SET007 Axioms: SET007+781.ax
%------------------------------------------------------------------------------
% File : SET007+781 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Convex Hull, Set of Convex Combinations and Convex Cone
% 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 : convex3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 18 ( 2 unt; 0 def)
% Number of atoms : 186 ( 11 equ)
% Maximal formula atoms : 20 ( 10 avg)
% Number of connectives : 201 ( 33 ~; 2 |; 110 &)
% ( 9 <=>; 47 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 17 ( 17 usr; 3 con; 0-4 aty)
% Number of variables : 53 ( 46 !; 7 ?)
% 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(rc1_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_convex3(B,A) ) ) ).
fof(rc2_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v1_convex3(B,A) ) ) ).
fof(rc3_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_convex3(B,A) ) ) ).
fof(d1_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( B = k1_convex3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v2_convex1(C,A)
& m1_rlvect_2(C,A) ) ) ) ) ).
fof(d2_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( C = k2_convex3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( v2_convex1(D,A)
& m2_rlvect_2(D,A,B) ) ) ) ) ) ).
fof(t1_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( v2_convex1(C,A)
& m1_rlvect_2(C,A)
& k13_rlvect_2(A,C) = B
& ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(B,D)
=> ( v2_convex1(C,A)
& m2_rlvect_2(C,A,D) ) ) ) ) ) ) ).
fof(t2_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ! [D] :
( ( v2_convex1(D,A)
& m1_rlvect_2(D,A) )
=> ? [E] :
( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(k8_rlvect_2(A,B,C),E)
& ~ ( v2_convex1(D,A)
& m2_rlvect_2(D,A,E) ) ) ) ) ) ) ) ).
fof(t3_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( B != C
& B != D
& C != D
& ! [E] :
( ( v2_convex1(E,A)
& m1_rlvect_2(E,A) )
=> ? [F] :
( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(k9_rlvect_2(A,B,C,D),F)
& ~ ( v2_convex1(E,A)
& m2_rlvect_2(E,A,F) ) ) ) ) ) ) ) ) ).
fof(d3_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_convex3(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> ( r1_xreal_0(C,np__0)
| r2_hidden(k3_rlvect_1(A,D,C),B) ) ) ) ) ) ) ) ).
fof(t6_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k1_xboole_0
=> v1_convex3(B,A) ) ) ) ).
fof(t7_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( v1_convex3(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v7_rlvect_1(A)
=> ( v1_convex1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_rlvect_1(A,C,D),B) ) ) ) ) ) ) ) ).
fof(t8_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_convex1(B,A)
& v1_convex3(B,A) )
<=> ! [C] :
( m2_rlvect_2(C,A,B)
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,k10_rlvect_2(A,C))
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,C,D),np__0) ) )
=> ( k10_rlvect_2(A,C) = k1_xboole_0
| r2_hidden(k13_rlvect_2(A,C),B) ) ) ) ) ) ) ).
fof(t9_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_convex3(B,A)
& v1_convex3(C,A) )
=> v1_convex3(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(dt_k1_convex3,axiom,
$true ).
fof(dt_k2_convex3,axiom,
$true ).
fof(t4_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_convex1(B,A)
<=> r1_tarski(a_2_0_convex3(A,B),B) ) ) ) ).
fof(t5_convex3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k3_convex1(A,B) = a_2_0_convex3(A,B) ) ) ).
fof(fraenkel_a_2_0_convex3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_convex3(B,C))
<=> ? [D] :
( v2_convex1(D,B)
& m2_rlvect_2(D,B,C)
& A = k13_rlvect_2(B,D)
& r2_hidden(D,k1_convex3(B)) ) ) ) ).
%------------------------------------------------------------------------------