SET007 Axioms: SET007+722.ax
%------------------------------------------------------------------------------
% File : SET007+722 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Preparing the Internal Approximations of Simple Closed Curves
% 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 : jordan11 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 17 ( 0 unt; 0 def)
% Number of atoms : 96 ( 4 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 84 ( 5 ~; 1 |; 28 &)
% ( 3 <=>; 47 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 4 con; 0-3 aty)
% Number of variables : 35 ( 35 !; 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_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k1_jordan11(A)
<=> ( r1_jordan1h(A,B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_jordan1h(A,C)
=> r1_xreal_0(B,C) ) ) ) ) ) ) ).
fof(t1_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_xreal_0(np__1,k1_jordan11(A)) ) ).
fof(d2_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k2_jordan11(A)
<=> ( ~ r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),B)
& r1_tarski(k3_goboard5(k1_jordan8(A,k1_jordan11(A)),k5_binarith(k3_jordan1h(A,k1_jordan11(A)),np__1),B),k1_jordan2c(np__2,A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_tarski(k3_goboard5(k1_jordan8(A,k1_jordan11(A)),k5_binarith(k3_jordan1h(A,k1_jordan11(A)),np__1),C),k1_jordan2c(np__2,A))
=> ( r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),C)
| r1_xreal_0(B,C) ) ) ) ) ) ) ) ).
fof(t2_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r1_xreal_0(k2_jordan11(A),np__1) ) ).
fof(t3_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),k1_nat_1(k2_jordan11(A),np__1)) ) ).
fof(d3_jordan11,axiom,
! [A] :
( ( v1_topreal2(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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k3_jordan11(A,B)
<=> ( r1_xreal_0(C,k1_matrix_1(k1_jordan8(A,B)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,D)
& r1_xreal_0(D,k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(B,k1_jordan11(A))),k5_binarith(k2_jordan11(A),np__2)),np__2)) )
=> r1_tarski(k3_goboard5(k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),D),k1_jordan2c(np__2,A)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(D,k1_matrix_1(k1_jordan8(A,B)))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(D,E)
& r1_xreal_0(E,k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(B,k1_jordan11(A))),k5_binarith(k2_jordan11(A),np__2)),np__2)) )
=> r1_tarski(k3_goboard5(k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),E),k1_jordan2c(np__2,A)) ) ) )
=> r1_xreal_0(C,D) ) ) ) ) ) ) ) ) ).
fof(t4_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> k3_jordan1h(B,A) = k3_real_1(k4_real_1(k1_card_4(np__2,k5_binarith(A,k1_jordan11(B))),k5_real_1(k3_jordan1h(B,k1_jordan11(B)),np__2)),np__2) ) ) ) ).
fof(t5_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> r1_xreal_0(k3_jordan11(B,A),k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(A,k1_jordan11(B))),k5_binarith(k2_jordan11(B),np__2)),np__2)) ) ) ) ).
fof(t6_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> r1_tarski(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),k1_jordan2c(np__2,B)) ) ) ) ).
fof(t7_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> ( ~ r1_xreal_0(k3_jordan11(B,A),np__1)
& r1_xreal_0(k3_jordan11(B,A),k1_matrix_1(k1_jordan8(B,A))) ) ) ) ) ).
fof(t8_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> r2_hidden(k4_tarski(k3_jordan1h(B,A),k3_jordan11(B,A)),k2_matrix_1(k1_jordan8(B,A))) ) ) ) ).
fof(t9_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> r2_hidden(k4_tarski(k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),k2_matrix_1(k1_jordan8(B,A))) ) ) ) ).
fof(t10_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( r1_jordan1h(B,A)
& r1_xboole_0(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k5_binarith(k3_jordan11(B,A),np__1)),B) ) ) ) ).
fof(t11_jordan11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_topreal2(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_jordan1h(B,A)
=> r1_xboole_0(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),B) ) ) ) ).
fof(dt_k1_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m2_subset_1(k1_jordan11(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_jordan11,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> m2_subset_1(k2_jordan11(A),k1_numbers,k5_numbers) ) ).
fof(dt_k3_jordan11,axiom,
! [A,B] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k3_jordan11(A,B),k1_numbers,k5_numbers) ) ).
%------------------------------------------------------------------------------