Quelle integral_pulse.pvs Sprache: PVS

integral_pulse[T: TYPE FROM real]: THEORY
%----------------------------------------------------------------------------
%
%  Integral of:
%
%                    |-------------------|    c
%                    |                   |
%                    |                   |
%     ---------------|                   |-----------------
%                   xl                   xh
%
%
%  Author:  Rick Butler               NASA Langley
%----------------------------------------------------------------------------

BEGIN

   ASSUMING
      IMPORTING deriv_domain_def[T]

      connected_domain : ASSUMPTION connected?[T]

      not_one_element : ASSUMPTION not_one_element?[T]

   ENDASSUMING

   IMPORTING integral_prep[T], reals@sigma_upto

   a,b,x,y,z: VAR T
   c:         VAR real
   f,g:       VAR [T -> real]
   xl,xh:     VAR T

   EX3p: LEMMA  a < b IMPLIES (FORALL (P: partition(a,b),p,q:nat):
                  0 < p AND p <= q AND q <= length(P) - 1 IMPLIES
                  sigma(p, q, (LAMBDA (n: upto(length(P) - 1)):
                  IF n = 0 THEN 0 ELSE seq(P)(n) - seq(P)(n - 1) ENDIF)) =
                         seq(P)(q) - seq(P)(p-1))

   ii: VAR nat
   EE: VAR posreal
   EXse: LEMMA a < b IMPLIES FORALL (P: partition(a,b),
                                     xis: (xis?(a,b,P))):
               0 < ii AND ii < length(P)
               AND (FORALL z:
                   (IF xl < z AND z < xh THEN f(z) = 1 ELSE f(z) = 0 ENDIF))
               AND width(a, b, P) < EE
               IMPLIES abs(Rie_sec(a, b, P, xis, f, ii)) < EE

%  Rosenlicht page 114

   Example_3: LEMMA a <= xl AND xl < xh AND xh <= b AND
                  (FORALL z: (IF xl < z AND z < xh THEN f(z) = 1
                                ELSE f(z) = 0 ENDIF))
                           IMPLIES integrable?(a,b,f) AND
                                   integral(a,b,f) = xh-xl

%
%  only one point where function is nonzero: Rosenlicht Ex 2 pg 114
%

   zero_except?(a,b,f): bool = a < b AND
                          (EXISTS z,c: a <= z AND z <= b AND f(z) = c AND
                                 (FORALL x: x /= z IMPLIES f(x) = 0))

   zero_except?_integrable: LEMMA a < b AND zero_except?(a,b,f) IMPLIES
                               integrable?(a,b,f) AND integral(a,b,f) = 0

   zero_not_intv?(xl,xh,f,c): bool =
                         (FORALL z:
                             IF xl < z AND z < xh THEN f(z) = c
                             ELSE  f(z) = 0
                             ENDIF)

   zero_not_intv?_integrable: LEMMA a <= xl AND xl < xh AND xh <= b AND
               zero_not_intv?(xl,xh,f,c) IMPLIES integrable?(a,b,f)
               AND integral(a,b,f) = c*(xh-xl)

%  ---------- doesn't matter what the value is at xl and xh -------------

   zero_except_intv?(xl,xh,c,f): bool =
                         (FORALL z:
                             (xl < z AND z < xh IMPLIES f(z) = c) AND
                             (z < xl IMPLIES f(z) = 0) AND
                             (z > xh IMPLIES f(z) = 0)  )

%  generalization of Example 3

   zero_except_intv?_integrable: LEMMA a <= xl AND xl < xh AND xh <= b AND
               zero_except_intv?(xl,xh,c,f) IMPLIES
                     integrable?(a,b,f) AND
                     integral(a,b,f) = c*(xh-xl)

END integral_pulse

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