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|
#|
This book shows that the sum and product of two differentiable
functions is differentiable. (See exercise6.lisp for the analogous
result about continuous functions.)
|#
(in-package "ACL2")
(include-book "derivatives")
;; First, we define a generic differentiable function, f1. This is
;; just a rewrite of the definition in derivatives.lisp for rdfn.
(encapsulate
((f1 (x) t))
;; Our witness continuous function is the identity function.
(local (defun f1 (x) x))
;; The function returns standard values for standard arguments.
(defthm f1-standard
(implies (standard-numberp x)
(standard-numberp (f1 x)))
:rule-classes (:rewrite :type-prescription))
;; For real arguments, the function returns real values.
(defthm f1-real
(implies (realp x)
(realp (f1 x)))
:rule-classes (:rewrite :type-prescription))
;; If x is a standard real and y1 and y2 are two arbitrary reals
;; close to x, then (f1(x)-f1(y1))/(x-y1) is close to
;; (f1(x)-f1(y2))/(x-y2). Also, (f1(x)-f1(y1))/(x-y1) is
;; limited. What this means is that the standard-part of that is a
;; standard number, and we'll call that the derivative of f1 at x.
(defthm f1-differentiable
(implies (and (standard-numberp x)
(realp x)
(realp y1)
(realp y2)
(i-close x y1) (not (= x y1))
(i-close x y2) (not (= x y2)))
(and (i-limited (/ (- (f1 x) (f1 y1)) (- x y1)))
(i-close (/ (- (f1 x) (f1 y1)) (- x y1))
(/ (- (f1 x) (f1 y2)) (- x y2))))))
)
;; Now, we do the same for another generic differentiable function,
;; f2.
(encapsulate
((f2 (x) t))
;; Our witness continuous function is the identity function.
(local (defun f2 (x) x))
;; The function returns standard values for standard arguments.
(defthm f2-standard
(implies (standard-numberp x)
(standard-numberp (f2 x)))
:rule-classes (:rewrite :type-prescription))
;; For real arguments, the function returns real values.
(defthm f2-real
(implies (realp x)
(realp (f2 x)))
:rule-classes (:rewrite :type-prescription))
;; If x is a standard real and y1 and y2 are two arbitrary reals
;; close to x, then (f2(x)-f2(y1))/(x-y1) is close to
;; (f2(x)-f2(y2))/(x-y2). Also, (f2(x)-f2(y1))/(x-y1) is
;; limited. What this means is that the standard-part of that is a
;; standard number, and we'll call that the derivative of f2 at x.
(defthm f2-differentiable
(implies (and (standard-numberp x)
(realp x)
(realp y1)
(realp y2)
(i-close x y1) (not (= x y1))
(i-close x y2) (not (= x y2)))
(and (i-limited (/ (- (f2 x) (f2 y1)) (- x y1)))
(i-close (/ (- (f2 x) (f2 y1)) (- x y1))
(/ (- (f2 x) (f2 y2)) (- x y2))))))
)
;; Now, we define the sum of f1 and f2.
(defun f1+f2 (x)
(+ (f1 x) (f2 x)))
;; To show that f1+f2 is differentiable we concentrate on showing that
;; it satisfies the constraint rdfn-differentiable, since the others
;; are trivial. First, we have to show that under suitable
;; hypothesis, the slope of the secant chord is limited. This follows
;; from the fact that the slope of the chord at f1+f2 is the sum of
;; the separate slopes at f1 and f2.
(defthm f1+f2-differentiable-part1
(implies (and (standard-numberp x)
(realp x)
(realp y1)
(i-close x y1) (not (= x y1)))
(i-limited (/ (- (f1+f2 x) (f1+f2 y1)) (- x y1))))
:hints (("Goal"
:use ((:instance i-limited-plus
(x (/ (- (f1 x) (f1 y1)) (- x y1)))
(y (/ (- (f2 x) (f2 y1)) (- x y1))))
(:instance f1-differentiable (y2 y1))
(:instance f2-differentiable (y2 y1)))
:in-theory (disable i-limited-plus f1-differentiable f2-differentiable))))
;; Next we need to show that if two chords at x are "close", so is their slope.
(encapsulate
()
(local
(defthm lemma-1
(equal (i-close (+ a x) (+ a y))
(i-close (fix x) (fix y)))
:hints (("Goal" :in-theory (enable i-close)))))
(local
(defthm lemma-2
(equal (i-close (+ x a) (+ y a))
(i-close (fix x) (fix y)))
:hints (("Goal" :in-theory (enable i-close)))))
(defthm congruence-of-close-over-+
(implies (and (i-close x1 x2)
(i-close y1 y2))
(i-close (+ x1 y1) (+ x2 y2)))
:hints (("Goal" :use ((:instance i-close-transitive
(x (+ x1 y1))
(y (+ x1 y2))
(z (+ x2 y2))))
:in-theory (enable-disable (i-close) (i-close-transitive)))))
)
(defthm f1+f2-differentiable-part2
(implies (and (standard-numberp x)
(realp x)
(realp y1)
(realp y2)
(i-close x y1) (not (= x y1))
(i-close x y2) (not (= x y2)))
(i-close (/ (- (f1+f2 x) (f1+f2 y1)) (- x y1))
(/ (- (f1+f2 x) (f1+f2 y2)) (- x y2))))
:hints (("Goal"
:use ((:instance congruence-of-close-over-+
(x1 (+ (* (F1 X) (/ (+ X (- Y1))))
(- (* (F1 Y1) (/ (+ X (- Y1)))))))
(x2 (+ (* (F1 X) (/ (+ X (- Y2))))
(- (* (F1 Y2) (/ (+ X (- Y2)))))))
(y1 (+ (* (F2 X) (/ (+ X (- Y1))))
(- (* (F2 Y1) (/ (+ X (- Y1)))))))
(y2 (+ (* (F2 X) (/ (+ X (- Y2))))
(- (* (F2 Y2) (/ (+ X (- Y2))))))))
(:instance f1-differentiable)
(:instance f2-differentiable))
:in-theory (disable f1-differentiable f2-differentiable))))
;; Now, to show that indeed we have the f1+f2 is differentiable, we
;; force ACL2 to functionally instantiate rdfn with f1+f2. The
;; specific theorem we use doesn't matter as much as the :hint.
(encapsulate
()
(local
(defthm f1+f2-continuous
(implies (and (standard-numberp x)
(realp x)
(i-close x y)
(realp y))
(i-close (f1+f2 x) (f1+f2 y)))
:hints (("Goal"
:by (:functional-instance rdfn-continuous
(rdfn f1+f2)))
("Subgoal 3"
:use ((:instance f1+f2-differentiable-part1)
(:instance f1+f2-differentiable-part2))
:in-theory (disable f1+f2-differentiable-part1
f1+f2-differentiable-part2
f1+f2)))))
)
;; Now, we do the same for f1*f2. First, we define this function:
(defun f1*f2 (x)
(* (f1 x) (f2 x)))
;; Again, we concentrate on showing that it satisfies the constraint
;; rdfn-differentiable, since the others are trivial. First, we have
;; to show that under suitable hypothesis, the slope of the secant
;; chord is limited. This follows from the fact that the slope of the
;; chord at f1*f2 is f1*slope(f2) + slope(f1)*f2. Both slopes are
;; limited, and so are the f1, f2, since f1&f2 are standard functions
;; and we're taking their values at either a standard point or a point
;; close to a standard point.
(defthm f2-continuous
(implies (and (standard-numberp x)
(realp x)
(i-close x y)
(realp y))
(i-close (f2 x) (f2 y)))
:hints (("Goal"
:by (:functional-instance rdfn-continuous
(rdfn f2)))
; Changed by Matt K. after v4-3 for tau-system to include these under "Goal'"
; instead of under "Subgoal 3", then again after 6.0 for (most likely)
; tau-system changing "Goal'" to "Subgoal 2".
("Subgoal 2"
:use (:instance f2-differentiable)
:in-theory (disable f2-differentiable))))
(local
(defthm f2-y1-limited
(implies (and (realp x)
(standard-numberp x)
(realp y1)
(i-close x y1))
(i-limited (f2 y1)))
:hints (("Goal"
:use ((:instance i-close-limited
(x (f2 x))
(y (f2 y1)))
(:instance f2-differentiable (y2 y1)))
:in-theory (enable-disable (standards-are-limited)
(i-close-limited f2-differentiable
;; The following are
;; also needed for v2-6.
I-CLOSE-SYMMETRIC
I-CLOSE-LARGE-2
))))))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (realp x)
(standard-numberp x)
(realp y1)
(i-close x y1))
(i-limited (/ (* (f2 y1) (- (f1 x) (f1 y1))) (- x y1))))
:hints (("Goal"
:use ((:instance f2-y1-limited)
(:instance i-limited-times
(x (f2 y1))
(y (/ (- (f1 x) (f1 y1)) (- x y1))))
(:instance f1-differentiable (y2 y1)))
:in-theory (disable f2-y1-limited i-limited-times f1-differentiable)))))
(local
(defthm lemma-2
(implies (and (realp x)
(standard-numberp x)
(realp y1)
(i-close x y1))
(i-limited (/ (* (f1 x) (- (f2 x) (f2 y1))) (- x y1))))
:hints (("Goal"
:use ((:instance standards-are-limited
(x (f1 x)))
(:instance i-limited-times
(x (f1 x))
(y (/ (- (f2 x) (f2 y1)) (- x y1))))
(:instance f2-differentiable (y2 y1)))
:in-theory (disable i-limited-times f2-differentiable)))))
(defthm f1*f2-differentiable-part1
(implies (and (realp x)
(standard-numberp x)
(realp y1)
(i-close x y1) (not (= x y1)))
(i-limited (/ (- (f1*f2 x) (f1*f2 y1)) (- x y1))))
:hints (("Goal"
:use ((:instance i-limited-plus
(x (/ (* (f1 x) (- (f2 x) (f2 y1))) (- x y1)))
(y (/ (* (f2 y1) (- (f1 x) (f1 y1))) (- x y1))))
(:instance lemma-1)
(:instance lemma-2))
:in-theory (disable i-limited-plus lemma-1 lemma-2))))
)
;; Next we need to show that if two chords at x are "close", so is their slope.
(defthm f2-uniformly-continuous
(implies (and (standard-numberp x)
(realp x)
(i-close x y1)
(realp y1)
(i-close x y2)
(realp y2))
(i-close (f2 y1) (f2 y2)))
:hints (("Goal"
:use ((:instance i-close-transitive
(x (f2 y1))
(y (f2 x))
(z (f2 y2))))
:in-theory (disable i-close-transitive))))
(defthm congruence-of-close-over-*
(implies (and (i-close x1 x2)
(i-close y1 y2)
(i-limited x1)
(i-limited y1))
(i-close (* x1 y1) (* x2 y2)))
:hints (("Goal"
:use ((:instance i-small-plus
(x (* x1 (- y1 y2)))
(y (* y2 (- x1 x2))))
(:instance small*limited->small
(x (- y1 y2))
(y x1))
(:instance small*limited->small
(x (- x1 x2))
(y y2)))
:in-theory (enable-disable (i-close) (i-small-plus small*limited->small)))))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (standard-numberp x)
(realp x)
(i-close x y1)
(not (= x y1))
(realp y1)
(i-close x y2)
(not (= x y2))
(realp y2))
(i-close (/ (* (f2 y1) (- (f1 x) (f1 y1))) (- x y1))
(/ (* (f2 y2) (- (f1 x) (f1 y2))) (- x y2))))
:hints (("Goal"
:use ((:instance congruence-of-close-over-*
(x1 (f2 y1))
(x2 (f2 y2))
(y1 (/ (- (f1 x) (f1 y1)) (- x y1)))
(y2 (/ (- (f1 x) (f1 y2)) (- x y2))))
(:instance f2-uniformly-continuous)
(:instance f1-differentiable)
(:instance f2-y1-limited))
:in-theory (disable congruence-of-close-over-* f2-uniformly-continuous
f1-differentiable f2-y1-limited)))))
(local
(defthm lemma-2
(implies (and (standard-numberp x)
(realp x)
(i-close x y1)
(not (= x y1))
(realp y1)
(i-close x y2)
(not (= x y2))
(realp y2))
(i-close (/ (* (f1 x) (- (f2 x) (f2 y1))) (- x y1))
(/ (* (f1 x) (- (f2 x) (f2 y2))) (- x y2))))
:hints (("Goal"
:use ((:instance congruence-of-close-over-*
(x1 (f1 x))
(x2 (f1 x))
(y1 (/ (- (f2 x) (f2 y1)) (- x y1)))
(y2 (/ (- (f2 x) (f2 y2)) (- x y2))))
(:instance f2-differentiable)
(:instance standards-are-limited
(x (f1 x))))
:in-theory (disable congruence-of-close-over-* f2-differentiable)))))
(defthm f1*f2-differentiable-part2
(implies (and (standard-numberp x)
(realp x)
(realp y1)
(realp y2)
(i-close x y1) (not (= x y1))
(i-close x y2) (not (= x y2)))
(i-close (/ (- (f1*f2 x) (f1*f2 y1)) (- x y1))
(/ (- (f1*f2 x) (f1*f2 y2)) (- x y2))))
:hints (("Goal"
:use ((:instance congruence-of-close-over-+
(x1 (/ (* (f1 x) (- (f2 x) (f2 y1))) (- x y1)))
(x2 (/ (* (f1 x) (- (f2 x) (f2 y2))) (- x y2)))
(y1 (/ (* (f2 y1) (- (f1 x) (f1 y1))) (- x y1)))
(y2 (/ (* (f2 y2) (- (f1 x) (f1 y2))) (- x y2))))
(:instance lemma-1)
(:instance lemma-2))
:in-theory (disable congruence-of-close-over-+ lemma-1 lemma-2))))
)
;; Now, to show that indeed we have the f1*f2 is differentiable, we
;; force ACL2 to functionally instantiate rdfn with f1*f2. The
;; specific theorem we use doesn't matter as much as the :hint.
(encapsulate
()
(local
(defthm f1*f2-continuous
(implies (and (standard-numberp x)
(realp x)
(i-close x y)
(realp y))
(i-close (f1*f2 x) (f1*f2 y)))
:hints (("Goal"
:by (:functional-instance rdfn-continuous
(rdfn f1*f2)))
("Subgoal 3"
:use ((:instance f1*f2-differentiable-part1)
(:instance f1*f2-differentiable-part2))
:in-theory (disable f1*f2-differentiable-part1
f1*f2-differentiable-part2
f1*f2)))))
)
|
