SET007 Axioms: SET007+431.ax
%------------------------------------------------------------------------------
% File : SET007+431 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Vertex Sequences Induced by 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 90 ( 5 unt; 0 def)
% Number of atoms : 743 ( 124 equ)
% Maximal formula atoms : 24 ( 8 avg)
% Number of connectives : 701 ( 48 ~; 14 |; 346 &)
% ( 18 <=>; 275 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-4 aty)
% Number of functors : 45 ( 45 usr; 9 con; 0-4 aty)
% Number of variables : 280 ( 265 !; 15 ?)
% 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_2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_graph_2(A)
& v2_graph_2(A) ) ).
fof(fc1_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ~ v1_xboole_0(u1_graph_1(A)) ) ).
fof(rc2_graph_2,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_xboole_0(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ).
fof(rc3_graph_2,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)
& v3_graph_2(B,A) ) ) ).
fof(cc1_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ( v1_xboole_0(B)
=> v2_funct_1(B) ) ) ) ).
fof(rc4_graph_2,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)
& v3_graph_2(B,A) ) ) ).
fof(cc2_graph_2,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) ) ) ) ).
fof(t1_graph_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k1_nat_1(A,np__1),B)
& r1_xreal_0(B,C) )
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(A,D)
& ~ r1_xreal_0(C,D)
& B = k1_nat_1(D,np__1) ) ) ) ) ) ).
fof(t2_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( A = k7_relat_1(B,k2_finseq_1(C))
=> ( r1_xreal_0(k3_finseq_1(A),k3_finseq_1(B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(A)) )
=> k1_funct_1(B,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ).
fof(t3_graph_2,axiom,
! [A,B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_tarski(A,k2_finseq_1(C))
& r1_tarski(B,k4_finseq_1(k14_finseq_1(A))) )
=> k5_relat_1(k14_finseq_1(B),k14_finseq_1(A)) = k14_finseq_1(k2_relat_1(k7_relat_1(k14_finseq_1(A),B))) ) ) ).
fof(d1_graph_2,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] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,k1_nat_1(C,np__1))
& r1_xreal_0(C,k3_finseq_1(A)) )
=> ( D = k1_graph_2(A,B,C)
<=> ( k1_nat_1(k3_finseq_1(D),B) = k1_nat_1(C,np__1)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(D),E)
=> k1_funct_1(D,k1_nat_1(E,np__1)) = k1_funct_1(A,k1_nat_1(B,E)) ) ) ) ) )
& ( ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,k1_nat_1(C,np__1))
& r1_xreal_0(C,k3_finseq_1(A)) )
=> ( D = k1_graph_2(A,B,C)
<=> D = k1_xboole_0 ) ) ) ) ) ) ) ).
fof(t6_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,k3_finseq_1(A)) )
=> k1_graph_2(A,B,B) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) ).
fof(t7_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k1_graph_2(A,np__1,k3_finseq_1(A)) = A ) ).
fof(t8_graph_2,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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,C)
& r1_xreal_0(C,D)
& r1_xreal_0(D,k3_finseq_1(A)) )
=> k7_finseq_1(k1_graph_2(A,k1_nat_1(B,np__1),C),k1_graph_2(A,k1_nat_1(C,np__1),D)) = k1_graph_2(A,k1_nat_1(B,np__1),D) ) ) ) ) ) ).
fof(t9_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,k3_finseq_1(A))
=> k7_finseq_1(k1_graph_2(A,np__1,B),k1_graph_2(A,k1_nat_1(B,np__1),k3_finseq_1(A))) = A ) ) ) ).
fof(t10_graph_2,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)
=> ( ( r1_xreal_0(B,C)
& r1_xreal_0(C,k3_finseq_1(A)) )
=> k7_finseq_1(k7_finseq_1(k1_graph_2(A,np__1,B),k1_graph_2(A,k1_nat_1(B,np__1),C)),k1_graph_2(A,k1_nat_1(C,np__1),k3_finseq_1(A))) = A ) ) ) ) ).
fof(t11_graph_2,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)
=> r1_tarski(k2_relat_1(k1_graph_2(A,B,C)),k2_relat_1(A)) ) ) ) ).
fof(t12_graph_2,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)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,C)
& r1_xreal_0(C,k3_finseq_1(A)) )
=> ( k1_funct_1(k1_graph_2(A,B,C),np__1) = k1_funct_1(A,B)
& k1_funct_1(k1_graph_2(A,B,C),k3_finseq_1(k1_graph_2(A,B,C))) = k1_funct_1(A,C) ) ) ) ) ) ).
fof(d2_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k3_graph_2(A,B) = k7_finseq_1(A,k1_graph_2(B,np__2,k3_finseq_1(B))) ) ) ).
fof(t13_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( A != k1_xboole_0
=> k1_nat_1(k3_finseq_1(k3_graph_2(B,A)),np__1) = k1_nat_1(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) ).
fof(t14_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(A)) )
=> k1_funct_1(k3_graph_2(A,B),C) = k1_funct_1(A,C) ) ) ) ) ).
fof(t15_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(A),C)
| k1_funct_1(k3_graph_2(B,A),k1_nat_1(k3_finseq_1(B),C)) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) ).
fof(t16_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__1)
=> k1_funct_1(k3_graph_2(B,A),k3_finseq_1(k3_graph_2(B,A))) = k1_funct_1(A,k3_finseq_1(A)) ) ) ) ).
fof(t17_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> r1_tarski(k2_relat_1(k3_graph_2(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ).
fof(t18_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( k1_funct_1(A,k3_finseq_1(A)) = k1_funct_1(B,np__1)
=> ( A = k1_xboole_0
| B = k1_xboole_0
| k2_relat_1(k3_graph_2(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) ).
fof(d3_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_graph_2(A)
<=> k4_card_1(k2_relat_1(A)) = np__2 ) ) ).
fof(t19_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_graph_2(A)
<=> ( ~ r1_xreal_0(k3_finseq_1(A),np__1)
& ? [B,C] :
( B != C
& k2_relat_1(A) = k2_tarski(B,C) ) ) ) ) ).
fof(d4_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v2_graph_2(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
& k1_funct_1(A,B) = k1_funct_1(A,k1_nat_1(B,np__1)) ) ) ) ) ).
fof(t20_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_graph_2(A)
& v2_graph_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_graph_2(B)
& v2_graph_2(B) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k2_relat_1(A) = k2_relat_1(B)
& k1_funct_1(A,np__1) = k1_funct_1(B,np__1) )
=> A = B ) ) ) ).
fof(t21_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_graph_2(A)
& v2_graph_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_graph_2(B)
& v2_graph_2(B) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k2_relat_1(A) = k2_relat_1(B) )
=> ( A = B
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(A))
& k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ) ) ).
fof(t22_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_graph_2(A)
& v2_graph_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_graph_2(B)
& v2_graph_2(B) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k2_relat_1(A) = k2_relat_1(B) )
=> ( A = B
| ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_graph_2(C)
& v2_graph_2(C) )
=> ~ ( k3_finseq_1(C) = k3_finseq_1(A)
& k2_relat_1(C) = k2_relat_1(A)
& C != A
& C != B ) ) ) ) ) ) ).
fof(t23_graph_2,axiom,
! [A,B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( A != B
& ~ r1_xreal_0(C,np__1)
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v1_graph_2(D)
& v2_graph_2(D) )
=> ~ ( k2_relat_1(D) = k2_tarski(A,B)
& k3_finseq_1(D) = C
& k1_funct_1(D,np__1) = A ) ) ) ) ).
fof(d5_graph_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( m1_graph_2(C,A,B)
<=> r1_tarski(C,B) ) ) ) ).
fof(t24_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> ( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k15_finseq_1(A) = A ) ) ).
fof(t25_graph_2,axiom,
$true ).
fof(t26_graph_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F) )
=> ( ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
& r1_tarski(k2_relat_1(C),k1_relat_1(A))
& D = k5_relat_1(B,A)
& E = k5_relat_1(C,A)
& F = k5_relat_1(k7_finseq_1(B,C),A) )
=> F = k7_finseq_1(D,E) ) ) ) ) ) ) ) ).
fof(t27_graph_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_graph_2(C,A,B)
=> ( r1_tarski(k1_relat_1(C),k4_finseq_1(B))
& r1_tarski(k2_relat_1(C),k2_relat_1(B)) ) ) ) ).
fof(t28_graph_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> m1_graph_2(B,A,B) ) ).
fof(t29_graph_2,axiom,
! [A,B,C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m1_graph_2(D,A,C)
=> m1_graph_2(k7_relat_1(D,B),A,C) ) ) ).
fof(t30_graph_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ! [E] :
( m1_graph_2(E,A,B)
=> ! [F] :
( m1_graph_2(F,A,B)
=> ! [G] :
( m1_graph_2(G,A,C)
=> ( ( k15_finseq_1(E) = C
& k15_finseq_1(G) = D
& F = k7_relat_1(E,k2_relat_1(k7_relat_1(k14_finseq_1(k1_relat_1(E)),k1_relat_1(G)))) )
=> k15_finseq_1(F) = D ) ) ) ) ) ) ) ).
fof(t31_graph_2,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)
=> r2_graph_1(A,C,B,D) ) ) ) ) ).
fof(t32_graph_2,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] :
( m1_subset_1(D,u1_graph_1(A))
=> ! [E] :
( m1_subset_1(E,u1_graph_1(A))
=> ! [F] :
~ ( r2_graph_1(A,B,C,F)
& r2_graph_1(A,D,E,F)
& ~ ( B = D
& C = E )
& ~ ( B = E
& C = D ) ) ) ) ) ) ) ).
fof(d7_graph_2,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) )
=> ( r1_graph_2(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)) )
=> r2_graph_1(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(t33_graph_2,axiom,
$true ).
fof(t34_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ( r1_graph_2(A,B,C)
=> ( C = k1_xboole_0
| k5_graph_2(A,k2_relat_1(C)) = k2_relat_1(B) ) ) ) ) ) ).
fof(t35_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> r1_graph_2(A,k13_binarith(u1_graph_1(A),B),k1_xboole_0) ) ) ).
fof(t36_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r1_graph_2(A,C,B) ) ) ) ).
fof(t37_graph_2,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_graph_1(D,A)
=> ( ( r1_graph_2(A,B,D)
& r1_graph_2(A,C,D) )
=> ( D = k1_xboole_0
| B = C
| ( k1_funct_1(B,np__1) != k1_funct_1(C,np__1)
& ! [E] :
( m2_finseq_1(E,u1_graph_1(A))
=> ~ ( r1_graph_2(A,E,D)
& E != B
& E != C ) ) ) ) ) ) ) ) ) ).
fof(d8_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_graph_2(A,B)
<=> ( r1_xreal_0(np__1,k3_finseq_1(B))
& k1_card_1(k5_graph_2(A,k2_relat_1(B))) = np__2
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& k1_funct_1(u3_graph_1(A),k1_funct_1(B,C)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) ) ).
fof(t38_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ( ( r2_graph_2(A,C)
& r1_graph_2(A,B,C) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(C))
& k1_funct_1(B,D) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) ).
fof(t39_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ( ( r2_graph_2(A,C)
& r1_graph_2(A,B,C) )
=> k2_relat_1(B) = k2_tarski(k1_funct_1(u3_graph_1(A),k1_funct_1(C,np__1)),k1_funct_1(u4_graph_1(A),k1_funct_1(C,np__1))) ) ) ) ) ).
fof(t40_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ( ( r2_graph_2(A,C)
& r1_graph_2(A,B,C) )
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_graph_2(B)
& v2_graph_2(B) ) ) ) ) ) ).
fof(t41_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ~ ( r2_graph_2(A,B)
& ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ! [D] :
( m2_finseq_1(D,u1_graph_1(A))
=> ~ ( C != D
& r1_graph_2(A,C,B)
& r1_graph_2(A,D,B)
& ! [E] :
( m2_finseq_1(E,u1_graph_1(A))
=> ~ ( r1_graph_2(A,E,B)
& E != C
& E != D ) ) ) ) ) ) ) ) ).
fof(t42_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ( r1_graph_2(A,B,C)
=> ( ( k1_card_1(u1_graph_1(A)) = np__1
| ( C != k1_xboole_0
& ~ r2_graph_2(A,C) ) )
<=> ! [D] :
( m2_finseq_1(D,u1_graph_1(A))
=> ( r1_graph_2(A,D,C)
=> D = B ) ) ) ) ) ) ) ).
fof(d9_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ( ( k1_card_1(u1_graph_1(A)) = np__1
| ( B != k1_xboole_0
& ~ r2_graph_2(A,B) ) )
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ( C = k6_graph_2(A,B)
<=> r1_graph_2(A,C,B) ) ) ) ) ) ).
fof(t43_graph_2,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] :
( m2_graph_1(E,B)
=> ! [F] :
( m2_graph_1(F,B)
=> ( ( r1_graph_2(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))) )
=> r1_graph_2(B,D,F) ) ) ) ) ) ) ) ).
fof(t44_graph_2,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] :
( 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) )
=> m2_graph_1(A,D) ) ) ) ) ) ) ).
fof(t45_graph_2,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] :
( m2_graph_1(F,C)
=> ! [G] :
( 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)
& r1_graph_2(C,D,F)
& E = k2_graph_2(u1_graph_1(C),D,A,k1_nat_1(B,np__1)) )
=> r1_graph_2(C,E,G) ) ) ) ) ) ) ) ) ).
fof(t46_graph_2,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_graph_1(D,A)
=> ! [E] :
( m2_graph_1(E,A)
=> ( ( r1_graph_2(A,B,D)
& r1_graph_2(A,C,E)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1) )
=> m2_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A) ) ) ) ) ) ) ).
fof(t47_graph_2,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] :
( m2_graph_1(E,A)
=> ! [F] :
( m2_graph_1(F,A)
=> ! [G] :
( m2_graph_1(G,A)
=> ( ( r1_graph_2(A,B,E)
& r1_graph_2(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) )
=> r1_graph_2(A,D,G) ) ) ) ) ) ) ) ) ).
fof(d10_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_graph_1(B,A)
=> ( v3_graph_2(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r1_graph_2(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(t48_graph_2,axiom,
$true ).
fof(t49_graph_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ! [C] :
( ( v3_graph_2(C,B)
& m2_graph_1(C,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(t50_graph_2,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] :
( ( v3_graph_2(D,A)
& m2_graph_1(D,A) )
=> ( ( r1_graph_2(A,B,D)
& r1_graph_2(A,C,D) )
=> ( r1_xreal_0(k3_finseq_1(D),np__2)
| B = C ) ) ) ) ) ) ).
fof(t51_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( ( v3_graph_2(C,A)
& m2_graph_1(C,A) )
=> ( r1_graph_2(A,B,C)
=> ! [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(B))
& k1_funct_1(B,D) = k1_funct_1(B,E) )
=> ( r1_xreal_0(E,D)
| ( D = np__1
& E = k3_finseq_1(B) ) ) ) ) ) ) ) ) ) ).
fof(t52_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ~ ( ~ ( v3_graph_2(C,A)
& m2_graph_1(C,A) )
& r1_graph_2(A,B,C)
& ! [D] :
( m1_graph_2(D,u2_graph_1(A),C)
=> ! [E] :
( m1_graph_2(E,u1_graph_1(A),B)
=> ! [F] :
( 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))
& r1_graph_2(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(t53_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_graph_1(A))
=> ! [C] :
( m2_graph_1(C,A)
=> ~ ( r1_graph_2(A,B,C)
& ! [D] :
( m1_graph_2(D,u2_graph_1(A),C)
=> ! [E] :
( m1_graph_2(E,u1_graph_1(A),B)
=> ! [F] :
( ( 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
& r1_graph_2(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(t54_graph_2,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)
& m2_graph_1(A,C) )
=> ( v2_funct_1(k7_relat_1(A,k2_finseq_1(B)))
& m2_graph_1(k7_relat_1(A,k2_finseq_1(B)),C) ) ) ) ) ) ).
fof(t55_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ( ~ r1_xreal_0(k3_finseq_1(B),np__2)
=> ( v2_funct_1(B)
& m2_graph_1(B,A) ) ) ) ) ).
fof(t56_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v3_graph_2(B,A)
& m2_graph_1(B,A) )
=> ( ( v2_funct_1(B)
& m2_graph_1(B,A) )
<=> ~ ( k3_finseq_1(B) != np__0
& k3_finseq_1(B) != np__1
& k1_funct_1(B,np__1) = k1_funct_1(B,np__2) ) ) ) ) ).
fof(d11_graph_2,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 = k7_graph_2(A,B)
<=> ( r1_graph_2(A,C,B)
& k1_funct_1(C,np__1) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,np__1)) ) ) ) ) ) ) ).
fof(dt_m1_graph_2,axiom,
! [A,B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_graph_2(C,A,B)
=> ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) ) ) ) ).
fof(existence_m1_graph_2,axiom,
! [A,B] :
( m1_finseq_1(B,A)
=> ? [C] : m1_graph_2(C,A,B) ) ).
fof(dt_k1_graph_2,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> ( v1_relat_1(k1_graph_2(A,B,C))
& v1_funct_1(k1_graph_2(A,B,C))
& v1_finseq_1(k1_graph_2(A,B,C)) ) ) ).
fof(dt_k2_graph_2,axiom,
! [A,B,C,D] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> m2_finseq_1(k2_graph_2(A,B,C,D),A) ) ).
fof(redefinition_k2_graph_2,axiom,
! [A,B,C,D] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> k2_graph_2(A,B,C,D) = k1_graph_2(B,C,D) ) ).
fof(dt_k3_graph_2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k3_graph_2(A,B))
& v1_funct_1(k3_graph_2(A,B))
& v1_finseq_1(k3_graph_2(A,B)) ) ) ).
fof(dt_k4_graph_2,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,A)
& m1_finseq_1(C,A) )
=> m2_finseq_1(k4_graph_2(A,B,C),A) ) ).
fof(redefinition_k4_graph_2,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,A)
& m1_finseq_1(C,A) )
=> k4_graph_2(A,B,C) = k3_graph_2(B,C) ) ).
fof(dt_k5_graph_2,axiom,
$true ).
fof(dt_k6_graph_2,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& m1_graph_1(B,A) )
=> m2_finseq_1(k6_graph_2(A,B),u1_graph_1(A)) ) ).
fof(dt_k7_graph_2,axiom,
! [A,B] :
( ( v2_graph_1(A)
& l1_graph_1(A)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> m2_finseq_1(k7_graph_2(A,B),u1_graph_1(A)) ) ).
fof(t4_graph_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_card_1(a_2_0_graph_2(A,B)) = k1_nat_1(B,np__1) ) ) ).
fof(t5_graph_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,A) )
=> k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),C) = k1_nat_1(B,C) ) ) ) ) ).
fof(d6_graph_2,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] : k5_graph_2(A,B) = a_2_2_graph_2(A,B) ) ).
fof(s1_graph_2,axiom,
v1_finset_1(a_0_0_graph_2) ).
fof(fraenkel_a_2_0_graph_2,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_graph_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = D
& r1_xreal_0(B,D)
& r1_xreal_0(D,k1_nat_1(B,C)) ) ) ) ).
fof(fraenkel_a_2_1_graph_2,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_1_graph_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = D
& r1_xreal_0(k1_nat_1(C,np__1),D)
& r1_xreal_0(D,k1_nat_1(C,B)) ) ) ) ).
fof(fraenkel_a_2_2_graph_2,axiom,
! [A,B,C] :
( ( v2_graph_1(B)
& l1_graph_1(B) )
=> ( r2_hidden(A,a_2_2_graph_2(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)
| D = k1_funct_1(u4_graph_1(B),E) ) ) ) ) ) ).
fof(fraenkel_a_0_0_graph_2,axiom,
! [A] :
( r2_hidden(A,a_0_0_graph_2)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s1_graph_2(B)
& r1_xreal_0(f1_s1_graph_2,B)
& r1_xreal_0(B,f2_s1_graph_2)
& p1_s1_graph_2(B) ) ) ).
%------------------------------------------------------------------------------