SET007 Axioms: SET007+446.ax
%------------------------------------------------------------------------------
% File : SET007+446 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Left and Right Component of the Complement of a Closed Curve
% 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 : goboard9 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 37 ( 2 unt; 0 def)
% Number of atoms : 338 ( 19 equ)
% Maximal formula atoms : 16 ( 9 avg)
% Number of connectives : 359 ( 58 ~; 1 |; 214 &)
% ( 2 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 5 con; 0-3 aty)
% Number of variables : 74 ( 72 !; 2 ?)
% 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_goboard9,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ~ v1_xboole_0(A)
& v1_jordan1(A,np__2) ) ).
fof(rc2_goboard9,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& ~ v5_seqm_3(A)
& v1_finset_1(A)
& v1_finseq_1(A) ) ).
fof(fc1_goboard9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& ~ v5_seqm_3(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k3_finseq_5(A))
& v1_funct_1(k3_finseq_5(A))
& ~ v5_seqm_3(k3_finseq_5(A))
& v1_finset_1(k3_finseq_5(A))
& v1_finseq_1(k3_finseq_5(A)) ) ) ).
fof(t1_goboard9,axiom,
$true ).
fof(t2_goboard9,axiom,
$true ).
fof(t3_goboard9,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r4_connsp_1(A,D,B)
& r4_connsp_1(A,D,C)
& B != C
& ~ r1_xboole_0(B,C) ) ) ) ) ) ).
fof(t4_goboard9,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C))))
=> ( B = D
=> k3_pre_topc(A,B) = k3_pre_topc(k3_pre_topc(A,C),D) ) ) ) ) ) ).
fof(t5_goboard9,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(B,C)
& v2_connsp_1(B,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r4_connsp_1(A,C,D)
& r1_tarski(B,D) ) ) ) ) ) ) ).
fof(t6_goboard9,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(C,A)
& r4_connsp_1(A,E,D)
& r1_tarski(B,D)
& r1_tarski(C,E) )
=> ( r1_xboole_0(B,C)
| r1_tarski(C,D) ) ) ) ) ) ) ) ).
fof(t7_goboard9,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)))
=> v1_jordan1(k3_topreal1(np__2,A,B),np__2) ) ) ).
fof(t8_goboard9,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)))
=> v2_connsp_1(k3_topreal1(np__2,A,B),k15_euclid(np__2)) ) ) ).
fof(t9_goboard9,axiom,
! [A] :
( ( v1_jordan1(A,np__2)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_jordan1(B,np__2)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> v1_jordan1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),A,B),np__2) ) ) ).
fof(t10_goboard9,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> k4_finseq_5(k1_numbers,k2_goboard1(A)) = k2_goboard1(k4_finseq_5(u1_struct_0(k15_euclid(np__2)),A)) ) ).
fof(t11_goboard9,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> k4_finseq_5(k1_numbers,k3_goboard1(A)) = k3_goboard1(k4_finseq_5(u1_struct_0(k15_euclid(np__2)),A)) ) ).
fof(t12_goboard9,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(np__1,B)
& r1_xreal_0(np__1,C)
& k1_nat_1(B,C) = k3_finseq_1(A) )
=> k5_goboard5(A,B) = k4_goboard5(k1_goboard9(A),C) ) ) ) ) ).
fof(t13_goboard9,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(np__1,B)
& r1_xreal_0(np__1,C)
& k1_nat_1(B,C) = k3_finseq_1(A) )
=> k5_goboard5(k1_goboard9(A),B) = k4_goboard5(A,C) ) ) ) ) ).
fof(t14_goboard9,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)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,k3_finseq_1(k3_goboard2(A)))
& r1_xreal_0(D,k1_matrix_1(k3_goboard2(A)))
& k3_goboard5(k3_goboard2(A),C,D) = k5_goboard5(A,B) ) ) ) ) ) ) ).
fof(t15_goboard9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& v5_goboard1(B)
& v6_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(A,k1_matrix_1(B))
=> v1_jordan1(k1_tops_1(k15_euclid(np__2),k2_goboard5(B,A)),np__2) ) ) ) ).
fof(t16_goboard9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& v5_goboard1(B)
& v6_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(A,k3_finseq_1(B))
=> v1_jordan1(k1_tops_1(k15_euclid(np__2),k1_goboard5(B,A)),np__2) ) ) ) ).
fof(t17_goboard9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& v5_goboard1(C)
& v6_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(B,k1_matrix_1(C))
& k1_tops_1(k15_euclid(np__2),k3_goboard5(C,A,B)) = k1_xboole_0 ) ) ) ) ).
fof(t18_goboard9,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)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
& k1_tops_1(k15_euclid(np__2),k5_goboard5(A,B)) = k1_xboole_0 ) ) ) ).
fof(t19_goboard9,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)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
& k1_tops_1(k15_euclid(np__2),k4_goboard5(A,B)) = k1_xboole_0 ) ) ) ).
fof(t20_goboard9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& v5_goboard1(C)
& v6_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(B,k1_matrix_1(C)) )
=> v1_jordan1(k1_tops_1(k15_euclid(np__2),k3_goboard5(C,A,B)),np__2) ) ) ) ) ).
fof(t21_goboard9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& v5_goboard1(C)
& v6_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(B,k1_matrix_1(C)) )
=> v2_connsp_1(k1_tops_1(k15_euclid(np__2),k3_goboard5(C,A,B)),k15_euclid(np__2)) ) ) ) ) ).
fof(t22_goboard9,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)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> v2_connsp_1(k1_tops_1(k15_euclid(np__2),k5_goboard5(A,B)),k15_euclid(np__2)) ) ) ) ).
fof(t23_goboard9,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)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> v2_connsp_1(k1_tops_1(k15_euclid(np__2),k4_goboard5(A,B)),k15_euclid(np__2)) ) ) ) ).
fof(d1_goboard9,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] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = k2_goboard9(A)
<=> ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),B)
& r1_tarski(k1_tops_1(k15_euclid(np__2),k5_goboard5(A,np__1)),B) ) ) ) ) ).
fof(d2_goboard9,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] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = k3_goboard9(A)
<=> ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),B)
& r1_tarski(k1_tops_1(k15_euclid(np__2),k4_goboard5(A,np__1)),B) ) ) ) ) ).
fof(t24_goboard9,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)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> r1_tarski(k1_tops_1(k15_euclid(np__2),k5_goboard5(A,B)),k2_goboard9(A)) ) ) ) ).
fof(t25_goboard9,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_goboard2(k1_goboard9(A)) = k3_goboard2(A) ) ).
fof(t26_goboard9,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_goboard9(A) = k2_goboard9(k1_goboard9(A)) ) ).
fof(t27_goboard9,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_goboard9(k1_goboard9(A)) = k2_goboard9(A) ) ).
fof(t28_goboard9,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)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> r1_tarski(k1_tops_1(k15_euclid(np__2),k4_goboard5(A,B)),k3_goboard9(A)) ) ) ) ).
fof(dt_k1_goboard9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ( ~ v1_xboole_0(k1_goboard9(A))
& v1_topreal1(k1_goboard9(A))
& v2_topreal1(k1_goboard9(A))
& v1_finseq_6(k1_goboard9(A),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k1_goboard9(A))
& v2_goboard5(k1_goboard9(A))
& m2_finseq_1(k1_goboard9(A),u1_struct_0(k15_euclid(np__2))) ) ) ).
fof(redefinition_k1_goboard9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k1_goboard9(A) = k3_finseq_5(A) ) ).
fof(dt_k2_goboard9,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)
& m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> m1_subset_1(k2_goboard9(A),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ).
fof(dt_k3_goboard9,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)
& m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> m1_subset_1(k3_goboard9(A),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ).
%------------------------------------------------------------------------------