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|
#|
This book uses the "increasing" version of the intermediate value
theorem to prove a "decreasing" version (i.e., when f(a) > f(b)).
The basic idea is to consider the "increasing" version for the
function g(x)=-f(x).
|#
(in-package "ACL2")
(include-book "continuity")
;; First, we find the root. It would be nice if we could define this
;; using something akin to a :functional-instance (:lambda-instance,
;; anyone?)
(defun find-zero-n-2 (a z i n eps)
(declare (xargs :measure (nfix (1+ (- n i)))))
(if (and (realp a)
(integerp i)
(integerp n)
(< i n)
(realp eps)
(< 0 eps)
(< z (rcfn (+ a eps))))
(find-zero-n-2 (+ a eps) z (1+ i) n eps)
(realfix a)))
;; The key lemma -- if -x is close to -y, then x is close to y.
(defthm close-uminus
(implies (and (acl2-numberp x)
(acl2-numberp y))
(equal (i-close (- x) (- y))
(i-close x y)))
:hints (("Goal"
:use ((:instance i-small-uminus (x (+ x (- y)))))
:in-theory (enable i-close i-small-uminus))))
;; We prove that this function returns a limited number for limited
;; arguments.
(defthm limited-find-zero-2-body
(implies (and (i-limited a)
(i-limited b)
(realp b)
(realp z))
(i-limited (find-zero-n-2 a
z
0
(i-large-integer)
(+ (- (* (/ (i-large-integer)) a))
(* (/ (i-large-integer)) b)))))
:hints (("Goal"
:use ((:instance
(:functional-instance
limited-find-zero-body
(rcfn (lambda (x) (- (rcfn x))))
(find-zero-n (lambda (a z i n
eps)
(find-zero-n-2
a (- z) i n eps))))
(z (- z))))
:in-theory (disable limited-find-zero-body))))
;; We define the root we want in the range [a,b)
(defun-std find-zero-2 (a b z)
(if (and (realp a)
(realp b)
(realp z)
(< a b))
(standard-part
(find-zero-n-2 a
z
0
(i-large-integer)
(/ (- b a) (i-large-integer))))
0))
;; And here is the second version of the intermediate value theorem.
(local
(defthm standardp-minus-z
(implies (and (realp z)
(standardp z))
(standardp (- z)))
:rule-classes (:type-prescription :forward-chaining)))
(local
(defthmd definition-of-find-zero-2-lemma
(implies (and (standardp a)
(standardp b)
(standardp z))
(equal (find-zero-2 a b z)
(if (and (realp a)
(realp b)
(realp z)
(< a b))
(standard-part
(find-zero-n-2 a
z
0
(i-large-integer)
(/ (- b a) (i-large-integer))))
0)))))
(local
(defthmd definition-of-find-zero-2-uminus-z
(implies (and (standardp a)
(standardp b)
(standardp z))
(equal (find-zero-2 a b (- z))
(if (and (realp a)
(realp b)
(realp (- z))
(< a b))
(standard-part
(find-zero-n-2 a
(- z)
0
(i-large-integer)
(/ (- b a) (i-large-integer))))
0)))
:hints (("Goal"
:use ((:instance definition-of-find-zero-2-lemma
(z (- z))))))
))
(defthm intermediate-value-theorem-2
(implies (and (realp a)
(realp b)
(realp z)
(< a b)
(< z (rcfn a))
(< (rcfn b) z))
(and (realp (find-zero-2 a b z))
(< a (find-zero-2 a b z))
(< (find-zero-2 a b z) b)
(equal (rcfn (find-zero-2 a b z))
z)))
:hints (("Goal"
:use ((:instance
(:functional-instance
intermediate-value-theorem
(rcfn (lambda (x) (- (rcfn x))))
(find-zero (lambda (a b z)
(find-zero-2 a b
(if (realp z) (- z) z))))
(find-zero-n (lambda (a z i n
eps)
(find-zero-n-2
a (- z) i n eps))))
(z (if (realp z) (- z) z))
))
:in-theory
(disable intermediate-value-theorem))
))
|
