let F be PartFunc of REAL,REAL; :: thesis: for X being set
for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X
let X be set ; :: thesis: for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X
let r be Real; :: thesis: ( F is_convex_on X implies max+ (F - r) is_convex_on X )
assume A1: F is_convex_on X ; :: thesis: max+ (F - r) is_convex_on X
then A2: X c= dom F ;
A3: ( dom F = dom (F - r) & dom (max+ (F - r)) = dom (F - r) ) by Def10, VALUED_1:3;
hence X c= dom (max+ (F - r)) by A1; :: according to RFUNCT_3:def 12 :: thesis: for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s))
let p be Real; :: thesis: ( 0 <= p & p <= 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s)) )
assume that
A4: 0 <= p and
A5: p <= 1 ; :: thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
(max+ (F - r)) . ((p * r) + ((1 - p) * s)) <= (p * ((max+ (F - r)) . r)) + ((1 - p) * ((max+ (F - r)) . s))
let x, y be Real; :: thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) )
assume that
A6: x in X and
A7: y in X and
A8: (p * x) + ((1 - p) * y) in X ; :: thesis: (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y))
F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, A4, A5, A6, A7, A8;
then (F . ((p * x) + ((1 - p) * y))) - r <= ((p * (F . x)) + ((1 - p) * (F . y))) - r by XREAL_1:9;
then A9: max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= max ((((p * (F . x)) + ((1 - p) * (F . y))) - r),0) by XXREAL_0:26;
0 + p <= 1 by A5;
then 0 <= 1 - p by XREAL_1:19;
then A10: max+ ((1 - p) * ((F - r) . y)) = (1 - p) * (max+ ((F - r) . y)) by Th4;
A11: max+ ((p * ((F - r) . x)) + ((1 - p) * ((F - r) . y))) <= (max+ (p * ((F - r) . x))) + (max+ ((1 - p) * ((F - r) . y))) by Th5;
A12: max+ (p * ((F - r) . x)) = p * (max+ ((F - r) . x)) by A4, Th4;
reconsider pc = (p * x) + ((1 - p) * y) as Element of REAL by XREAL_0:def 1;
reconsider x = x, y = y as Element of REAL by XREAL_0:def 1;
((p * (F . x)) + ((1 - p) * (F . y))) - r = (p * ((F . x) - r)) + ((1 - p) * ((F . y) - r))
.= (p * ((F - r) . x)) + ((1 - p) * ((F . y) - r)) by A6, A2, VALUED_1:3
.= (p * ((F - r) . x)) + ((1 - p) * ((F - r) . y)) by A7, A2, VALUED_1:3 ;
then max+ ((F . ((p * x) + ((1 - p) * y))) - r) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A9, A11, A12, A10, XXREAL_0:2;
then max+ ((F - r) . pc) <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A8, A2, VALUED_1:3;
then (max+ (F - r)) . pc <= (p * (max+ ((F - r) . x))) + ((1 - p) * (max+ ((F - r) . y))) by A3, A8, A2, Def10;
then (max+ (F - r)) . pc <= (p * ((max+ (F - r)) . x)) + ((1 - p) * (max+ ((F - r) . y))) by A3, A6, A2, Def10;
hence (max+ (F - r)) . ((p * x) + ((1 - p) * y)) <= (p * ((max+ (F - r)) . x)) + ((1 - p) * ((max+ (F - r)) . y)) by A3, A7, A2, Def10; :: thesis: verum