SET007 Axioms: SET007+583.ax
%------------------------------------------------------------------------------
% File : SET007+583 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Components of the Complement of a Special Polygonal 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 : sprect_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 8 ( 1 unt; 0 def)
% Number of atoms : 55 ( 5 equ)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 60 ( 13 ~; 2 |; 25 &)
% ( 0 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 8 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-5 aty)
% Number of functors : 23 ( 23 usr; 5 con; 0-4 aty)
% Number of variables : 16 ( 16 !; 0 ?)
% 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(cc1_sprect_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_compts_1(B,A)
=> v4_pre_topc(B,A) ) ) ) ).
fof(t1_sprect_4,axiom,
! [A] :
( ( v4_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v4_pre_topc(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( ~ r1_xboole_0(k5_topreal1(np__2,A),B)
& ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),B)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_jordan3(A,k1_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))),B) != k1_struct_0(k15_euclid(np__2),k1_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)))) ) ) ) ).
fof(t2_sprect_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v4_topreal1(A)
& B = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) )
=> k5_topreal1(np__2,k3_jordan3(A,B)) = k1_xboole_0 ) ) ) ).
fof(t3_sprect_4,axiom,
$true ).
fof(t4_sprect_4,axiom,
! [A] :
( ( v4_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r2_hidden(B,k5_topreal1(np__2,k1_jordan3(u1_struct_0(k15_euclid(np__2)),A,C,k3_finseq_1(A)))) )
=> ( r1_xreal_0(k3_finseq_1(A),C)
| r1_jordan5c(k5_topreal1(np__2,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C),B) ) ) ) ) ) ).
fof(t5_sprect_4,axiom,
! [A] :
( ( v4_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r2_hidden(B,k4_topreal1(np__2,A,D))
& r2_hidden(C,k3_topreal1(np__2,B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(D,np__1)))) )
=> ( r1_xreal_0(k3_finseq_1(A),D)
| r1_jordan5c(k5_topreal1(np__2,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),B,C) ) ) ) ) ) ) ).
fof(t6_sprect_4,axiom,
! [A] :
( ( v4_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v4_pre_topc(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( ~ r1_xboole_0(k5_topreal1(np__2,A),B)
& ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),B)
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k3_jordan3(A,k2_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))),B) != k1_struct_0(k15_euclid(np__2),k2_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)))) ) ) ) ).
fof(t7_sprect_4,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))) )
=> k2_goboard9(A) != k3_goboard9(A) ) ).
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