Quellcode-Bibliothek convex_functions.pvs Sprache: PVS

convex_functions: THEORY
%------------------------------------------------------------------------------
% In This Theory, We Define Convex And Stricly Convex Functions. We Prove
%    That The Maximum Of Two Convex Functions Is Convex And That
%    A Convex Function Has At Most One Minimum
%
%     Authors: Anthony Narkawicz,  NASA Langley
%
%     Version 1.0         9/23/2009  Initial Version
%     Version 1.1   10/21/2009
%------------------------------------------------------------------------------
BEGIN

   IMPORTING quadratic, real_fun_ops

   f,g: VAR [real -> real]
   A,B,C,v,w,x,y,z,t,b,c,d: VAR real
   k  : VAR nat
   a,H: VAR posreal
   aaa: VAR nnreal

   % Convex

   convex?(f): bool = FORALL (x,y,t): 0 <= t and t <= 1 IMPLIES
                f(t*x + (1-t)*y) <= t*f(x) + (1-t)*f(y)

   % Basic Properties

   composition_of_convex: LEMMA (FORALL (x,y): x<=y IMPLIES g(x)<=g(y))
          AND convex?(f) AND convex?(g) IMPLIES convex?(g o f)

   max_of_convex: LEMMA convex?(f) AND convex?(g) IMPLIES
       convex?((LAMBDA (z): max(f(z),g(z))))

   sum_of_convex: LEMMA convex?(f) AND convex?(g) IMPLIES convex?(f+g)

   scal_convex:   LEMMA convex?(f) IMPLIES convex?(aaa*f)

   convex_function_right_lt: LEMMA convex?(f) AND A<C AND C<B
     IMPLIES f(B) >= ((B-A)/(C-A))*f(C) - ((B-C)/(C-A))*f(A)

   convex_function_left_lt: LEMMA convex?(f) AND A<C AND C<B
     IMPLIES f(A) >= ((B-A)/(B-C))*f(C) - ((C-A)/(B-C))*f(B)

   % The Following Theorem Can Help In Proofs Where Convexity Is Used
   % By Allowing The User To Escape The Construction Of A Specific t-value

   between_point_is_wtd_av: LEMMA FORALL (x,y,z: real): x<=z AND z<=y
          IMPLIES (EXISTS (t: real): 0<=t AND t<=1
            AND z = t*x + (1-t)*y)

   % Sometimes, it may be convenient to have the t as the coeff of x

   between_point_is_wtd_av2: LEMMA FORALL (x,y,z: real): x<=z AND z<=y
          IMPLIES (EXISTS (t: real): 0<=t AND t<=1
            AND z = (1-t)*x + t*y)

   % The Minima Of Convex Functions Have Certain Fun(ction) properties

   convex_const_on_connected_min: LEMMA convex?(f) AND x < y AND
     (FORALL (z): x<=z AND z<=y IMPLIES f(x) = f(z))
  IMPLIES
  (FORALL (w): f(x) <= f(w))

   convex_min_is_connected: LEMMA convex?(f) AND x < y AND f(x) = f(y)
          AND (FORALL (z): f(x) <= f(z))
       IMPLIES
       (FORALL (w): x<=w AND w<=y IMPLIES f(w) = f(x))

     % Part B: Local Minima Of Convex Functions....

   loc_convex_min_is_connected: LEMMA convex?(f) AND A<=x AND x<y AND y<=B AND
     f(x) = f(y) AND
     (FORALL (w): x<=w AND w<=y IMPLIES f(x) <= f(w)) IMPLIES
  (FORALL (w): x<=w AND w<=y IMPLIES f(w) = f(x))

   % Helpful results about convex functions

   convex_btw_pt_left_lt: LEMMA convex?(f) AND A<x AND x<=B AND f(A)<=c
           AND f(B)<c IMPLIES f(x) < c

   convex_btw_pt_right_lt: LEMMA convex?(f) AND A<=x AND x<B AND f(A)<c
           AND f(B)<=c IMPLIES f(x) < c

   convex_wtd_av_lt: LEMMA convex?(f) AND f(x)<c AND f(y) < c
             IMPLIES (FORALL (t): 0 <= t AND t <= 1 IMPLIES
                 f(t*x + (1-t)*y) < c)

   % If a convex function has x as its min, then it is increasing to the
   % right of x and decreasing to the left of x

   convex_increasing: LEMMA convex?(f) AND (FORALL (y): f(y)>=f(x))
           AND v>=x AND w>=v IMPLIES
        f(w)>=f(v)

   convex_decreasing: LEMMA convex?(f) AND (FORALL (y): f(y)>=f(x))
           AND w<=v AND v<=x IMPLIES
        f(w)>=f(v)

   % STRICTLY CONVEX FUNCTIONS

   strictly_convex?(f): bool = FORALL (x,y,t): 0 < t and t < 1 and x /= y IMPLIES
                 f(t*x + (1-t)*y) < t*f(x) + (1-t)*f(y)

   strictly_convex_implies_convex: LEMMA strictly_convex?(f) IMPLIES convex?(f)

   strictly_convex_unique_min: LEMMA strictly_convex?(f) AND (FORALL (z):
             f(x) <= f(z) AND f(y) <= f(z)) IMPLIES
          x = y

   strictly_conv_uniq_intv_min: LEMMA strictly_convex?(f) AND
       (A <= x AND x <= B) AND
    (A <= y AND y <= B) AND
    (FORALL (z): (A <= z and z <= B) IMPLIES
     f(x) <= f(z) AND f(y) <= f(z))
    IMPLIES
    x = y

   % Basic Properties

   composition_of_strictly_convex: LEMMA (FORALL (x,y): x<=y IMPLIES g(x)<g(y))
          AND convex?(f) AND strictly_convex?(g)
       IMPLIES strictly_convex?(g o f)

   max_of_strictly_convex: LEMMA strictly_convex?(f) AND
         strictly_convex?(g) IMPLIES
    strictly_convex?((LAMBDA (z): max(f(z),g(z))))

   sum_of_strictly_convex: LEMMA strictly_convex?(f) AND strictly_convex?(g)
         IMPLIES strictly_convex?(f+g)

   scal_strictly_convex:  LEMMA strictly_convex?(f) IMPLIES strictly_convex?(a*f)

   % Helpful results about strictly convex functions

   strictly_convex_btw_pt_lt: LEMMA strictly_convex?(f) AND A<x AND x<B AND f(A)<=c
           AND f(B)<=c IMPLIES f(x) < c

   % Special Case: Quadratics

   quad_convex: LEMMA convex?(quadratic(aaa,b,c))

   quad_strictly_convex: LEMMA strictly_convex?(quadratic(a,b,c))

   abs_linear_convex: LEMMA convex?((LAMBDA (t): abs(x + t*y)-H))

   quad_linear_max_convex: LEMMA LET maxfun = (LAMBDA (t): max(quadratic(a,b,c)(t),
            abs(x + t*y)-H))
          IN convex?(maxfun)

   quad_linear_max_glob_min: LEMMA LET maxfun = (LAMBDA (t): max(quadratic(a,b,c)(t),
            abs(x + t*y)-H))
          IN y /= 0 AND maxfun(w) = 0 AND maxfun(v) = 0 AND
             (FORALL (z): 0 <= maxfun(z))
      IMPLIES
      v = w

   quad_linear_max_loc_min: LEMMA LET maxfun = (LAMBDA (t): max(quadratic(a,b,c)(t),
            abs(x + t*y)-H))
          IN y /= 0 AND
             (A <= w AND w <= B) AND (A <= v AND v <= B) AND
             maxfun(w) = 0 AND maxfun(v) = 0 AND
             (FORALL (z): (A <= z AND z <= B) IMPLIES 0 <= maxfun(z))
      IMPLIES
      v = w

   % Special Case: Exponentials Of Convex Functions

   power_of_convex: LEMMA convex?(f) AND (FORALL (x): f(x)>=0)
     IMPLIES convex?(LAMBDA (x): f(x)^k)

   power_of_strictly_convex: LEMMA strictly_convex?(f) AND (FORALL (x): f(x)>=0)
     IMPLIES strictly_convex?(LAMBDA (x): f(x)^(k+1))

END convex_functions

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