Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Definition of Convex Function and Jensen's Inequality

**by****Grigory E. Ivanov**- Received July 17, 2003
- MML identifier: CONVFUN1

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, SUPINF_1, FUNCT_1, FUNCT_2, ARYTM_3, MEASURE6, JORDAN1, BINOP_1, VECTSP_1, ARYTM_1, RELAT_1, PARTFUN1, RFUNCT_3, FINSEQ_1, FINSEQ_4, BOOLE, FINSET_1, CARD_1, SGRAPH1, RFINSEQ, SQUARE_1, CONVFUN1; notation TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, NUMBERS, XREAL_0, FUNCT_1, REAL_1, FUNCT_2, FINSEQ_1, RLVECT_1, CONVEX1, SUPINF_1, RELSET_1, ZFMISC_1, MEASURE6, BINOP_1, SEQ_1, DOMAIN_1, VECTSP_1, EXTREAL1, SUPINF_2, PARTFUN1, RFUNCT_3, MESFUNC1, NAT_1, RVSUM_1, FINSEQ_4, TOPREAL1, FINSET_1, CARD_1, RFINSEQ, BINARITH; constructors REAL_1, MEASURE6, CONVEX1, DOMAIN_1, VECTSP_1, EXTREAL1, SUPINF_2, RFUNCT_3, MESFUNC1, TOPREAL1, FINSEQ_4, BINARITH, MEMBERED, INT_1; clusters STRUCT_0, RLVECT_1, FUNCT_2, SUPINF_1, XREAL_0, RELSET_1, SUBSET_1, BINARITH, FINSET_1, FINSEQ_1, INT_1, FUNCT_1, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Product of Two Real Linear Spaces definition let X,Y be non empty RLSStruct; func Add_in_Prod_of_RLS(X,Y) -> BinOp of [:the carrier of X, the carrier of Y:] means :: CONVFUN1:def 1 for z1, z2 being Element of [:the carrier of X, the carrier of Y:], x1, x2 being VECTOR of X, y1, y2 being VECTOR of Y st z1 = [x1,y1] & z2 = [x2,y2] holds it.(z1,z2) = [(the add of X).[x1,x2],(the add of Y).[y1,y2]]; end; definition let X,Y be non empty RLSStruct; func Mult_in_Prod_of_RLS(X,Y) -> Function of [:REAL, [:the carrier of X, the carrier of Y:]:], [:the carrier of X, the carrier of Y:] means :: CONVFUN1:def 2 for a being Real, z being Element of [:the carrier of X, the carrier of Y:], x being VECTOR of X, y being VECTOR of Y st z = [x,y] holds it.[a,z] = [(the Mult of X).[a,x],(the Mult of Y).[a,y]]; end; definition let X,Y be non empty RLSStruct; func Prod_of_RLS(X,Y) -> RLSStruct equals :: CONVFUN1:def 3 RLSStruct(# [:the carrier of X, the carrier of Y:], [0.X,0.Y], Add_in_Prod_of_RLS(X,Y), Mult_in_Prod_of_RLS(X,Y) #); end; definition let X,Y be non empty RLSStruct; cluster Prod_of_RLS(X,Y) -> non empty; end; theorem :: CONVFUN1:1 for X,Y being non empty RLSStruct, x being VECTOR of X, y being VECTOR of Y, u being VECTOR of Prod_of_RLS(X,Y), p being Real st u = [x,y] holds p*u = [p*x,p*y]; theorem :: CONVFUN1:2 for X,Y being non empty RLSStruct, x1, x2 being VECTOR of X, y1, y2 being VECTOR of Y, u1, u2 being VECTOR of Prod_of_RLS(X,Y) st u1 = [x1,y1] & u2 = [x2,y2] holds u1+u2 = [x1+x2,y1+y2]; definition let X,Y be Abelian (non empty RLSStruct); cluster Prod_of_RLS(X,Y) -> Abelian; end; definition let X,Y be add-associative (non empty RLSStruct); cluster Prod_of_RLS(X,Y) -> add-associative; end; definition let X,Y be right_zeroed (non empty RLSStruct); cluster Prod_of_RLS(X,Y) -> right_zeroed; end; definition let X,Y be right_complementable (non empty RLSStruct); cluster Prod_of_RLS(X,Y) -> right_complementable; end; definition let X,Y be RealLinearSpace-like (non empty RLSStruct); cluster Prod_of_RLS(X,Y) -> RealLinearSpace-like; end; theorem :: CONVFUN1:3 for X,Y being RealLinearSpace, n being Nat, x being FinSequence of the carrier of X, y being FinSequence of the carrier of Y, z being FinSequence of the carrier of Prod_of_RLS(X,Y) st len x = n & len y = n & len z = n & (for i being Nat st i in Seg n holds z.i = [x.i,y.i]) holds Sum z = [Sum x, Sum y]; begin :: Real Linear Space of Real Numbers definition func RLS_Real -> non empty RLSStruct equals :: CONVFUN1:def 4 RLSStruct(# REAL,0,addreal,multreal #); end; definition cluster RLS_Real -> Abelian add-associative right_zeroed right_complementable RealLinearSpace-like; end; begin :: Sum of Finite Sequence of Extended Real Numbers definition let F be FinSequence of ExtREAL; func Sum(F) -> R_eal means :: CONVFUN1:def 5 ex f being Function of NAT, ExtREAL st it = f.(len F) & f.0 = 0. & for i being Nat st i < len F holds f.(i+1)=f.i+F.(i+1); end; theorem :: CONVFUN1:4 Sum(<*> ExtREAL) = 0.; theorem :: CONVFUN1:5 for a being R_eal holds Sum<*a*> = a; theorem :: CONVFUN1:6 for a,b being R_eal holds Sum<*a,b*> = a+b; theorem :: CONVFUN1:7 for F,G being FinSequence of ExtREAL st not -infty in rng F & not -infty in rng G holds Sum(F^G) = Sum(F)+Sum(G); theorem :: CONVFUN1:8 for F,G being FinSequence of ExtREAL, s being Permutation of dom F st G = F*s & not -infty in rng F holds Sum(F) = Sum(G); begin :: Definition of Convex Function definition let X be non empty RLSStruct, f be Function of the carrier of X,ExtREAL; func epigraph f -> Subset of Prod_of_RLS(X,RLS_Real) equals :: CONVFUN1:def 6 {[x,y] where x is Element of X, y is Element of REAL: f.x <=' R_EAL(y)}; end; definition let X be non empty RLSStruct, f be Function of the carrier of X,ExtREAL; attr f is convex means :: CONVFUN1:def 7 epigraph f is convex; end; theorem :: CONVFUN1:9 for X being non empty RLSStruct, f being Function of the carrier of X,ExtREAL st (for x being VECTOR of X holds f.x <> -infty) holds f is convex iff for x1, x2 being VECTOR of X, p being Real st 0<p & p<1 holds f.(p*x1+(1-p)*x2) <=' R_EAL(p)*f.x1+R_EAL(1-p)*f.x2; theorem :: CONVFUN1:10 for X being RealLinearSpace, f being Function of the carrier of X,ExtREAL st (for x being VECTOR of X holds f.x <> -infty) holds f is convex iff for x1, x2 being VECTOR of X, p being Real st 0<=p & p<=1 holds f.(p*x1+(1-p)*x2) <=' R_EAL(p)*f.x1+R_EAL(1-p)*f.x2; begin :: Relation between notions "function is convex" and "function is convex on set" theorem :: CONVFUN1:11 for f being PartFunc of REAL,REAL, g being Function of the carrier of RLS_Real,ExtREAL, X being Subset of RLS_Real st X c= dom f & for x being Real holds (x in X implies g.x=f.x) & (not x in X implies g.x=+infty) holds g is convex iff f is_convex_on X & X is convex; begin :: CONVEX2:6 in other words theorem :: CONVFUN1:12 for X being RealLinearSpace, M being Subset of X holds M is convex iff (for n being non empty Nat, p being FinSequence of REAL, y,z being FinSequence of the carrier of X st len p = n & len y = n & len z = n & Sum p = 1 & (for i being Nat st i in Seg n holds p.i>0 & z.i=p.i*y/.i & y/.i in M) holds Sum(z) in M); begin theorem :: CONVFUN1:13 for X being RealLinearSpace, f being Function of the carrier of X,ExtREAL st (for x being VECTOR of X holds f.x <> -infty) holds f is convex iff (for n being non empty Nat, p being FinSequence of REAL, F being FinSequence of ExtREAL, y,z being FinSequence of the carrier of X st len p = n & len F = n & len y = n & len z = n & Sum p = 1 & (for i being Nat st i in Seg n holds p.i>0 & z.i=p.i*y/.i & F.i = R_EAL(p.i)*f.(y/.i)) holds f.Sum(z) <=' Sum F); theorem :: CONVFUN1:14 for X being RealLinearSpace, f being Function of the carrier of X,ExtREAL st (for x being VECTOR of X holds f.x <> -infty) holds f is convex iff (for n being non empty Nat, p being FinSequence of REAL, F being FinSequence of ExtREAL, y,z being FinSequence of the carrier of X st len p = n & len F = n & len y = n & len z = n & Sum p = 1 & (for i being Nat st i in Seg n holds p.i>=0 & z.i=p.i*y/.i & F.i = R_EAL(p.i)*f.(y/.i)) holds f.Sum(z) <=' Sum F);

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