:: The Geometric Interior in Real Linear Spaces :: by Karol P\c{a}k :: :: Received February 9, 2010 :: Copyright (c) 2010-2018 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 ARYTM_1, ARYTM_3, XBOOLE_0, CARD_1, CONVEX1, CONVEX2, CONVEX3, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSET_1, FUNCOP_1, FUNCT_1, FUNCT_2, FUNCT_4, MEMBERED, ORDERS_1, RELAT_1, RLVECT_1, RLVECT_2, RUSUB_4, SEMI_AF1, SETFAM_1, TARSKI, TOPS_1, RLAFFIN1, RLAFFIN2, ZFMISC_1, REAL_1, CARD_3, XXREAL_0, NAT_1, SUBSET_1, NUMBERS, ORDINAL1, STRUCT_0, SUPINF_2, ORDINAL4, VALUED_1, XREAL_0, PARTFUN1; notations TARSKI, XBOOLE_0, SUBSET_1, ZFMISC_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, ORDERS_1, CARD_1, XREAL_0, REAL_1, FINSET_1, SETFAM_1, DOMAIN_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, RELSET_1, FINSEQ_1, STRUCT_0, FINSEQ_2, FINSEQOP, RVSUM_1, RLVECT_1, RLVECT_2, RUSUB_4, CONVEX1, CONVEX2, CONVEX3, RLAFFIN1, FUNCT_4, FUNCOP_1, MEMBERED, XXREAL_2; constructors BINOP_2, CONVEX1, CONVEX3, DOMAIN_1, FINSEQOP, REAL_1, RVSUM_1, RUSUB_5, SETFAM_1, RLAFFIN1, SIMPLEX0, XXREAL_2, FUNCT_4, RELSET_1; registrations CARD_1, FINSET_1, FINSEQ_2, FUNCT_2, MEMBERED, NAT_1, NUMBERS, RELAT_1, RLVECT_1, STRUCT_0, SUBSET_1, VALUED_0, XCMPLX_0, XREAL_0, XXREAL_0, RLAFFIN1, FUNCOP_1, SETFAM_1, XXREAL_2, ABIAN, RELSET_1, RLVECT_2, ORDINAL1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Preliminaries reserve x,y for set, r,s for Real, n for Nat, V for RealLinearSpace, v,u,w,p for VECTOR of V, A,B for Subset of V, Af for finite Subset of V, I for affinely-independent Subset of V, If for finite affinely-independent Subset of V, F for Subset-Family of V, L1,L2 for Linear_Combination of V; theorem :: RLAFFIN2:1 for L be Linear_Combination of A st L is convex & v <> Sum L & L.v <> 0 ex p st p in conv(A\{v}) & Sum L = L.v*v + (1-L.v)*p & 1/L.v*(Sum L) + (1-1/L.v)*p = v; theorem :: RLAFFIN2:2 for p1,p2,w1,w2 be Element of V st v in conv I & u in conv I & not u in conv(I\{p1}) & not u in conv(I\{p2}) & w1 in conv(I\{p1}) & w2 in conv(I\{p2}) & r*u+(1-r)*w1 = v & s*u + (1-s)*w2 = v & r < 1 & s < 1 holds w1 = w2 & r = s; theorem :: RLAFFIN2:3 for L be Linear_Combination of Af st Af c=conv If & sum L=1 holds Sum L in Affin If & for x be Element of V ex F be FinSequence of REAL,G be FinSequence of V st (Sum L|--If).x = Sum F & len G = len F & G is one-to-one & rng G = Carrier L & for n st n in dom F holds F.n = L.(G.n)*(G.n|--If).x; theorem :: RLAFFIN2:4 for Aff be Subset of V st Aff is Affine & conv A /\ conv B c= Aff & conv(A\{v}) c= Aff & not v in Aff holds conv (A\{v}) /\ conv B = conv A /\ conv B; begin :: The Geometric Interior definition let V be non empty RLSStruct; let A be Subset of V; func Int A -> Subset of V means :: RLAFFIN2:def 1 x in it iff x in conv A & not ex B be Subset of V st B c< A & x in conv B; end; registration let V be non empty RLSStruct; let A be empty Subset of V; cluster Int A -> empty; end; theorem :: RLAFFIN2:5 for V be non empty RLSStruct for A be Subset of V holds Int A c= conv A; theorem :: RLAFFIN2:6 for V be vector-distributive scalar-distributive scalar-associative scalar-unitalnon empty RLSStruct for A be Subset of V holds Int A = A iff A is trivial; theorem :: RLAFFIN2:7 A c< B implies conv A misses Int B; theorem :: RLAFFIN2:8 conv A = union {Int B : B c= A}; theorem :: RLAFFIN2:9 conv A = Int A \/ union{conv (A\{v}) : v in A}; theorem :: RLAFFIN2:10 x in Int A implies ex L be Linear_Combination of A st L is convex & x = Sum L ; theorem :: RLAFFIN2:11 for L be Linear_Combination of A st L is convex & Sum L in Int A holds Carrier L = A; theorem :: RLAFFIN2:12 for L be Linear_Combination of I st L is convex & Carrier L = I holds Sum L in Int I; theorem :: RLAFFIN2:13 Int A is non empty implies A is finite; theorem :: RLAFFIN2:14 v in I & u in Int I & p in conv(I\{v}) & r*v + (1-r)*p = u implies p in Int (I\{v}); begin :: The Center of Mass definition let V; func center_of_mass V -> Function of BOOL the carrier of V,the carrier of V means :: RLAFFIN2:def 2 (for A be non empty finite Subset of V holds it.A = 1/card A * Sum(A)) & for A st A is infinite holds it.A = 0.V; end; theorem :: RLAFFIN2:15 ex L be Linear_Combination of Af st Sum L = r*Sum Af & sum L = r * card Af & L = (ZeroLC V) +* (Af-->r); theorem :: RLAFFIN2:16 Af is non empty implies (center_of_mass V).Af in conv Af; theorem :: RLAFFIN2:17 union F is finite implies (center_of_mass V).:F c= conv union F; theorem :: RLAFFIN2:18 v in If implies ((center_of_mass V).If |-- If).v = 1/card If; theorem :: RLAFFIN2:19 (center_of_mass V).If in If iff card If=1; theorem :: RLAFFIN2:20 If is non empty implies (center_of_mass V).If in Int If; theorem :: RLAFFIN2:21 A c= If & (center_of_mass V).If in Affin A implies If = A; theorem :: RLAFFIN2:22 v in Af & Af\{v} is non empty implies (center_of_mass V).Af = (1-1/card Af) * (center_of_mass V)/.(Af\{v}) + 1/card Af*v; theorem :: RLAFFIN2:23 conv A c=conv If & If is non empty & conv A misses Int If implies ex B be Subset of V st B c< If & conv A c= conv B; theorem :: RLAFFIN2:24 Sum L1 <> Sum L2 & sum L1 = sum L2 implies ex v st L1.v > L2.v; theorem :: RLAFFIN2:25 for p be Real st (r*L1+(1-r)*L2).v <= p & p <= (s*L1+(1-s)*L2).v ex rs be Real st (rs*L1+(1-rs)*L2).v = p & (r <= s implies r <= rs & rs <= s) & (s <= r implies s <= rs & rs <= r); theorem :: RLAFFIN2:26 v in conv A & u in conv A & v <> u implies ex p,w,r st p in A & w in conv(A\{p}) & 0<=r & r<1 & r*u+(1-r)*w = v; theorem :: RLAFFIN2:27 A\/{v} is affinely-independent iff A is affinely-independent & (v in A or not v in Affin A); theorem :: RLAFFIN2:28 Af c= I & v in Af implies I\{v}\/{(center_of_mass V).Af} is affinely-independent Subset of V; theorem :: RLAFFIN2:29 for F be c=-linear Subset-Family of V st union F is finite affinely-independent holds (center_of_mass V).:F is affinely-independent Subset of V; theorem :: RLAFFIN2:30 for F be c=-linear Subset-Family of V st union F is affinely-independent finite holds Int ((center_of_mass V).:F) c= Int union F;