SET007 Axioms: SET007+706.ax
%------------------------------------------------------------------------------
% File : SET007+706 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Remarks on Finite Sequences on Go-Boards
% 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 : jordan1f [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 13 ( 0 unt; 0 def)
% Number of atoms : 165 ( 19 equ)
% Maximal formula atoms : 24 ( 12 avg)
% Number of connectives : 189 ( 37 ~; 2 |; 97 &)
% ( 0 <=>; 53 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 13 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-3 aty)
% Number of functors : 42 ( 42 usr; 6 con; 0-4 aty)
% Number of variables : 51 ( 51 !; 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(t1_jordan1f,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_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( ( ~ v3_relat_1(E)
& v1_matrix_1(E)
& v3_goboard1(E)
& v4_goboard1(E)
& v5_goboard1(E)
& v6_goboard1(E)
& m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
& ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D))
& r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
& r2_hidden(k4_tarski(A,C),k2_matrix_1(E))
& r1_xreal_0(B,C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,F)
& r1_xreal_0(F,C)
& k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,F)) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) ).
fof(t2_jordan1f,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_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( ( ~ v3_relat_1(E)
& v1_matrix_1(E)
& v3_goboard1(E)
& v4_goboard1(E)
& v5_goboard1(E)
& v6_goboard1(E)
& m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
& ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D))
& r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
& r2_hidden(k4_tarski(A,C),k2_matrix_1(E))
& r1_xreal_0(B,C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,F)
& r1_xreal_0(F,C)
& k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,F)) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) ).
fof(t3_jordan1f,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_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( ( ~ v3_relat_1(E)
& v1_matrix_1(E)
& v3_goboard1(E)
& v4_goboard1(E)
& v5_goboard1(E)
& v6_goboard1(E)
& m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
& ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D))
& r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
& r2_hidden(k4_tarski(C,B),k2_matrix_1(E))
& r1_xreal_0(A,C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(A,F)
& r1_xreal_0(F,C)
& k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,F,B)) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) ).
fof(t4_jordan1f,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_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( ( ~ v3_relat_1(E)
& v1_matrix_1(E)
& v3_goboard1(E)
& v4_goboard1(E)
& v5_goboard1(E)
& v6_goboard1(E)
& m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
& ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D))
& r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
& r2_hidden(k4_tarski(C,B),k2_matrix_1(E))
& r1_xreal_0(A,C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(A,F)
& r1_xreal_0(F,C)
& k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,F,B)) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) ).
fof(t5_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),np__1) = k30_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ).
fof(t6_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),np__1) = k34_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ).
fof(t7_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),k3_finseq_1(k1_jordan1e(A,B))) = k34_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ).
fof(t8_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),k3_finseq_1(k2_jordan1e(A,B))) = k30_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ).
fof(t9_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( k5_topreal1(np__2,k1_jordan1e(A,B)) = k8_jordan6(k5_topreal1(np__2,k1_jordan9(A,B)))
& k5_topreal1(np__2,k2_jordan1e(A,B)) = k9_jordan6(k5_topreal1(np__2,k1_jordan9(A,B))) )
| ( k5_topreal1(np__2,k1_jordan1e(A,B)) = k9_jordan6(k5_topreal1(np__2,k1_jordan9(A,B)))
& k5_topreal1(np__2,k2_jordan1e(A,B)) = k8_jordan6(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ) ) ).
fof(t10_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_goboard1(u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),k1_jordan8(A,B)) ) ) ).
fof(t11_jordan1f,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
& ! [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)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
& r2_hidden(k4_tarski(F,G),k2_matrix_1(A))
& B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,F,G) )
=> k2_xcmplx_0(k18_complex1(k6_xcmplx_0(F,D)),k18_complex1(k6_xcmplx_0(G,E))) = np__1 ) ) ) ) ) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
& B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E) ) ) )
| r1_goboard1(u1_struct_0(k15_euclid(np__2)),k8_finseq_1(u1_struct_0(k15_euclid(np__2)),k12_finseq_1(u1_struct_0(k15_euclid(np__2)),B),C),A) ) ) ) ) ) ).
fof(t12_jordan1f,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)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_goboard1(u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),k1_jordan8(A,B)) ) ) ).
fof(t13_jordan1f,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_topreal2(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)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k21_euclid(C) = k7_xcmplx_0(k2_xcmplx_0(k18_pscomp_1(B),k20_pscomp_1(B)),np__2)
& k22_euclid(C) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,np__1),k1_jordan1a(k1_jordan8(B,np__1)),np__1),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,np__1),k1_jordan1a(k1_jordan8(B,np__1)),k1_matrix_1(k1_jordan8(B,np__1)))),k8_jordan6(k5_topreal1(np__2,k1_jordan9(B,k1_nat_1(A,np__1)))))))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k1_matrix_1(k1_jordan8(B,k1_nat_1(A,np__1))))
& C = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,k1_nat_1(A,np__1)),k1_jordan1a(k1_jordan8(B,k1_nat_1(A,np__1))),D) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------