SET007 Axioms: SET007+475.ax
%------------------------------------------------------------------------------
% File : SET007+475 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The First Part of Jordan's Theorem for Special Polygons
% 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 : gobrd12 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 12 ( 1 unt; 0 def)
% Number of atoms : 158 ( 10 equ)
% Maximal formula atoms : 29 ( 13 avg)
% Number of connectives : 169 ( 23 ~; 2 |; 102 &)
% ( 2 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 11 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-4 aty)
% Number of functors : 26 ( 26 usr; 4 con; 0-4 aty)
% Number of variables : 36 ( 32 !; 4 ?)
% 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(t1_gobrd12,axiom,
$true ).
fof(t2_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
=> r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) ) ) ) ) ).
fof(t3_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
=> k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))) = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_goboard5(k3_goboard2(A),B,C),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) ) ) ) ) ).
fof(t4_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
=> ( v2_connsp_1(k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))))
& k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)) ) ) ) ) ) ).
fof(t6_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ( m1_connsp_3(k4_subset_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k4_connsp_3(k15_euclid(np__2),k2_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k3_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))))
& k4_connsp_3(k15_euclid(np__2),k2_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k2_goboard9(A)
& k4_connsp_3(k15_euclid(np__2),k3_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k3_goboard9(A) ) ) ).
fof(t7_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A)))
& r1_xreal_0(D,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(E,k1_matrix_1(k3_goboard2(A)))
& r2_gobrd10(B,C,D,E) )
=> ( r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A)))
<=> r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),D,E)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) ) ) ) ).
fof(t8_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_finseq_1(B,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(B),D)
| r2_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D),k4_finseq_4(k5_numbers,k5_numbers,B,k1_nat_1(D,np__1)),k4_finseq_4(k5_numbers,k5_numbers,C,k1_nat_1(D,np__1))) ) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(D,k4_finseq_1(B))
& E = k1_funct_1(B,D)
& F = k1_funct_1(C,D) )
=> ( r1_xreal_0(E,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(F,k1_matrix_1(k3_goboard2(A))) ) ) ) ) ) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(B))
& r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) )
| ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) ) ) ) ).
fof(t9_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A)))
& r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ).
fof(t10_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
=> r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) ).
fof(t11_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A)) ) ).
fof(t5_gobrd12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)) = k3_tarski(a_1_0_gobrd12(A)) ) ).
fof(fraenkel_a_1_0_gobrd12,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& ~ v5_seqm_3(B)
& v1_topreal1(B)
& v2_topreal1(B)
& v1_finseq_6(B,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(B)
& v2_goboard5(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ( r2_hidden(A,a_1_0_gobrd12(B))
<=> ? [C,D] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& m2_subset_1(D,k1_numbers,k5_numbers)
& A = k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(B),C,D)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B))))
& r1_xreal_0(C,k3_finseq_1(k3_goboard2(B)))
& r1_xreal_0(D,k1_matrix_1(k3_goboard2(B))) ) ) ) ).
%------------------------------------------------------------------------------