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|
(in-package "ACL2")
(local (include-book "arithmetic-5/top" :dir :system))
(local (include-book "nonstd/nsa/nsa" :dir :system))
(include-book "nonstd/nsa/derivatives" :dir :system)
(include-book "continuous-function")
(include-book "ftc-1")
(encapsulate
( ((rcdfn *) => *)
((rcdfn-prime *) => *)
((rcdfn-domain) => *)
)
(local (defun rcdfn (x) (declare (ignore x)) 0))
(local (defun rcdfn-prime (x) (declare (ignore x)) 0))
(local (defun rcdfn-domain () (interval 0 1)))
(defthm intervalp-rcdfn-domain
(interval-p (rcdfn-domain))
:rule-classes (:type-prescription :rewrite))
(defthm rcdfn-domain-real
(implies (inside-interval-p x (rcdfn-domain))
(realp x))
:rule-classes (:forward-chaining))
(defthm rcdfn-domain-non-trivial
(or (null (interval-left-endpoint (rcdfn-domain)))
(null (interval-right-endpoint (rcdfn-domain)))
(< (interval-left-endpoint (rcdfn-domain))
(interval-right-endpoint (rcdfn-domain))))
:rule-classes nil)
(defthm rcdfn-real
(realp (rcdfn x))
:rule-classes (:rewrite :type-prescription))
(defthm rcdfn-prime-real
(realp (rcdfn-prime x))
:rule-classes (:rewrite :type-prescription))
(defthm rcdfn-prime-is-derivative
(implies (and (standardp x)
(inside-interval-p x (rcdfn-domain))
(inside-interval-p x1 (rcdfn-domain))
(i-close x x1) (not (= x x1)))
(i-close (/ (- (rcdfn x) (rcdfn x1)) (- x x1))
(rcdfn-prime x))))
(defthm rcdfn-prime-continuous
(implies (and (standardp x)
(inside-interval-p x (rcdfn-domain))
(i-close x x1)
(inside-interval-p x1 (rcdfn-domain)))
(i-close (rcdfn-prime x)
(rcdfn-prime x1))))
)
(defun map-rcdfn-prime (p)
(if (consp p)
(cons (rcdfn-prime (car p))
(map-rcdfn-prime (cdr p)))
nil))
(defun riemann-rcdfn-prime (p)
(dotprod (deltas p)
(map-rcdfn-prime (cdr p))))
(defthm realp-riemann-rcdfn-prime
(implies (partitionp p)
(realp (riemann-rcdfn-prime p))))
(encapsulate
nil
(local
(defthm limited-riemann-rcdfn-prime-small-partition
(implies (and (realp a) (standardp a)
(realp b) (standardp b)
(inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(< a b))
(i-limited (riemann-rcdfn-prime (make-small-partition a b))))
:hints (("Goal"
:by (:functional-instance limited-riemann-rcfn-small-partition
(rcfn-domain rcdfn-domain)
(rcfn rcdfn-prime)
(map-rcfn map-rcdfn-prime)
(riemann-rcfn riemann-rcdfn-prime)))
("Subgoal 2"
:use ((:instance rcdfn-domain-non-trivial))))))
(local (in-theory (disable riemann-rcdfn-prime)))
(defun-std strict-int-rcdfn-prime (a b)
(if (and (realp a)
(realp b)
(inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(< a b))
(standard-part (riemann-rcdfn-prime (make-small-partition a b)))
0))
)
(defun int-rcdfn-prime (a b)
(if (<= a b)
(strict-int-rcdfn-prime a b)
(- (strict-int-rcdfn-prime b a))))
(defthm strict-int-rcdfn-prime-is-integral-of-rcdfn-prime
(implies (and (standardp a)
(standardp b)
(<= a b)
(inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(partitionp p)
(equal (car p) a)
(equal (car (last p)) b)
(i-small (mesh p)))
(i-close (riemann-rcdfn-prime p)
(strict-int-rcdfn-prime a b)))
:hints (("Goal"
:do-not-induct t
:by (:functional-instance strict-int-rcfn-is-integral-of-rcfn
(rcfn-domain rcdfn-domain)
(int-rcfn int-rcdfn-prime)
(rcfn rcdfn-prime)
(map-rcfn map-rcdfn-prime)
(riemann-rcfn riemann-rcdfn-prime)
(strict-int-rcfn strict-int-rcdfn-prime))
)
("Subgoal 2"
:use ((:instance rcdfn-domain-non-trivial)))
))
(defthmd ftc-1-for-rcdfn
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(inside-interval-p c (rcdfn-domain))
(standardp b)
(i-close b c)
(not (equal b c)))
(i-close (/ (- (int-rcdfn-prime a b) (int-rcdfn-prime a c))
(- b c))
(rcdfn-prime b)))
:hints (("Goal"
:by (:functional-instance ftc-1
(rcfn-domain rcdfn-domain)
(int-rcfn int-rcdfn-prime)
(rcfn rcdfn-prime)
(map-rcfn map-rcdfn-prime)
(riemann-rcfn riemann-rcdfn-prime)
(strict-int-rcfn strict-int-rcdfn-prime)
))
))
(local
(defun diff-fn (a x)
(if (inside-interval-p a (rcdfn-domain))
(- (int-rcdfn-prime a x) (rcdfn x))
0
)))
(local
(defthmd close-plus-constant
(implies (i-close x y)
(i-close (+ x z) (+ y z)))
:hints (("Goal"
:use ((:instance i-small-plus
(x (- x y))
(y z)))
:in-theory (e/d (i-close)
(i-small-plus))))
))
(local
(defthmd close-plus
(implies (and (i-close x1 x2)
(i-close y1 y2))
(i-close (+ x1 y1) (+ x2 y2)))
:hints (("Goal"
:use ((:instance i-small-plus
(x (- x1 x2))
(y (- y1 y2))))
:in-theory (e/d (i-close)
(i-small-plus))))
))
(local
(defthmd close-minus
(implies (and (i-close x1 x2)
(i-close y1 y2))
(i-close (- x1 y1) (- x2 y2)))
:hints (("Goal"
:use ((:instance i-small-plus
(x (- x1 x2))
(y (- (- y1 y2))))
(:instance i-small-uminus
(x (- y1 y2)))
)
:in-theory (e/d (i-close)
(i-small-plus i-small-uminus))))
))
(local
(defthmd diff-fn-has-zero-derivative-1
(implies (and (standardp x)
(inside-interval-p x (rcdfn-domain))
(i-close x x1)
(inside-interval-p x1 (rcdfn-domain))
(not (equal x x1)))
(i-close (/ (- (diff-fn a x) (diff-fn a x1)) (- x x1))
0))
:hints (("Goal"
:use ((:instance ftc-1-for-rcdfn
(a a)
(b x)
(c x1))
(:instance rcdfn-prime-is-derivative)
(:instance close-minus
(y1 (+ (* (rcdfn x) (/ (+ x (- x1))))
(- (* (rcdfn x1) (/ (+ x (- x1)))))))
(y2 (rcdfn-prime x))
(x1 (+ (* (int-rcdfn-prime a x)
(/ (+ x (- x1))))
(- (* (int-rcdfn-prime a x1)
(/ (+ x (- x1)))))))
(x2 (rcdfn-prime x))))
:in-theory (disable rcdfn-prime-is-derivative
i-small-uminus
int-rcdfn-prime
)))
))
(local
(encapsulate
nil
(local
(defthm derivative-diff-fn-lemma
(IMPLIES
(AND (STANDARDP A) (STANDARDP X))
(STANDARDP
(IF (INSIDE-INTERVAL-P X (RCDFN-DOMAIN))
(COND ((INSIDE-INTERVAL-P (+ X (/ (I-LARGE-INTEGER)))
(RCDFN-DOMAIN))
(STANDARD-PART (* (+ (DIFF-FN A (+ X (/ (I-LARGE-INTEGER))))
(- (DIFF-FN A X)))
(/ (/ (I-LARGE-INTEGER))))))
((INSIDE-INTERVAL-P (+ X (- (/ (I-LARGE-INTEGER))))
(RCDFN-DOMAIN))
(STANDARD-PART (* (+ (DIFF-FN A (+ X (- (/ (I-LARGE-INTEGER)))))
(- (DIFF-FN A X)))
(/ (- (/ (I-LARGE-INTEGER)))))))
(T 'error))
'error)))
:hints (("Goal"
:use ((:instance diff-fn-has-zero-derivative-1
(a a)
(x x)
(x1 (+ x (/ (i-large-integer)))))
(:instance diff-fn-has-zero-derivative-1
(a a)
(x x)
(x1 (- x (/ (i-large-integer)))))
(:instance i-close-to-small-sum
(x x)
(eps (/ (i-large-integer))))
(:instance i-close-to-small-sum
(x x)
(eps (- (/ (i-large-integer)))))
(:instance i-close-limited
(x 0)
(y (/ (+ (DIFF-FN A X)
(- (DIFF-FN A (+ X (/ (I-LARGE-INTEGER))))))
(- (/ (I-LARGE-INTEGER))))))
(:instance i-close-limited
(x 0)
(y (/ (+ (DIFF-FN A X)
(- (DIFF-FN A (+ X (- (/ (I-LARGE-INTEGER)))))))
(/ (I-LARGE-INTEGER)))))
)
:in-theory (disable diff-fn
i-close-to-small-sum
i-close-limited
standard-part-of-plus
standard-part-of-uminus)
))
;:rule-classes :rewrite
))
(defun differential-diff-fn (a x eps)
(/ (- (diff-fn a (+ x eps)) (diff-fn a x)) eps))
(local
(in-theory '(derivative-diff-fn-lemma
differential-diff-fn)))
(defun-std derivative-diff-fn (a x)
(if (inside-interval-p x (rcdfn-domain))
(if (inside-interval-p (+ x (/ (i-large-integer))) (rcdfn-domain))
(standard-part (differential-diff-fn a x (/ (i-large-integer))))
(if (inside-interval-p (- x (/ (i-large-integer))) (rcdfn-domain))
(standard-part (differential-diff-fn a x (- (/ (i-large-integer)))))
'error))
'error)
)
)
)
(local
(defthm-std standard-rcdfn-domain
(standardp (rcdfn-domain))))
(local
(defthm-std standard-rcdfn-domain-left-endpoint
(standardp (interval-left-endpoint (rcdfn-domain)))))
(local
(defthm-std standard-rcdfn-domain-right-endpoint
(standardp (interval-right-endpoint (rcdfn-domain)))))
(local
(defthm standard-part-eps
(implies (i-small eps)
(equal (standard-part eps) 0))
:hints (("Goal"
:in-theory (enable i-small)))
))
(local
(defthmd x-in-interval-implies-x+-eps-in-interval-1
(implies (and (realp x)
(standardp x)
(realp x1)
(standardp x1)
(< x1 x)
(realp eps)
(i-small eps))
(< x1
(+ x eps)))
:hints (("Goal"
:use ((:instance standard-part-<-2
(x x1)
(y (+ x eps))))))))
(local
(defthmd x-in-interval-implies-x+-eps-in-interval-2
(implies (and (realp x)
(standardp x)
(realp x2)
(standardp x2)
(< x x2)
(realp eps)
(i-small eps))
(< (+ x eps)
x2))
:hints (("Goal"
:use ((:instance standard-part-<-2
(x (+ x eps))
(y x2)))))))
(local
(defthm x-in-trivial-interval
(implies (and (realp x)
(interval-p interval)
(not (realp (interval-left-endpoint interval)))
(not (realp (interval-right-endpoint interval))))
(inside-interval-p x interval))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))
))
(local
(defthm x-in-left-trivial-interval
(implies (and (realp x)
(interval-p interval)
(not (realp (interval-left-endpoint interval)))
(inside-interval-p y interval)
(< x y))
(inside-interval-p x interval))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))
))
(local
(defthm x-in-right-trivial-interval
(implies (and (realp x)
(interval-p interval)
(not (realp (interval-right-endpoint interval)))
(inside-interval-p y interval)
(> x y))
(inside-interval-p x interval))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))
))
(local
(defthm nil-not-in-interval
(implies (and (not x)
(interval-p interval))
(not (inside-interval-p x interval)))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))
))
(local
(defthm x-in-interval-implies-x+-eps-in-interval
(implies (and (inside-interval-p x (rcdfn-domain))
(standardp x)
(realp eps)
(i-small eps)
(< 0 eps))
(or (inside-interval-p (+ x eps) (rcdfn-domain))
(inside-interval-p (- x eps) (rcdfn-domain))))
:hints (("Goal"
:use ((:instance rcdfn-domain-non-trivial)
(:instance x-in-interval-implies-x+-eps-in-interval-1
(x x)
(eps eps)
(x1 (interval-left-endpoint (rcdfn-domain))))
(:instance x-in-interval-implies-x+-eps-in-interval-1
(x x)
(eps (- eps))
(x1 (interval-left-endpoint (rcdfn-domain))))
(:instance x-in-interval-implies-x+-eps-in-interval-2
(x x)
(eps eps)
(x2 (interval-right-endpoint (rcdfn-domain))))
(:instance x-in-interval-implies-x+-eps-in-interval-2
(x x)
(eps (- eps))
(x2 (interval-right-endpoint (rcdfn-domain))))
)
)
("Subgoal 7"
:in-theory (enable interval-definition-theory))
("Subgoal 4"
:in-theory (enable interval-definition-theory))
("Subgoal 3"
:in-theory (enable interval-definition-theory))
("Subgoal 1"
:in-theory (enable interval-definition-theory))
)
:rule-classes nil
))
(local
(defthmd close-minus-0
(equal (i-close (- x) 0)
(i-close (fix x) 0))
:hints (("Goal"
:in-theory (enable i-close)))))
(local
(defthm close-0-standard-part-0
(implies (i-close x 0)
(equal (standard-part x) 0))
:hints (("Goal"
:use ((:instance close-x-y->same-standard-part
(x 0)
(y x)))
:in-theory (disable close-x-y->same-standard-part)))))
(local
(defthm-std derivative-diff-is-zero
(implies (inside-interval-p x (rcdfn-domain))
(equal (derivative-diff-fn a x) 0))
:hints (("Goal"
:use ((:instance x-in-interval-implies-x+-eps-in-interval
(eps (/ (i-large-integer))))
(:instance diff-fn-has-zero-derivative-1
(x x)
(x1 (+ x (/ (i-large-integer)))))
(:instance diff-fn-has-zero-derivative-1
(x x)
(x1 (+ x (- (/ (i-large-integer))))))
(:instance i-close-to-small-sum
(x x)
(eps (/ (i-large-integer))))
(:instance i-close-to-small-sum
(x x)
(eps (- (/ (i-large-integer)))))
(:instance close-minus-0
(x (+ (- (* (i-large-integer) (diff-fn a x)))
(* (i-large-integer)
(diff-fn a (+ (/ (i-large-integer)) x))))))
(:instance close-minus-0
(x (+ (- (* (i-large-integer) (diff-fn a x)))
(* (i-large-integer)
(diff-fn a (+ (- (/ (i-large-integer))) x)))))))
:in-theory (disable diff-fn
standard-part-of-plus
i-close-to-small-sum)
))))
(encapsulate
((rcdfn-subdomain () t))
(local
(defun rcdfn-subdomain ()
(let ((left (interval-left-endpoint (rcdfn-domain)))
(right (interval-right-endpoint (rcdfn-domain))))
(if (null left)
(if (null right)
(interval 0 1)
(interval (- right 2) (- right 1)))
(if (null right)
(interval (+ left 1) (+ left 2))
(interval (+ left (* 1/3 (- right left)))
(+ left (* 2/3 (- right left)))))))))
(defthm rcdfn-subdomain-is-subdomain
(subinterval-p (rcdfn-subdomain) (rcdfn-domain))
:hints (("Goal"
:use ((:instance interval-p (interval (rcdfn-domain))))
:in-theory (enable interval-definition-theory))
))
(defthm rcdfn-subdomain-closed-bounded
(and (interval-left-inclusive-p (rcdfn-subdomain))
(interval-right-inclusive-p (rcdfn-subdomain))))
(defthm rcdfn-subdomain-real-left
(realp (interval-left-endpoint (rcdfn-subdomain)))
:rule-classes (:rewrite :type-prescription))
(defthm rcdfn-subdomain-real-right
(realp (interval-right-endpoint (rcdfn-subdomain)))
:rule-classes (:rewrite :type-prescription))
(defthm rcdfn-subdomain-non-trivial
(< (interval-left-endpoint (rcdfn-subdomain))
(interval-right-endpoint (rcdfn-subdomain)))
:hints (("Goal"
:use (:instance rcdfn-domain-non-trivial)))
)
)
(local
(defun-sk exists-mvt-point-for-diff-fn ()
(exists (x)
(and (inside-interval-p x (rcdfn-subdomain))
(not (equal x (interval-left-endpoint (rcdfn-subdomain))))
(not (equal x (interval-right-endpoint (rcdfn-subdomain))))
(equal (derivative-diff-fn (interval-left-endpoint (rcdfn-subdomain)) x)
(/ (- (diff-fn (interval-left-endpoint (rcdfn-subdomain))
(interval-right-endpoint (rcdfn-subdomain)))
(diff-fn (interval-left-endpoint (rcdfn-subdomain))
(interval-left-endpoint (rcdfn-subdomain))))
(- (interval-right-endpoint (rcdfn-subdomain))
(interval-left-endpoint (rcdfn-subdomain)))))))))
(local
(defthm-std rcdfn-prime-standard
(implies (standardp x)
(standardp (rcdfn-prime x)))))
(local
(defthm left-endpoint-in-domain
(inside-interval-p (interval-left-endpoint (rcdfn-subdomain))
(rcdfn-domain))
:hints (("Goal"
:use ((:instance rcdfn-subdomain-is-subdomain)
(:instance rcdfn-subdomain-closed-bounded)
(:instance rcdfn-subdomain-real-left))
:in-theory (e/d (interval-definition-theory)
(rcdfn-subdomain-is-subdomain
rcdfn-subdomain-closed-bounded
rcdfn-subdomain-real-left))))))
(local
(defthm close-0-not-large
(implies (i-close x 0)
(not (i-large x)))
:hints (("Goal"
:in-theory (enable i-close)))))
(local
(defthm realp-strict-int-rcdfn-prime
(implies (and (realp a)
(realp b))
(realp (strict-int-rcdfn-prime a b)))
:hints (("Goal"
:by (:functional-instance realp-strict-int-rcfn
(strict-int-rcfn strict-int-rcdfn-prime)
(rcfn-domain rcdfn-domain)
(rcfn rcdfn-prime)
(riemann-rcfn riemann-rcdfn-prime)
(map-rcfn map-rcdfn-prime))))))
(local
(defthm-std realp-strict-int-rcdfn-prime-stronger
(realp (strict-int-rcdfn-prime a b))
:hints (("Goal"
:cases ((not (realp a))
(not (realp b))))
("Subgoal 3"
:use ((:instance realp-strict-int-rcdfn-prime)))
)))
(local
(defthm realp-int-rcdfn-prime
(realp (int-rcdfn-prime a b))
))
(local
(defthm realp-left-endpoint
(realp (interval-left-endpoint (rcdfn-subdomain)))
))
(local
(defthm-std standard-left-endpoint-subdomain
(standardp (INTERVAL-LEFT-ENDPOINT (RCDFN-SUBDOMAIN)))))
(local
(defthm mvt-theorem-sk-for-diff-fn
(exists-mvt-point-for-diff-fn)
:hints (("Goal"
:by (:functional-instance mvt-theorem-sk
(exists-mvt-point exists-mvt-point-for-diff-fn)
(exists-mvt-point-witness exists-mvt-point-for-diff-fn-witness)
(rdfn-domain rcdfn-domain)
(rdfn-subdomain rcdfn-subdomain)
(rdfn (lambda (x)
(diff-fn (interval-left-endpoint (rcdfn-subdomain))
x)))
(derivative-rdfn (lambda (x)
(derivative-diff-fn
(interval-left-endpoint (rcdfn-subdomain))
x)))
(differential-rdfn
(lambda (x eps)
(differential-diff-fn
(interval-left-endpoint (rcdfn-subdomain))
x
eps)))
)
)
("Subgoal 8"
:use ((:instance exists-mvt-point-for-diff-fn-suff))
:in-theory (disable exists-mvt-point-for-diff-fn-suff)
)
("Subgoal 6"
:use ((:instance diff-fn-has-zero-derivative-1
(a (interval-left-endpoint (rcdfn-subdomain)))
(x x)
(x1 y1))
(:instance diff-fn-has-zero-derivative-1
(a (interval-left-endpoint (rcdfn-subdomain)))
(x x)
(x1 y2))
)
:in-theory (disable diff-fn-has-zero-derivative-1
diff-fn)
)
("Subgoal 4"
:use ((:instance rcdfn-domain-non-trivial)))
("Subgoal 2"
:in-theory (disable differential-diff-fn
derivative-diff-is-zero))
)))
(local
(defthm strict-int-rcdfn-prime-a-a
(implies (inside-interval-p a (rcdfn-domain))
(equal (strict-int-rcdfn-prime a a) 0))
:hints (("Goal"
:by (:functional-instance strict-int-a-a
(strict-int-rcfn strict-int-rcdfn-prime)
(rcfn-domain rcdfn-domain)
(rcfn rcdfn-prime)
(riemann-rcfn riemann-rcdfn-prime)
(map-rcfn map-rcdfn-prime))))))
(local
(defthm diff-fn-of-a
(equal (diff-fn a a)
(if (inside-interval-p a (rcdfn-domain))
(- (rcdfn a))
0))))
(local
(defthm int-rcdfn-prime-a-a
(implies (inside-interval-p a (rcdfn-domain))
(equal (int-rcdfn-prime a a) 0))
:hints (("Goal"
:use ((:instance strict-int-rcdfn-prime-a-a))
:in-theory (e/d (int-rcdfn-prime)
(strict-int-rcdfn-prime-a-a
strict-int-rcdfn-prime
(int-rcdfn-prime)))))
))
(local
(defthm diff-fn-inside-interval
(equal (diff-fn (interval-left-endpoint (rcdfn-subdomain))
(interval-right-endpoint (rcdfn-subdomain)))
(- (rcdfn (interval-left-endpoint (rcdfn-subdomain)))))
:hints (("Goal"
:use ((:instance mvt-theorem-sk-for-diff-fn)
(:instance exists-mvt-point-for-diff-fn)
(:instance derivative-diff-is-zero
(a (interval-left-endpoint (rcdfn-subdomain)))
(x (exists-mvt-point-for-diff-fn-witness))))
:in-theory (disable int-rcdfn-prime))
("Subgoal 2"
:use ((:instance rcdfn-subdomain-non-trivial))
:in-theory (disable rcdfn-subdomain-non-trivial))
)
))
(local
(defthm diff-fn-expander
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(< a b))
(equal (diff-fn a b)
(- (rcdfn a))))
:hints (("Goal"
:use ((:functional-instance diff-fn-inside-interval
(rcdfn-subdomain (lambda ()
(if (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(< a b))
(interval a b)
(rcdfn-subdomain))
))))))))
(local
(defthm ftc-2-lemma
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(< a b))
(equal (int-rcdfn-prime a b)
(- (rcdfn b)
(rcdfn a))))
:hints (("Goal"
:use ((:instance diff-fn-expander))
:in-theory (disable int-rcdfn-prime
diff-fn-expander)
))))
(local
(defthmd int-rcdfn-reverse-bounds
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain)))
(equal (- (int-rcdfn-prime b a))
(int-rcdfn-prime a b)))))
(local
(defthm ftc-2-lemma-2
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain))
(> a b))
(equal (int-rcdfn-prime a b)
(- (rcdfn b)
(rcdfn a))))
:hints (("Goal"
:use ((:instance ftc-2-lemma (a b) (b a))
(:instance int-rcdfn-reverse-bounds)
)
:in-theory (disable int-rcdfn-prime
ftc-2-lemma)
))))
(defthm ftc-2
(implies (and (inside-interval-p a (rcdfn-domain))
(inside-interval-p b (rcdfn-domain)))
(equal (int-rcdfn-prime a b)
(- (rcdfn b)
(rcdfn a))))
:hints (("Goal"
:cases ((< a b) (> a b))
:in-theory (disable int-rcdfn-prime
(int-rcdfn-prime)))))
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