SET007 Axioms: SET007+584.ax
%------------------------------------------------------------------------------
% File : SET007+584 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Gauges
% 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 : jordan8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 1 unt; 0 def)
% Number of atoms : 225 ( 24 equ)
% Maximal formula atoms : 25 ( 9 avg)
% Number of connectives : 242 ( 42 ~; 4 |; 109 &)
% ( 3 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 10 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 42 ( 42 usr; 7 con; 0-4 aty)
% Number of variables : 75 ( 75 !; 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(fc1_jordan8,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k1_jordan8(A,B))
& ~ v3_relat_1(k1_jordan8(A,B))
& v1_funct_1(k1_jordan8(A,B))
& v1_finset_1(k1_jordan8(A,B))
& v1_finseq_1(k1_jordan8(A,B))
& v1_matrix_1(k1_jordan8(A,B))
& v3_goboard1(k1_jordan8(A,B))
& v4_goboard1(k1_jordan8(A,B)) ) ) ).
fof(fc2_jordan8,axiom,
! [A,B] :
( ( ~ v1_xboole_0(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(B,k5_numbers) )
=> ( v1_relat_1(k1_jordan8(A,B))
& ~ v3_relat_1(k1_jordan8(A,B))
& v1_funct_1(k1_jordan8(A,B))
& v1_finset_1(k1_jordan8(A,B))
& v1_finseq_1(k1_jordan8(A,B))
& v1_matrix_1(k1_jordan8(A,B))
& v3_goboard1(k1_jordan8(A,B))
& v4_goboard1(k1_jordan8(A,B))
& v5_goboard1(k1_jordan8(A,B))
& v6_goboard1(k1_jordan8(A,B)) ) ) ).
fof(t1_jordan8,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ( r1_xreal_0(np__2,k3_finseq_1(B))
=> k16_finseq_1(A,B,np__2) = k10_finseq_1(k4_finseq_4(k5_numbers,A,B,np__1),k4_finseq_4(k5_numbers,A,B,np__2)) ) ) ).
fof(t2_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ( r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
=> k16_finseq_1(B,C,k1_nat_1(A,np__1)) = k7_finseq_1(k16_finseq_1(B,C,A),k9_finseq_1(k4_finseq_4(k5_numbers,B,C,k1_nat_1(A,np__1)))) ) ) ) ).
fof(t3_jordan8,axiom,
! [A,B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> r1_goboard1(A,k6_finseq_1(A),B) ) ).
fof(t4_jordan8,axiom,
$true ).
fof(t5_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(B)) )
=> ( r1_goboard1(B,C,D)
=> r1_goboard1(B,k1_rfinseq(B,C,A),D) ) ) ) ) ) ).
fof(t6_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(B)) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r1_goboard1(B,C,D)
& ! [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)
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(k4_tarski(E,F),k2_matrix_1(D))
& k4_finseq_4(k5_numbers,B,C,A) = k3_matrix_1(B,D,E,F)
& r2_hidden(k4_tarski(G,H),k2_matrix_1(D))
& k4_finseq_4(k5_numbers,B,C,k1_nat_1(A,np__1)) = k3_matrix_1(B,D,G,H)
& ~ ( ~ ( E = G
& k1_nat_1(F,np__1) = H )
& ~ ( k1_nat_1(E,np__1) = G
& F = H )
& ~ ( E = k1_nat_1(G,np__1)
& F = H )
& ~ ( E = G
& F = k1_nat_1(H,np__1) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_jordan8,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] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
=> ( v2_goboard5(B)
& v1_topreal1(B) ) ) ) ) ).
fof(t8_jordan8,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] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
& v5_seqm_3(B) ) ) ) ).
fof(t9_jordan8,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] :
( ( ~ v1_xboole_0(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))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E)
& B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,F,G) )
=> k3_real_1(k18_complex1(k5_real_1(F,D)),k18_complex1(k5_real_1(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)),C,k12_finseq_1(u1_struct_0(k15_euclid(np__2)),B)),A) ) ) ) ) ) ).
fof(t10_jordan8,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] :
( ( ~ v3_relat_1(D)
& v1_matrix_1(D)
& v3_goboard1(D)
& v4_goboard1(D)
& v5_goboard1(D)
& v6_goboard1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(D),k1_nat_1(A,B))
| r1_xreal_0(k1_matrix_1(D),C)
| r1_xboole_0(k3_goboard5(D,A,C),k3_goboard5(D,k1_nat_1(A,B),C))
| r1_xreal_0(B,np__1) ) ) ) ) ) ) ).
fof(t11_jordan8,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v2_sppol_1(A)
<=> r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) ) ).
fof(t12_jordan8,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_sppol_1(A)
<=> r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) ) ).
fof(d1_jordan8,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( C = k1_jordan8(A,B)
<=> ( k3_finseq_1(C) = k1_nat_1(k1_card_4(np__2,B),np__3)
& k3_finseq_1(C) = k1_matrix_1(C)
& ! [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(C))
=> k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,D,E) = k23_euclid(k3_real_1(k18_pscomp_1(A),k4_real_1(k6_real_1(k5_real_1(k20_pscomp_1(A),k18_pscomp_1(A)),k1_card_4(np__2,B)),k5_real_1(D,np__2))),k3_real_1(k21_pscomp_1(A),k4_real_1(k6_real_1(k5_real_1(k19_pscomp_1(A),k21_pscomp_1(A)),k1_card_4(np__2,B)),k5_real_1(E,np__2)))) ) ) ) ) ) ) ) ) ).
fof(t13_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_xreal_0(np__4,k3_finseq_1(k1_jordan8(B,A))) ) ) ).
fof(t14_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
=> k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),np__2,A)) = k18_pscomp_1(C) ) ) ) ) ).
fof(t15_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
=> k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),k5_binarith(k3_finseq_1(k1_jordan8(C,B)),np__1),A)) = k20_pscomp_1(C) ) ) ) ) ).
fof(t16_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
=> k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),A,np__2)) = k21_pscomp_1(C) ) ) ) ) ).
fof(t17_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
=> k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),A,k5_binarith(k3_finseq_1(k1_jordan8(C,B)),np__1))) = k19_pscomp_1(C) ) ) ) ) ).
fof(t18_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& 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(A,k3_finseq_1(k1_jordan8(C,B)))
=> r1_xboole_0(k3_goboard5(k1_jordan8(C,B),A,k3_finseq_1(k1_jordan8(C,B))),C) ) ) ) ) ).
fof(t19_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& 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(A,k3_finseq_1(k1_jordan8(C,B)))
=> r1_xboole_0(k3_goboard5(k1_jordan8(C,B),k3_finseq_1(k1_jordan8(C,B)),A),C) ) ) ) ) ).
fof(t20_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& 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(A,k3_finseq_1(k1_jordan8(C,B)))
=> r1_xboole_0(k3_goboard5(k1_jordan8(C,B),A,np__0),C) ) ) ) ) ).
fof(t21_jordan8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& 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(A,k3_finseq_1(k1_jordan8(C,B)))
=> r1_xboole_0(k3_goboard5(k1_jordan8(C,B),np__0,A),C) ) ) ) ) ).
fof(dt_k1_jordan8,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(B,k5_numbers) )
=> ( v1_matrix_1(k1_jordan8(A,B))
& m2_finseq_1(k1_jordan8(A,B),k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) ) ) ).
%------------------------------------------------------------------------------