& l1_graph_1(C) )
=> ( ( r2_hidden(A,k7_graph_1(B))
& r2_hidden(C,k7_graph_1(B)) )
=> r2_hidden(k1_graph_1(A,C),k7_graph_1(B)) ) ) ) ) ).
fof(s1_graph_1,axiom,
? [A] :
! [B] :
( r2_hidden(B,A)
<=> ( v1_graph_1(B)
& m3_graph_1(B,f1_s1_graph_1)
& p1_s1_graph_1(B) ) ) ).
fof(dt_m1_graph_1,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_finseq_1(B) ) ) ) ).
fof(existence_m1_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] : m1_graph_1(B,A) ) ).
fof(dt_m2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> m2_finseq_1(B,u2_graph_1(A)) ) ) ).
fof(existence_m2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] : m2_graph_1(B,A) ) ).
fof(redefinition_m2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
<=> m1_graph_1(B,A) ) ) ).
fof(dt_m3_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m3_graph_1(B,A)
=> ( v2_graph_1(B)
& l1_graph_1(B) ) ) ) ).
fof(existence_m3_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ? [B] : m3_graph_1(B,A) ) ).
fof(dt_l1_graph_1,axiom,
$true ).
fof(existence_l1_graph_1,axiom,
? [A] : l1_graph_1(A) ).
fof(abstractness_v1_graph_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( v1_graph_1(A)
=> A = g1_graph_1(u1_graph_1(A),u2_graph_1(A),u3_graph_1(A),u4_graph_1(A)) ) ) ).
fof(redefinition_v9_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_graph_1(B,A) )
=> ( v9_graph_1(B,A)
<=> v2_funct_1(B) ) ) ).
fof(reflexivity_r4_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& v2_graph_1(B)
& l1_graph_1(B) )
=> r4_graph_1(A,A) ) ).
fof(dt_k1_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& v2_graph_1(B)
& l1_graph_1(B) )
=> ( v1_graph_1(k1_graph_1(A,B))
& v2_graph_1(k1_graph_1(A,B))
& l1_graph_1(k1_graph_1(A,B)) ) ) ).
fof(dt_k2_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> m2_subset_1(k2_graph_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k3_graph_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A) )
=> m2_subset_1(k3_graph_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k4_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m2_subset_1(k4_graph_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k5_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m2_subset_1(k5_graph_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k6_graph_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& v7_graph_1(A)
& l1_graph_1(A)
& m1_subset_1(B,u1_graph_1(A)) )
=> m2_subset_1(k6_graph_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k7_graph_1,axiom,
$true ).
fof(dt_u1_graph_1,axiom,
$true ).
fof(dt_u2_graph_1,axiom,
$true ).
fof(dt_u3_graph_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( v1_funct_1(u3_graph_1(A))
& v1_funct_2(u3_graph_1(A),u2_graph_1(A),u1_graph_1(A))
& m2_relset_1(u3_graph_1(A),u2_graph_1(A),u1_graph_1(A)) ) ) ).
fof(dt_u4_graph_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( v1_funct_1(u4_graph_1(A))
& v1_funct_2(u4_graph_1(A),u2_graph_1(A),u1_graph_1(A))
& m2_relset_1(u4_graph_1(A),u2_graph_1(A),u1_graph_1(A)) ) ) ).
fof(dt_g1_graph_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A) )
=> ( v1_graph_1(g1_graph_1(A,B,C,D))
& l1_graph_1(g1_graph_1(A,B,C,D)) ) ) ).
fof(free_g1_graph_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A) )
=> ! [E,F,G,H] :
( g1_graph_1(A,B,C,D) = g1_graph_1(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
%------------------------------------------------------------------------------