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|
(in-package "ACL2")
(local (include-book "arithmetic-5/top" :dir :system))
(local (include-book "nonstd/nsa/equivalence-continuity" :dir :system))
(local (include-book "nonstd/nsa/equivalence-derivatives" :dir :system))
(include-book "make-partition")
(local (include-book "equivalence-integrals"))
(local (include-book "ftc-2"))
(include-book "integrable-functions")
(encapsulate
( ((rcdfn-classical *) => *)
((rcdfn-classical-prime *) => *)
((rcdfn-classical-domain) => *)
)
(local (defun rcdfn-classical (x) (declare (ignore x)) 0))
(local (defun rcdfn-classical-prime (x) (declare (ignore x)) 0))
(local (defun rcdfn-classical-domain () (interval 0 1)))
(defthm intervalp-rcdfn-classical-domain
(interval-p (rcdfn-classical-domain))
:rule-classes (:type-prescription :rewrite))
(defthm rcdfn-classical-domain-real
(implies (inside-interval-p x (rcdfn-classical-domain))
(realp x))
:rule-classes (:forward-chaining))
(defthm rcdfn-classical-domain-non-trivial
(or (null (interval-left-endpoint (rcdfn-classical-domain)))
(null (interval-right-endpoint (rcdfn-classical-domain)))
(< (interval-left-endpoint (rcdfn-classical-domain))
(interval-right-endpoint (rcdfn-classical-domain))))
:rule-classes nil)
(defthm rcdfn-classical-real
(realp (rcdfn-classical x))
:rule-classes (:rewrite :type-prescription))
(defthm rcdfn-classical-prime-real
(realp (rcdfn-classical-prime x))
:rule-classes (:rewrite :type-prescription))
(defun-sk forall-x-eps-delta-in-range-deriv-rcdfn-classical-works (x eps delta)
(forall (x1)
(implies (and (inside-interval-p x1 (rcdfn-classical-domain))
(inside-interval-p x (rcdfn-classical-domain))
(realp delta)
(< 0 delta)
(realp eps)
(< 0 eps)
(not (equal x x1))
(< (abs (- x x1)) delta))
(< (abs (- (/ (- (rcdfn-classical x) (rcdfn-classical x1))
(- x x1))
(rcdfn-classical-prime x)))
eps))))
(defun-sk exists-delta-for-x-and-eps-so-deriv-rcdfn-classical-works (x eps)
(exists delta
(implies (and (inside-interval-p x (rcdfn-classical-domain))
(realp eps)
(< 0 eps))
(and (realp delta)
(< 0 delta)
(forall-x-eps-delta-in-range-deriv-rcdfn-classical-works x eps delta)))))
(defthm rcdfn-classical-prime-is-derivative
(implies (and (inside-interval-p x (rcdfn-classical-domain))
(realp eps)
(< 0 eps))
(exists-delta-for-x-and-eps-so-deriv-rcdfn-classical-works x eps))
:hints (("Goal"
:use ((:instance exists-delta-for-x-and-eps-so-deriv-rcdfn-classical-works-suff
(delta 1))
(:instance forall-x-eps-delta-in-range-deriv-rcdfn-classical-works
(x x)
(eps eps)
(delta 1)))
:in-theory (disable abs))))
(defun-sk forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime (x0 eps delta)
(forall (x)
(implies (and (inside-interval-p x (rcdfn-classical-domain))
(inside-interval-p x0 (rcdfn-classical-domain))
(realp delta)
(< 0 delta)
(realp eps)
(< 0 eps)
(< (abs (- x x0)) delta)
(not (equal x x0)))
(< (abs (- (rcdfn-classical-prime x) (rcdfn-classical-prime x0)))
eps)))
:rewrite :direct)
(defun-sk exists-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcdfn-classical-domain))
;(standardp x0)
(realp eps)
;(standardp eps)
(< 0 eps))
(and ;(standardp delta)
(realp delta)
(< 0 delta)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime x0 eps delta)))))
(defthmd rcdfn-classical-prime-is-continuous
(implies (and (inside-interval-p x0 (rcdfn-classical-domain))
;(standardp x0)
(realp eps)
;(standardp eps)
(< 0 eps))
(exists-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime x0 eps))
:hints (("Goal"
:use ((:instance exists-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-suff
(delta 1))
(:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime
(x0 x0)
(eps eps)
(delta 1)))
:in-theory (disable abs))))
)
(defun-sk exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcdfn-classical-domain))
(standardp x0)
(realp eps)
(standardp eps)
(< 0 eps))
(and (standardp delta)
(realp delta)
(< 0 delta)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime x0 eps delta))))
:classicalp nil)
(encapsulate
nil
(local
(defthm-std lemma-1
(implies (and (standardp x0)
(standardp eps))
(standardp (exists-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-witness x0 eps)))))
(defthmd rcdfn-classical-prime-is-continuous-classically
(implies (and (inside-interval-p x0 (rcdfn-classical-domain))
(standardp x0)
(realp eps)
(standardp eps)
(< 0 eps))
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime x0 eps))
:hints (("Goal"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-suff
(delta (exists-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-witness x0 eps)))
(:instance rcdfn-classical-prime-is-continuous))
:in-theory (disable abs))))
)
(defthm rcdfn-classical-prime-is-derivative-nonstd
(implies (and (standardp x)
(inside-interval-p x (rcdfn-classical-domain))
(inside-interval-p x1 (rcdfn-classical-domain))
(i-close x x1) (not (= x x1)))
(i-close (/ (- (rcdfn-classical x) (rcdfn-classical x1)) (- x x1))
(rcdfn-classical-prime x)))
:hints (("Goal"
:use ((:functional-instance rdfn-classic-is-differentiable
(rdfn-classical-domain rcdfn-classical-domain)
(rdfn-classical rcdfn-classical)
(derivative-rdfn-classical rcdfn-classical-prime)
(exists-delta-for-x-and-eps-so-deriv-classical-works
exists-delta-for-x-and-eps-so-deriv-rcdfn-classical-works)
(exists-delta-for-x-and-eps-so-deriv-classical-works-witness
exists-delta-for-x-and-eps-so-deriv-rcdfn-classical-works-witness)
(forall-x-eps-delta-in-range-deriv-classical-works
forall-x-eps-delta-in-range-deriv-rcdfn-classical-works)
(forall-x-eps-delta-in-range-deriv-classical-works-witness
forall-x-eps-delta-in-range-deriv-rcdfn-classical-works-witness)
)))
("Subgoal 5"
:use ((:instance rcdfn-classical-prime-is-derivative)))
("Subgoal 3"
:use ((:instance forall-x-eps-delta-in-range-deriv-rcdfn-classical-works-necc)))
("Subgoal 2"
:use ((:instance rcdfn-classical-domain-non-trivial)))
)
)
(defthm rcdfn-classical-prime-continuous-nonstd
(implies (and (standardp x)
(inside-interval-p x (rcdfn-classical-domain))
(i-close x x1)
(inside-interval-p x1 (rcdfn-classical-domain)))
(i-close (rcdfn-classical-prime x)
(rcdfn-classical-prime x1)))
:hints (("Goal"
:use ((:instance
(:functional-instance rcfn-classical-is-continuous-using-nonstandard-criterion
(rcfn-classical rcdfn-classical-prime)
(rcfn-classical-domain rcdfn-classical-domain)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f
exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f-witness
exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-witness)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f
forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f-witness
forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime-witness))
(x0 x)
(x x1)
)))
("Subgoal 7"
:use ((:instance rcdfn-classical-prime-is-continuous-classically)))
("Subgoal 5"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-rcdfn-classical-prime-suff)))
("Subgoal 3"
:use ((:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-rcdfn-classical-prime-necc)))
("Subgoal 2"
:use ((:instance rcdfn-classical-domain-non-trivial)))
))
(defun map-rcdfn-classical-prime (p)
(if (consp p)
(cons (rcdfn-classical-prime (car p))
(map-rcdfn-classical-prime (cdr p)))
nil))
(defun riemann-rcdfn-classical-prime (p)
(dotprod (deltas p)
(map-rcdfn-classical-prime (cdr p))))
(defthm realp-riemann-rcdfn-classical-prime
(implies (partitionp p)
(realp (riemann-rcdfn-classical-prime p))))
(encapsulate
nil
(local
(defthm limited-riemann-rcdfn-classical-prime-small-partition
(implies (and (realp a) (standardp a)
(realp b) (standardp b)
(inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain))
(< a b))
(i-limited (riemann-rcdfn-classical-prime (make-small-partition a b))))
:hints (("Goal"
:by (:functional-instance limited-riemann-rcfn-small-partition
(rcfn-domain rcdfn-classical-domain)
(rcfn rcdfn-classical-prime)
(map-rcfn map-rcdfn-classical-prime)
(riemann-rcfn riemann-rcdfn-classical-prime)))
("Subgoal 2"
:use ((:instance rcdfn-classical-domain-non-trivial))))))
(local (in-theory (disable riemann-rcdfn-classical-prime)))
(defun-std strict-int-rcdfn-classical-prime (a b)
(if (and (realp a)
(realp b)
(inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain))
(< a b))
(standard-part (riemann-rcdfn-classical-prime (make-small-partition a b)))
0))
)
(defun int-rcdfn-classical-prime (a b)
(if (<= a b)
(strict-int-rcdfn-classical-prime a b)
(- (strict-int-rcdfn-classical-prime b a))))
(defthm-std realp-strict-int-rcdfn-classical-prime
(IMPLIES (AND (REALP A) (REALP B))
(REALP (STRICT-INT-RCDFN-CLASSICAL-PRIME A B)))
; Matt K. v7-1 mod for ACL2 mod on 2/13/2015: "Goal'" changed to "Goal".
:hints (("Goal"
:use ((:instance realp-riemann-rcdfn-classical-prime
(p (make-small-partition a b))))
:in-theory (disable realp-riemann-rcdfn-classical-prime
riemann-rcdfn-classical-prime)))
)
(defun-sk forall-partitions-riemann-sum-is-close-to-int-rcdfn-classical-prime (a b eps delta)
(forall (p)
(implies (and (<= a b)
(partitionp p)
(equal (car p) a)
(equal (car (last p)) b)
(< (mesh p) delta))
(< (abs (- (riemann-rcdfn-classical-prime p)
(strict-int-rcdfn-classical-prime a b)))
eps)))
:rewrite :direct)
(defun-sk exists-delta-so-that-riemann-sum-is-close-to-int-rcdfn-classical-prime (a b eps)
(exists (delta)
(implies (and (inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain))
(<= a b)
(realp eps)
(< 0 eps))
(and (realp delta)
(< 0 delta)
(forall-partitions-riemann-sum-is-close-to-int-rcdfn-classical-prime a b eps delta)))))
(defthm strict-int-rcdfn-classical-prime-is-integral-of-rcdfn-classical-prime
(implies (and (standardp a)
(standardp b)
(<= a b)
(inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain))
(partitionp p)
(equal (car p) a)
(equal (car (last p)) b)
(i-small (mesh p)))
(i-close (riemann-rcdfn-classical-prime p)
(strict-int-rcdfn-classical-prime a b)))
:hints (("Goal"
:do-not-induct t
:by (:functional-instance strict-int-rcdfn-prime-is-integral-of-rcdfn-prime
(rcdfn rcdfn-classical)
(rcdfn-prime rcdfn-classical-prime)
(rcdfn-domain rcdfn-classical-domain)
(map-rcdfn-prime map-rcdfn-classical-prime)
(riemann-rcdfn-prime riemann-rcdfn-classical-prime)
(strict-int-rcdfn-prime strict-int-rcdfn-classical-prime)
(int-rcdfn-prime int-rcdfn-classical-prime)))
("Subgoal 3"
:use ((:instance rcdfn-classical-prime-is-derivative-nonstd)))
("Subgoal 2"
:use ((:instance rcdfn-classical-domain-non-trivial)))
))
(defthm rcdfn-classical-prime-is-integrable-hyperreal
(implies (and (<= a b)
(inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain))
(realp eps)
(< 0 eps))
(exists-delta-so-that-riemann-sum-is-close-to-int-rcdfn-classical-prime a b eps))
:hints (("Goal" :do-not-induct t
:by (:functional-instance rifn-is-integrable-hyperreal
(rifn rcdfn-classical-prime)
(domain-rifn rcdfn-classical-domain)
(map-rifn map-rcdfn-classical-prime)
(riemann-rifn riemann-rcdfn-classical-prime)
(strict-int-rifn strict-int-rcdfn-classical-prime)
(int-rifn int-rcdfn-classical-prime)
(forall-partitions-riemann-sum-is-close-to-int-rifn
forall-partitions-riemann-sum-is-close-to-int-rcdfn-classical-prime)
(forall-partitions-riemann-sum-is-close-to-int-rifn-witness
forall-partitions-riemann-sum-is-close-to-int-rcdfn-classical-prime-witness)
(exists-delta-so-that-riemann-sum-is-close-to-int-rifn
exists-delta-so-that-riemann-sum-is-close-to-int-rcdfn-classical-prime)
(exists-delta-so-that-riemann-sum-is-close-to-int-rifn-witness
exists-delta-so-that-riemann-sum-is-close-to-int-rcdfn-classical-prime-witness)))
("Subgoal 8"
:use ((:instance exists-delta-so-that-riemann-sum-is-close-to-int-rcdfn-classical-prime-suff)))
("Subgoal 6"
:use ((:instance forall-partitions-riemann-sum-is-close-to-int-rcdfn-classical-prime-necc)))
("Subgoal 4"
:use ((:instance strict-int-rcdfn-classical-prime-is-integral-of-rcdfn-classical-prime)))
("Subgoal 3"
:use ((:instance rcdfn-classical-domain-non-trivial)))
))
(defthm ftc-2-for-rcdfn-classical
(implies (and (inside-interval-p a (rcdfn-classical-domain))
(inside-interval-p b (rcdfn-classical-domain)))
(equal (int-rcdfn-classical-prime a b)
(- (rcdfn-classical b)
(rcdfn-classical a))))
:hints (("Goal"
:by (:functional-instance ftc-2
(rcdfn rcdfn-classical)
(rcdfn-prime rcdfn-classical-prime)
(rcdfn-domain rcdfn-classical-domain)
(map-rcdfn-prime map-rcdfn-classical-prime)
(riemann-rcdfn-prime riemann-rcdfn-classical-prime)
(strict-int-rcdfn-prime strict-int-rcdfn-classical-prime)
(int-rcdfn-prime int-rcdfn-classical-prime)))
)
)
|
