SET007 Axioms: SET007+563.ax
%------------------------------------------------------------------------------
% File : SET007+563 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Graph Theoretical Properties of Arcs in the Plane
% Version : [Urb08] axioms.
% English : Graph Theoretical Properties of Arcs in the Plane and
% Fashoda Meet Theorem
% 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 : jgraph_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 18 unt; 0 def)
% Number of atoms : 500 ( 75 equ)
% Maximal formula atoms : 32 ( 6 avg)
% Number of connectives : 495 ( 72 ~; 13 |; 207 &)
% ( 5 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 49 ( 47 usr; 1 prp; 0-4 aty)
% Number of functors : 58 ( 58 usr; 7 con; 0-4 aty)
% Number of variables : 177 ( 172 !; 5 ?)
% 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(fc1_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> v2_graph_1(k1_jgraph_1(A)) ) ).
fof(t1_jgraph_1,axiom,
$true ).
fof(t2_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> r1_xreal_0(k9_square_1(k3_real_1(k7_square_1(A),k7_square_1(B))),k3_real_1(k18_complex1(A),k18_complex1(B))) ) ) ).
fof(t3_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(k18_complex1(A),k9_square_1(k3_real_1(k7_square_1(A),k7_square_1(B))))
& r1_xreal_0(k18_complex1(B),k9_square_1(k3_real_1(k7_square_1(A),k7_square_1(B)))) ) ) ) ).
fof(t4_jgraph_1,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))
=> ( ( v1_graph_4(B,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(d1_jgraph_1,axiom,
! [A] : k1_jgraph_1(A) = g1_graph_1(A,k2_zfmisc_1(A,A),k9_funct_3(A,A),k10_funct_3(A,A)) ).
fof(t5_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_graph_1(k1_jgraph_1(A))
& l1_graph_1(k1_jgraph_1(A)) ) ) ).
fof(t6_jgraph_1,axiom,
! [A] : u1_graph_1(k1_jgraph_1(A)) = A ).
fof(d2_jgraph_1,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) )
=> ( B = k2_jgraph_1(A)
<=> ( k3_finseq_1(B) = k5_binarith(k3_finseq_1(A),np__1)
& ! [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(B,C) = k4_tarski(k1_funct_1(A,C),k1_funct_1(A,k1_nat_1(C,np__1))) ) ) ) ) ) ) ) ).
fof(t7_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> m2_finseq_1(B,u1_graph_1(k1_jgraph_1(A))) ) ) ).
fof(t8_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> m2_finseq_1(k2_jgraph_1(B),u2_graph_1(k1_jgraph_1(A))) ) ) ).
fof(t9_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,k3_finseq_1(k3_jgraph_1(A,C))) )
=> r2_hidden(k1_funct_1(k3_jgraph_1(A,C),B),u2_graph_1(k1_jgraph_1(A))) ) ) ) ) ).
fof(t10_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( v8_graph_1(k3_jgraph_1(A,B),k1_jgraph_1(A))
& m2_graph_1(k3_jgraph_1(A,B),k1_jgraph_1(A)) ) ) ) ).
fof(t11_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(k1_jgraph_1(A)))
=> ( ( r1_xreal_0(np__1,k3_finseq_1(B))
& B = C )
=> r3_graph_4(k1_jgraph_1(A),C,k4_jgraph_1(A,B)) ) ) ) ) ).
fof(d3_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_jgraph_1(A,B,C)
<=> ( k1_funct_1(B,np__1) = k1_funct_1(C,np__1)
& k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,k3_finseq_1(C))
& ? [D] :
( m1_graph_2(D,u2_graph_1(k1_jgraph_1(A)),k4_jgraph_1(A,B))
& ? [E] :
( m1_graph_2(E,A,B)
& ? [F] :
( v8_graph_1(F,k1_jgraph_1(A))
& v3_graph_2(F,k1_jgraph_1(A))
& m2_graph_1(F,k1_jgraph_1(A))
& ? [G] :
( m2_finseq_1(G,u1_graph_1(k1_jgraph_1(A)))
& k15_finseq_1(D) = F
& k15_finseq_1(E) = C
& G = C
& r3_graph_4(k1_jgraph_1(A),G,F) ) ) ) ) ) ) ) ) ) ).
fof(t12_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_jgraph_1(A,B,C)
=> ( r1_xreal_0(np__1,k3_finseq_1(C))
& r1_xreal_0(k3_finseq_1(C),k3_finseq_1(B)) ) ) ) ) ) ).
fof(t13_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
& ! [C] :
( m2_finseq_1(C,A)
=> ~ r1_jgraph_1(A,B,C) ) ) ) ) ).
fof(t14_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_jgraph_1(A,B,C)
=> r1_tarski(k2_relat_1(k4_jgraph_1(A,C)),k2_relat_1(k4_jgraph_1(A,B))) ) ) ) ) ).
fof(t15_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_jgraph_1(A,B,C)
=> ( k1_funct_1(B,np__1) = k1_funct_1(B,k3_finseq_1(B))
| ( v2_funct_1(C)
& r1_tarski(k2_relat_1(k4_jgraph_1(A,C)),k2_relat_1(k4_jgraph_1(A,B)))
& k1_funct_1(C,np__1) = k1_funct_1(B,np__1)
& k1_funct_1(C,k3_finseq_1(C)) = k1_funct_1(B,k3_finseq_1(B)) ) ) ) ) ) ) ).
fof(d4_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ( v1_jgraph_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xboole_0(k4_topreal1(A,B,C),k4_topreal1(A,B,D))
& ~ ( k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal1(A,B,C),k4_topreal1(A,B,D)) = k1_tarski(k1_funct_1(B,C))
& ( k1_funct_1(B,C) = k1_funct_1(B,D)
| k1_funct_1(B,C) = k1_funct_1(B,k1_nat_1(D,np__1)) ) )
& ~ ( k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal1(A,B,C),k4_topreal1(A,B,D)) = k1_tarski(k1_funct_1(B,k1_nat_1(C,np__1)))
& ( k1_funct_1(B,k1_nat_1(C,np__1)) = k1_funct_1(B,D)
| k1_funct_1(B,k1_nat_1(C,np__1)) = k1_funct_1(B,k1_nat_1(D,np__1)) ) )
& k4_topreal1(A,B,C) != k4_topreal1(A,B,D) ) ) ) ) ) ) ).
fof(t16_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v3_topreal1(A)
=> v1_goboard5(A) ) ) ).
fof(t17_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_goboard5(A)
& r1_xboole_0(k4_topreal1(np__2,A,np__1),k4_topreal1(np__2,A,k5_binarith(k3_finseq_1(A),np__1))) )
=> v3_topreal1(A) ) ) ).
fof(t18_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_jgraph_1(A,np__2)
& v1_graph_4(k4_jgraph_1(u1_struct_0(k15_euclid(np__2)),A),k1_jgraph_1(u1_struct_0(k15_euclid(np__2)))) )
=> v1_goboard5(A) ) ) ).
fof(t19_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_jgraph_1(A,np__2)
& v1_graph_4(k4_jgraph_1(u1_struct_0(k15_euclid(np__2)),A),k1_jgraph_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( k1_funct_1(A,np__1) = k1_funct_1(A,k3_finseq_1(A))
| v3_topreal1(A) ) ) ) ).
fof(t20_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ~ ( ? [E] :
( E != C
& r2_hidden(E,k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D))) )
& ~ r2_hidden(B,k3_topreal1(A,C,D))
& ~ r2_hidden(D,k3_topreal1(A,B,C)) ) ) ) ) ) ).
fof(t21_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ( v3_topreal1(A)
& r1_tarski(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,A,np__1),k4_topreal1(np__2,A,k1_nat_1(np__1,np__1))),k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(np__1,np__1))))
& r1_tarski(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,A,k5_binarith(k3_finseq_1(A),np__2)),k4_topreal1(np__2,A,k5_binarith(k3_finseq_1(A),np__1))),k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k5_binarith(k3_finseq_1(A),np__1)))) )
=> v2_topreal1(A) ) ) ).
fof(t22_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( v1_graph_4(k4_jgraph_1(A,B),k1_jgraph_1(A))
=> ( k1_funct_1(B,np__1) = k1_funct_1(B,k3_finseq_1(B))
| ( v2_funct_1(B)
& k3_finseq_1(B) != np__1 ) ) ) ) ) ).
fof(t23_jgraph_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( v2_funct_1(B)
=> ( r1_xreal_0(k3_finseq_1(B),np__1)
| ( v1_graph_4(k4_jgraph_1(A,B),k1_jgraph_1(A))
& k1_funct_1(B,np__1) != k1_funct_1(B,k3_finseq_1(B)) ) ) ) ) ) ).
fof(t24_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_jgraph_1(A,np__2)
& v1_graph_4(k4_jgraph_1(u1_struct_0(k15_euclid(np__2)),A),k1_jgraph_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( k1_funct_1(A,np__1) = k1_funct_1(A,k3_finseq_1(A))
| v2_topreal1(A) ) ) ) ).
fof(t25_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_jgraph_1(u1_struct_0(k15_euclid(np__2)),A,B)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(A))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,D) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(D,np__1)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1))
& k1_funct_1(A,D) = k1_funct_1(B,C)
& k1_funct_1(A,k1_nat_1(D,np__1)) = k1_funct_1(B,k1_nat_1(C,np__1)) ) ) ) ) ) ) ).
fof(t26_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jgraph_1(u1_struct_0(k15_euclid(np__2)),A,B)
=> r1_tarski(k2_relat_1(B),k2_relat_1(A)) ) ) ) ).
fof(t27_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jgraph_1(u1_struct_0(k15_euclid(np__2)),A,B)
=> r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A)) ) ) ) ).
fof(t28_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v1_topreal1(A)
& r1_jgraph_1(u1_struct_0(k15_euclid(np__2)),A,B) )
=> v1_topreal1(B) ) ) ) ).
fof(t29_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(A)
& r1_xreal_0(np__2,k3_finseq_1(A))
& k1_funct_1(A,np__1) != k1_funct_1(A,k3_finseq_1(A))
& ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& v1_topreal1(B)
& v2_funct_1(B)
& r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
& k1_funct_1(A,np__1) = k1_funct_1(B,np__1)
& k1_funct_1(A,k3_finseq_1(A)) = k1_funct_1(B,k3_finseq_1(B))
& r1_tarski(k2_relat_1(B),k2_relat_1(A)) ) ) ) ) ).
fof(t30_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(A)
& v1_topreal1(B)
& r1_xreal_0(np__2,k3_finseq_1(A))
& r1_xreal_0(np__2,k3_finseq_1(B))
& k1_funct_1(A,np__1) != k1_funct_1(A,k3_finseq_1(A))
& k1_funct_1(B,np__1) != k1_funct_1(B,k3_finseq_1(B))
& r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k2_goboard1(B),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_goboard4(k3_goboard1(B),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) ).
fof(t31_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(A,C)
& r1_xreal_0(C,B)
& r1_xreal_0(A,D)
& r1_xreal_0(D,B) )
=> r1_xreal_0(k18_complex1(k5_real_1(C,D)),k5_real_1(B,A)) ) ) ) ) ) ).
fof(d5_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k5_toprns_1(A,B)
<=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( B = D
=> C = k12_euclid(D) ) ) ) ) ) ) ).
fof(t32_jgraph_1,axiom,
$true ).
fof(t33_jgraph_1,axiom,
$true ).
fof(t34_jgraph_1,axiom,
$true ).
fof(t35_jgraph_1,axiom,
$true ).
fof(t36_jgraph_1,axiom,
$true ).
fof(t37_jgraph_1,axiom,
$true ).
fof(t38_jgraph_1,axiom,
$true ).
fof(t39_jgraph_1,axiom,
$true ).
fof(t40_jgraph_1,axiom,
$true ).
fof(t41_jgraph_1,axiom,
$true ).
fof(t42_jgraph_1,axiom,
$true ).
fof(t43_jgraph_1,axiom,
$true ).
fof(t44_jgraph_1,axiom,
$true ).
fof(t45_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(A)))
=> ( ( D = B
& E = C )
=> k5_toprns_1(A,k20_euclid(A,B,C)) = k4_metric_1(k14_euclid(A),D,E) ) ) ) ) ) ) ).
fof(t46_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k7_square_1(k5_toprns_1(np__2,A)) = k3_real_1(k7_square_1(k21_euclid(A)),k7_square_1(k22_euclid(A))) ) ).
fof(t47_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> k5_toprns_1(np__2,A) = k9_square_1(k3_real_1(k7_square_1(k21_euclid(A)),k7_square_1(k22_euclid(A)))) ) ).
fof(t48_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> r1_xreal_0(k5_toprns_1(np__2,A),k3_real_1(k18_complex1(k21_euclid(A)),k18_complex1(k22_euclid(A)))) ) ).
fof(t49_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> r1_xreal_0(k5_toprns_1(np__2,k20_euclid(np__2,A,B)),k3_real_1(k18_complex1(k5_real_1(k21_euclid(A),k21_euclid(B))),k18_complex1(k5_real_1(k22_euclid(A),k22_euclid(B))))) ) ) ).
fof(t50_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(k18_complex1(k21_euclid(A)),k5_toprns_1(np__2,A))
& r1_xreal_0(k18_complex1(k22_euclid(A)),k5_toprns_1(np__2,A)) ) ) ).
fof(t51_jgraph_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(k18_complex1(k5_real_1(k21_euclid(A),k21_euclid(B))),k5_toprns_1(np__2,k20_euclid(np__2,A,B)))
& r1_xreal_0(k18_complex1(k5_real_1(k22_euclid(A),k22_euclid(B))),k5_toprns_1(np__2,k20_euclid(np__2,A,B))) ) ) ) ).
fof(t52_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ~ ( r2_hidden(B,k3_topreal1(A,C,D))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1)
& B = k17_euclid(A,k18_euclid(k5_real_1(np__1,E),A,C),k18_euclid(E,A,D)) ) ) ) ) ) ) ) ).
fof(t53_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(B,k3_topreal1(A,C,D))
=> ( r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k5_toprns_1(A,k20_euclid(A,C,D)))
& r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,D)),k5_toprns_1(A,k20_euclid(A,C,D))) ) ) ) ) ) ) ).
fof(t54_jgraph_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> ( B = k1_xboole_0
| C = k1_xboole_0
| r1_xreal_0(np__0,k9_weierstr(A,B,C)) ) ) ) ) ) ).
fof(t55_jgraph_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( ( v6_compts_1(B,k5_pcomps_1(A))
& v6_compts_1(C,k5_pcomps_1(A)) )
=> ( B = k1_xboole_0
| C = k1_xboole_0
| ( r1_xboole_0(B,C)
<=> ~ r1_xreal_0(k9_weierstr(A,B,C),np__0) ) ) ) ) ) ) ).
fof(t56_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(A))
& r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k3_goboard1(A),C,D)
& ~ r1_xreal_0(B,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& r1_xreal_0(k1_nat_1(E,np__1),k3_finseq_1(A))
& r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,E),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(E,np__1))))) ) )
& ! [E] :
( m2_finseq_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(E)
& k1_funct_1(E,np__1) = k1_funct_1(A,np__1)
& k1_funct_1(E,k3_finseq_1(E)) = k1_funct_1(A,k3_finseq_1(A))
& r1_xreal_0(k3_finseq_1(A),k3_finseq_1(E))
& r1_goboard4(k2_goboard1(E),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k3_goboard1(E),C,D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(E))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(G,k4_finseq_1(A))
& ~ r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,F),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,G)))) ) ) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,F)
& r1_xreal_0(k1_nat_1(F,np__1),k3_finseq_1(E))
& r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,F),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,k1_nat_1(F,np__1))))) ) ) ) ) ) ) ) ) ) ).
fof(t57_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(A))
& r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(A),np__1),k1_goboard1(k3_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k2_goboard1(A),C,D)
& ~ r1_xreal_0(B,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& r1_xreal_0(k1_nat_1(E,np__1),k3_finseq_1(A))
& r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,E),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(E,np__1))))) ) )
& ! [E] :
( m2_finseq_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v1_topreal1(E)
& k1_funct_1(E,np__1) = k1_funct_1(A,np__1)
& k1_funct_1(E,k3_finseq_1(E)) = k1_funct_1(A,k3_finseq_1(A))
& r1_xreal_0(k3_finseq_1(A),k3_finseq_1(E))
& r1_goboard4(k3_goboard1(E),k1_goboard1(k3_goboard1(A),np__1),k1_goboard1(k3_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k2_goboard1(E),C,D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(E))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(G,k4_finseq_1(A))
& ~ r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,F),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,G)))) ) ) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,F)
& r1_xreal_0(k1_nat_1(F,np__1),k3_finseq_1(E))
& r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,F),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),E,k1_nat_1(F,np__1))))) ) ) ) ) ) ) ) ) ) ).
fof(t58_jgraph_1,axiom,
$true ).
fof(t59_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
=> ( k3_finseq_1(k2_goboard1(A)) = k3_finseq_1(A)
& k1_goboard1(k2_goboard1(A),np__1) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
& k1_goboard1(k2_goboard1(A),k3_finseq_1(A)) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))) ) ) ) ).
fof(t60_jgraph_1,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
=> ( k3_finseq_1(k3_goboard1(A)) = k3_finseq_1(A)
& k1_goboard1(k3_goboard1(A),np__1) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
& k1_goboard1(k3_goboard1(A),k3_finseq_1(A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))) ) ) ) ).
fof(t61_jgraph_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,k4_finseq_1(B))
=> ( k1_goboard1(k2_goboard1(B),A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A))
& k1_goboard1(k3_goboard1(B),A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) ) ) ) ) ).
fof(t62_jgraph_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal1(k15_euclid(np__2),C,D,A)
& r1_topreal1(k15_euclid(np__2),E,F,B)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,A)
=> ( r1_xreal_0(k21_euclid(C),k21_euclid(G))
& r1_xreal_0(k21_euclid(G),k21_euclid(D)) ) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,B)
=> ( r1_xreal_0(k21_euclid(C),k21_euclid(G))
& r1_xreal_0(k21_euclid(G),k21_euclid(D)) ) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,A)
=> ( r1_xreal_0(k22_euclid(E),k22_euclid(G))
& r1_xreal_0(k22_euclid(G),k22_euclid(F)) ) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,B)
=> ( r1_xreal_0(k22_euclid(E),k22_euclid(G))
& r1_xreal_0(k22_euclid(G),k22_euclid(F)) ) ) )
& r2_subset_1(A,B) ) ) ) ) ) ) ) ).
fof(t63_jgraph_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D)))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) )
=> ( ( C = E
& v5_pre_topc(C,A,B) )
=> v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ).
fof(t64_jgraph_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ~ ( v2_compts_1(A)
& v3_compts_1(B)
& v5_pre_topc(C,A,B)
& v2_funct_1(C)
& D = k1_pscomp_1(u1_struct_0(A),u1_struct_0(B),C)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D)))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) )
=> ~ ( C = E
& v3_tops_2(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) ).
fof(t65_jgraph_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(A,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2))) )
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k5_topmetr))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k5_topmetr))
=> ~ ( G = np__0
& H = np__1
& v5_pre_topc(A,k5_topmetr,k15_euclid(np__2))
& v2_funct_1(A)
& v5_pre_topc(B,k5_topmetr,k15_euclid(np__2))
& v2_funct_1(B)
& k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,G)) = C
& k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,H)) = D
& k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,G)) = E
& k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,H)) = F
& ! [I] :
( m1_subset_1(I,u1_struct_0(k5_topmetr))
=> ( r1_xreal_0(C,k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,I)))
& r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,I)),D)
& r1_xreal_0(C,k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,I)))
& r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,I)),D)
& r1_xreal_0(E,k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,I)))
& r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A,I)),F)
& r1_xreal_0(E,k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,I)))
& r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B,I)),F) ) )
& r1_xboole_0(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),A),k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),B)) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_jgraph_1,axiom,
! [A] : l1_graph_1(k1_jgraph_1(A)) ).
fof(dt_k2_jgraph_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k2_jgraph_1(A))
& v1_funct_1(k2_jgraph_1(A))
& v1_finseq_1(k2_jgraph_1(A)) ) ) ).
fof(dt_k3_jgraph_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> m2_finseq_1(k3_jgraph_1(A,B),u2_graph_1(k1_jgraph_1(A))) ) ).
fof(redefinition_k3_jgraph_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> k3_jgraph_1(A,B) = k2_jgraph_1(B) ) ).
fof(dt_k4_jgraph_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> ( v8_graph_1(k4_jgraph_1(A,B),k1_jgraph_1(A))
& m2_graph_1(k4_jgraph_1(A,B),k1_jgraph_1(A)) ) ) ).
fof(redefinition_k4_jgraph_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> k4_jgraph_1(A,B) = k2_jgraph_1(B) ) ).
%------------------------------------------------------------------------------