SET007 Axioms: SET007+467.ax
%------------------------------------------------------------------------------
% File : SET007+467 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Adjacency Concept for Pairs of Natural Numbers
% 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 : gobrd10 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 16 ( 1 unt; 0 def)
% Number of atoms : 157 ( 26 equ)
% Maximal formula atoms : 30 ( 9 avg)
% Number of connectives : 153 ( 12 ~; 5 |; 54 &)
% ( 4 <=>; 78 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 12 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-4 aty)
% Number of functors : 14 ( 14 usr; 3 con; 0-4 aty)
% Number of variables : 69 ( 69 !; 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_gobrd10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_gobrd10(A,B)
<=> ( B = k1_nat_1(A,np__1)
| A = k1_nat_1(B,np__1) ) ) ) ) ).
fof(t1_gobrd10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_gobrd10(A,B)
=> r1_gobrd10(k1_nat_1(A,np__1),k1_nat_1(B,np__1)) ) ) ) ).
fof(t2_gobrd10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_gobrd10(A,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(np__1,B) )
=> r1_gobrd10(k5_binarith(A,np__1),k5_binarith(B,np__1)) ) ) ) ).
fof(d2_gobrd10,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)
=> ( r2_gobrd10(A,B,C,D)
<=> ( ( r1_gobrd10(A,C)
& B = D )
| ( A = C
& r1_gobrd10(B,D) ) ) ) ) ) ) ) ).
fof(t3_gobrd10,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)
=> ( r2_gobrd10(A,C,B,D)
=> r2_gobrd10(k1_nat_1(A,np__1),k1_nat_1(C,np__1),k1_nat_1(B,np__1),k1_nat_1(D,np__1)) ) ) ) ) ) ).
fof(t4_gobrd10,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)
=> ( ( r2_gobrd10(A,C,B,D)
& r1_xreal_0(np__1,A)
& r1_xreal_0(np__1,B)
& r1_xreal_0(np__1,C)
& r1_xreal_0(np__1,D) )
=> r2_gobrd10(k5_binarith(A,np__1),k5_binarith(C,np__1),k5_binarith(B,np__1),k5_binarith(D,np__1)) ) ) ) ) ) ).
fof(d3_gobrd10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( C = k1_gobrd10(A,B)
<=> ( k3_finseq_1(C) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,A) )
=> k1_funct_1(C,D) = B ) ) ) ) ) ) ) ).
fof(t5_gobrd10,axiom,
$true ).
fof(t6_gobrd10,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)
=> ~ ( r1_xreal_0(B,A)
& r1_xreal_0(C,A)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( k1_funct_1(D,np__1) = B
& k1_funct_1(D,k3_finseq_1(D)) = C
& k3_finseq_1(D) = k1_nat_1(k1_nat_1(k5_binarith(B,C),k5_binarith(C,B)),np__1)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,k3_finseq_1(D))
& F = k1_funct_1(D,E) )
=> r1_xreal_0(F,A) ) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,E)
& ~ r1_xreal_0(k3_finseq_1(D),E)
& k1_funct_1(D,k1_nat_1(E,np__1)) != k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,D,E),np__1)
& k1_funct_1(D,E) != k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,D,k1_nat_1(E,np__1)),np__1) ) ) ) ) ) ) ) ) ).
fof(t7_gobrd10,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)
=> ~ ( r1_xreal_0(B,A)
& r1_xreal_0(C,A)
& ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( k1_funct_1(D,np__1) = B
& k1_funct_1(D,k3_finseq_1(D)) = C
& k3_finseq_1(D) = k1_nat_1(k1_nat_1(k5_binarith(B,C),k5_binarith(C,B)),np__1)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,k3_finseq_1(D))
& F = k1_funct_1(D,E) )
=> r1_xreal_0(F,A) ) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(k3_finseq_1(D),E)
| r1_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,D,E),k4_finseq_4(k5_numbers,k5_numbers,D,k1_nat_1(E,np__1))) ) ) ) ) ) ) ) ) ) ).
fof(t8_gobrd10,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] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,A)
& r1_xreal_0(D,B)
& r1_xreal_0(E,A)
& r1_xreal_0(F,B)
& ! [G] :
( m2_finseq_1(G,k5_numbers)
=> ! [H] :
( m2_finseq_1(H,k5_numbers)
=> ~ ( ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ! [J] :
( m2_subset_1(J,k1_numbers,k5_numbers)
=> ! [K] :
( m2_subset_1(K,k1_numbers,k5_numbers)
=> ( ( r2_hidden(I,k4_finseq_1(G))
& J = k1_funct_1(G,I)
& K = k1_funct_1(H,I) )
=> ( r1_xreal_0(J,A)
& r1_xreal_0(K,B) ) ) ) ) )
& k1_funct_1(G,np__1) = C
& k1_funct_1(G,k3_finseq_1(G)) = E
& k1_funct_1(H,np__1) = D
& k1_funct_1(H,k3_finseq_1(H)) = F
& k3_finseq_1(G) = k3_finseq_1(H)
& k3_finseq_1(G) = k1_nat_1(k1_nat_1(k1_nat_1(k1_nat_1(k5_binarith(C,E),k5_binarith(E,C)),k5_binarith(D,F)),k5_binarith(F,D)),np__1)
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,I)
=> ( r1_xreal_0(k3_finseq_1(G),I)
| r2_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,G,I),k4_finseq_4(k5_numbers,k5_numbers,H,I),k4_finseq_4(k5_numbers,k5_numbers,G,k1_nat_1(I,np__1)),k4_finseq_4(k5_numbers,k5_numbers,H,k1_nat_1(I,np__1))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_gobrd10,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C,D] :
( m1_subset_1(D,k1_zfmisc_1(C))
=> ! [E] :
( m1_matrix_1(E,k1_zfmisc_1(C),A,B)
=> ( ! [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)
=> ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(A))
& r2_hidden(H,k2_finseq_1(A))
& r2_hidden(G,k2_finseq_1(B))
& r2_hidden(I,k2_finseq_1(B))
& r2_gobrd10(F,G,H,I) )
=> ( r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D)
<=> r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,H,I),D) ) ) ) ) ) )
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k2_finseq_1(A))
& r2_hidden(G,k2_finseq_1(B))
& r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D) ) ) )
| ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(A))
& r2_hidden(G,k2_finseq_1(B)) )
=> r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D) ) ) ) ) ) ) ) ) ) ).
fof(symmetry_r1_gobrd10,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( r1_gobrd10(A,B)
=> r1_gobrd10(B,A) ) ) ).
fof(irreflexivity_r1_gobrd10,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ~ r1_gobrd10(A,A) ) ).
fof(dt_k1_gobrd10,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_finseq_1(k1_gobrd10(A,B),k5_numbers) ) ).
fof(redefinition_k1_gobrd10,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_gobrd10(A,B) = k2_finseq_2(A,B) ) ).
%------------------------------------------------------------------------------