:: Some Properties for Convex Combinations
:: by Noboru Endou , Yasumasa Suzuki and Yasunari Shidama
::
:: Received April 3, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, REAL_1, FINSEQ_1, XBOOLE_0, RLVECT_1, CONVEX1, SUBSET_1,
BHSP_1, STRUCT_0, RELAT_1, FUNCT_1, PROB_2, XXREAL_0, CARD_1, ARYTM_3,
ARYTM_1, RLVECT_2, TARSKI, CARD_3, NAT_1, VALUED_1, JORDAN3, PARTFUN1,
ORDINAL4, FUNCT_2, RLSUB_1, RUSUB_4, CLASSES1, FINSET_1, CONVEX2,
FUNCT_7, XCMPLX_0;
notations TARSKI, XBOOLE_0, SUBSET_1, CARD_1, ORDINAL1, NUMBERS, XCMPLX_0,
XXREAL_0, XREAL_0, REAL_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, FINSET_1,
NAT_1, FINSEQ_1, DOMAIN_1, STRUCT_0, ALGSTR_0, RLVECT_1, RLSUB_1,
RLSUB_2, FINSEQ_4, RLVECT_2, RVSUM_1, BHSP_1, RUSUB_4, CONVEX1, FINSEQ_6,
CLASSES1;
constructors WELLORD2, DOMAIN_1, REAL_1, BINOP_2, FINSEQ_4, FINSOP_1, RLSUB_2,
RUSUB_4, CONVEX1, MATRIX_1, FINSEQ_6, RVSUM_1, CLASSES1, RELSET_1,
NUMBERS;
registrations XBOOLE_0, NUMBERS, XXREAL_0, NAT_1, FINSEQ_1, STRUCT_0,
RLVECT_1, RUSUB_4, CONVEX1, VALUED_0, RELSET_1, RLVECT_2, XREAL_0,
ORDINAL1, RVSUM_1, CARD_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
begin
theorem :: CONVEX2:1
for V being non empty RLSStruct, M,N being convex Subset of V holds M
/\ N is convex;
theorem :: CONVEX2:2
for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M
= {u where u is VECTOR of V : for i being Element of NAT st i in (dom F /\ dom
B) holds ex v being VECTOR of V st v = F.i & u .|. v <= B.i} holds M is convex;
theorem :: CONVEX2:3
for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M
= {u where u is VECTOR of V : for i being Element of NAT st i in (dom F /\ dom
B) holds ex v being VECTOR of V st v = F.i & u .|. v < B.i} holds M is convex
;
theorem :: CONVEX2:4
for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M
= {u where u is VECTOR of V : for i being Element of NAT st i in (dom F /\ dom
B) holds ex v being VECTOR of V st v = F.i & u .|. v >= B.i} holds M is convex;
theorem :: CONVEX2:5
for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M
= {u where u is VECTOR of V : for i being Element of NAT st i in (dom F /\ dom
B) holds ex v being VECTOR of V st v = F.i & u .|. v > B.i} holds M is convex
;
theorem :: CONVEX2:6
for V being RealLinearSpace, M being Subset of V holds (for N being
Subset of V, L being Linear_Combination of N st L is convex & N c= M holds Sum(
L) in M) iff M is convex;
definition
let V be RealLinearSpace, M be Subset of V;
func LinComb(M) -> set means
:: CONVEX2:def 1
for L being object holds L in it iff L is Linear_Combination of M;
end;
registration
let V be RealLinearSpace;
cluster convex for Linear_Combination of V;
end;
definition
let V be RealLinearSpace;
mode Convex_Combination of V is convex Linear_Combination of V;
end;
registration
let V be RealLinearSpace, M be non empty Subset of V;
cluster convex for Linear_Combination of M;
end;
definition
let V be RealLinearSpace, M be non empty Subset of V;
mode Convex_Combination of M is convex Linear_Combination of M;
end;
theorem :: CONVEX2:7
for V being RealLinearSpace, M be Subset of V holds Convex-Family M <> {};
theorem :: CONVEX2:8
for V being RealLinearSpace, L1,L2 being Convex_Combination of V,
r being Real st 0 < r & r < 1 holds r*L1 + (1-r)*L2 is Convex_Combination of V;
theorem :: CONVEX2:9
for V being RealLinearSpace, M being non empty Subset of V, L1,L2
being Convex_Combination of M, r being Real st 0 < r & r < 1 holds r*L1 + (1-r)
*L2 is Convex_Combination of M;
begin :: Miscellaneous
theorem :: CONVEX2:10
for V being RealLinearSpace, W1,W2 being Subspace of V holds Up(W1+W2)
= Up(W1) + Up(W2);
theorem :: CONVEX2:11
for V being RealLinearSpace, W1,W2 being Subspace of V holds Up(W1 /\
W2) = Up(W1) /\ Up(W2);
theorem :: CONVEX2:12
for V being RealLinearSpace, L1, L2 being Convex_Combination of V, a,b
being Real st a * b > 0 holds Carrier(a*L1 + b*L2) = Carrier(a * L1) \/ Carrier
(b * L2);
theorem :: CONVEX2:13
for F,G being Function st F,G are_fiberwise_equipotent for x1,x2 being
set st x1 in dom F & x2 in dom F & x1 <> x2 holds ex z1,z2 being set st z1 in
dom G & z2 in dom G & z1 <> z2 & F.x1 = G.z1 & F.x2 = G.z2;
theorem :: CONVEX2:14
for V being RealLinearSpace, L being Linear_Combination of V, A being
Subset of V st A c= Carrier(L) holds ex L1 being Linear_Combination of V st
Carrier(L1) = A;