SET007 Axioms: SET007+615.ax
%------------------------------------------------------------------------------
% File : SET007+615 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of the External Approximation of Jordan's 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 : jordan10 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 2 unt; 0 def)
% Number of atoms : 256 ( 18 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 297 ( 76 ~; 0 |; 155 &)
% ( 2 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 39 ( 39 usr; 4 con; 0-4 aty)
% Number of variables : 69 ( 66 !; 3 ?)
% 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_jordan10,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ~ v1_xboole_0(A)
& v4_pre_topc(A,k15_euclid(np__2))
& v2_connsp_1(A,k15_euclid(np__2))
& v1_jordan2c(A,np__2)
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A) ) ).
fof(fc1_jordan10,axiom,
! [A,B] :
( ( v2_connsp_1(A,k15_euclid(np__2))
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(B,k5_numbers) )
=> ( v4_pre_topc(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),k15_euclid(np__2))
& v1_jordan2c(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),np__2)
& v6_compts_1(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),k15_euclid(np__2)) ) ) ).
fof(fc2_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ v1_xboole_0(k1_jordan10(A)) ) ).
fof(fc3_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ v1_xboole_0(k2_jordan10(A)) ) ).
fof(t1_jordan10,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v2_connsp_1(E,k15_euclid(np__2))
& v6_compts_1(E,k15_euclid(np__2))
& ~ v1_sppol_1(E)
& ~ v2_sppol_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
& r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
& r2_hidden(k4_tarski(C,k1_nat_1(D,np__1)),k2_matrix_1(k1_jordan8(E,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,k1_nat_1(D,np__1))
& r1_xreal_0(k3_finseq_1(k1_jordan8(E,B)),C) ) ) ) ) ) ) ).
fof(t2_jordan10,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v2_connsp_1(E,k15_euclid(np__2))
& v6_compts_1(E,k15_euclid(np__2))
& ~ v1_sppol_1(E)
& ~ v2_sppol_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
& r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
& r2_hidden(k4_tarski(C,k1_nat_1(D,np__1)),k2_matrix_1(k1_jordan8(E,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,k1_nat_1(D,np__1))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
& r1_xreal_0(C,np__1) ) ) ) ) ) ) ).
fof(t3_jordan10,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v2_connsp_1(E,k15_euclid(np__2))
& v6_compts_1(E,k15_euclid(np__2))
& ~ v1_sppol_1(E)
& ~ v2_sppol_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
& r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
& r2_hidden(k4_tarski(k1_nat_1(C,np__1),D),k2_matrix_1(k1_jordan8(E,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),k1_nat_1(C,np__1),D)
& r1_xreal_0(D,np__1) ) ) ) ) ) ) ).
fof(t4_jordan10,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v2_connsp_1(E,k15_euclid(np__2))
& v6_compts_1(E,k15_euclid(np__2))
& ~ v1_sppol_1(E)
& ~ v2_sppol_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
& r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
& r2_hidden(k4_tarski(k1_nat_1(C,np__1),D),k2_matrix_1(k1_jordan8(E,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),k1_nat_1(C,np__1),D)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
& r1_xreal_0(k1_matrix_1(k1_jordan8(E,B)),D) ) ) ) ) ) ) ).
fof(t5_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_subset_1(B,k5_topreal1(np__2,k1_jordan9(B,A))) ) ) ).
fof(t6_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(B,A))) = k3_real_1(k19_pscomp_1(B),k6_real_1(k5_real_1(k19_pscomp_1(B),k21_pscomp_1(B)),k3_newton(np__2,A))) ) ) ).
fof(t7_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_connsp_1(C,k15_euclid(np__2))
& v6_compts_1(C,k15_euclid(np__2))
& ~ v1_sppol_1(C)
& ~ v2_sppol_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(C,A))),k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(C,B)))) ) ) ) ) ).
fof(t8_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r2_hidden(k32_pscomp_1(B),k3_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t9_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r2_subset_1(B,k3_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t10_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_subset_1(B,k2_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t11_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_tarski(B,k3_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t12_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_tarski(B,k1_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,A)))) ) ) ).
fof(t13_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_tarski(k2_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,A))),k2_jordan2c(np__2,B)) ) ) ).
fof(t14_jordan10,axiom,
! [A] :
( ( v2_connsp_1(A,k15_euclid(np__2))
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k5_setfam_1(u1_struct_0(k15_euclid(np__2)),k3_jordan10(A)) = k2_jordan2c(np__2,A) ) ).
fof(t15_jordan10,axiom,
! [A] :
( ( v2_connsp_1(A,k15_euclid(np__2))
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_tarski(A,k6_setfam_1(u1_struct_0(k15_euclid(np__2)),k4_jordan10(A))) ) ).
fof(t16_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_xboole_0(k1_jordan2c(np__2,B),k2_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t17_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_tarski(k1_jordan2c(np__2,B),k3_goboard9(k1_jordan9(B,A))) ) ) ).
fof(t18_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( ( v2_connsp_1(C,k15_euclid(np__2))
& v6_compts_1(C,k15_euclid(np__2))
& ~ v1_sppol_1(C)
& ~ v2_sppol_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan2c(np__2,C,B)
=> r1_xboole_0(B,k5_topreal1(np__2,k1_jordan9(C,A))) ) ) ) ) ).
fof(t19_jordan10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_connsp_1(B,k15_euclid(np__2))
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_xboole_0(k1_jordan2c(np__2,B),k5_topreal1(np__2,k1_jordan9(B,A))) ) ) ).
fof(t20_jordan10,axiom,
! [A] :
( ( v2_connsp_1(A,k15_euclid(np__2))
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k7_setfam_1(u1_struct_0(k15_euclid(np__2)),k3_jordan10(A)) = k4_jordan10(A) ) ).
fof(t21_jordan10,axiom,
! [A] :
( ( v2_connsp_1(A,k15_euclid(np__2))
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k6_setfam_1(u1_struct_0(k15_euclid(np__2)),k4_jordan10(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,k1_jordan2c(np__2,A)) ) ).
fof(dt_k1_jordan10,axiom,
$true ).
fof(dt_k2_jordan10,axiom,
$true ).
fof(dt_k3_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m1_subset_1(k3_jordan10(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) ).
fof(redefinition_k3_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k3_jordan10(A) = k1_jordan10(A) ) ).
fof(dt_k4_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m1_subset_1(k4_jordan10(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) ).
fof(redefinition_k4_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k4_jordan10(A) = k2_jordan10(A) ) ).
fof(d1_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k1_jordan10(A) = a_1_0_jordan10(A) ) ).
fof(d2_jordan10,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> k2_jordan10(A) = a_1_1_jordan10(A) ) ).
fof(fraenkel_a_1_0_jordan10,axiom,
! [A,B] :
( ( v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_0_jordan10(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k2_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,C))) ) ) ) ).
fof(fraenkel_a_1_1_jordan10,axiom,
! [A,B] :
( ( v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_1_jordan10(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k6_pre_topc(k15_euclid(np__2),k1_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,C)))) ) ) ) ).
%------------------------------------------------------------------------------