SET007 Axioms: SET007+724.ax
%------------------------------------------------------------------------------
% File : SET007+724 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introducing Spans
% 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 : jordan13 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 2 ( 0 unt; 0 def)
% Number of atoms : 46 ( 3 equ)
% Maximal formula atoms : 31 ( 23 avg)
% Number of connectives : 55 ( 11 ~; 1 |; 32 &)
% ( 1 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 16 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-4 aty)
% Number of functors : 18 ( 18 usr; 4 con; 0-4 aty)
% Number of variables : 6 ( 6 !; 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(d1_jordan13,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_topreal2(A)
& ~ 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_jordan1h(A,B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_sprect_2(C)
& ~ v5_seqm_3(C)
& v1_topreal1(C)
& v2_topreal1(C)
& v1_finseq_6(C,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(C)
& v2_goboard5(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( C = k1_jordan13(A,B)
<=> ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B))
& 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)),k1_jordan8(A,B),k3_jordan1h(A,B),k3_jordan11(A,B))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),k3_jordan11(A,B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__2),k3_finseq_1(C)) )
=> ( ( ( r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
& r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A) )
=> r2_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) )
& ( r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
=> ( r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A)
| r3_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) )
& ( ~ r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
=> r1_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_jordan13,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_topreal2(A)
& ~ 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_xboole_0(k1_jordan13(A,B))
& v1_sprect_2(k1_jordan13(A,B))
& ~ v5_seqm_3(k1_jordan13(A,B))
& v1_topreal1(k1_jordan13(A,B))
& v2_topreal1(k1_jordan13(A,B))
& v1_finseq_6(k1_jordan13(A,B),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k1_jordan13(A,B))
& v2_goboard5(k1_jordan13(A,B))
& m2_finseq_1(k1_jordan13(A,B),u1_struct_0(k15_euclid(np__2))) ) ) ).
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