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|
(in-package "ACL2")
(include-book "continuity")
(include-book "intervals")
(local (include-book "equivalence-limits"))
(include-book "continuity")
;; Part 1: The nonstd definition of continuity implies the classical definition
(defun-sk forall-x-within-delta-of-x0-f-x-within-epsilon-of-f (x0 eps delta)
(forall (x)
(implies (and (inside-interval-p x (rcfn-domain))
(inside-interval-p x0 (rcfn-domain))
(realp delta)
(< 0 delta)
(realp eps)
(< 0 eps)
(< (abs (- x x0)) delta)
(not (equal x x0)))
(< (abs (- (rcfn x) (rcfn x0))) eps)))
:rewrite :direct)
(defun-sk exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcfn-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-f x0 eps delta))))
:classicalp nil)
(defthmd rcfn-is-continuous-using-classical-criterion
(implies (and (inside-interval-p x0 (rcfn-domain))
(standardp x0)
(realp eps)
(standardp eps)
(< 0 eps))
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f x0 eps))
:hints (("Goal"
:by (:functional-instance rlfn-classic-has-a-limit-using-classical-criterion
(rlfn rcfn)
(rlfn-limit rcfn)
(rlfn-domain rcfn-domain)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-limit exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-limit-witness exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f-witness)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-limit forall-x-within-delta-of-x0-f-x-within-epsilon-of-f)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-limit-witness forall-x-within-delta-of-x0-f-x-within-epsilon-of-f-witness)
)
)
("Subgoal 7"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f-suff))
:in-theory (disable exists-standard-delta-for-x0-to-make-x-within-epsilon-of-f-suff
abs))
("Subgoal 5"
:use ((:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-f-necc))
:in-theory (disable forall-x-within-delta-of-x0-f-x-within-epsilon-of-f-necc
abs))
("Subgoal 2"
:use ((:instance rcfn-domain-non-trivial)))
)
)
;; Part 2: The classical definition of continuity implies the nonstd definition
(encapsulate
((rcfn-classical (x) t)
(rcfn-classical-domain () t))
(local (defun rcfn-classical (x) (declare (ignore x)) 0))
(local (defun rcfn-classical-domain () (interval nil nil)))
(defthm intervalp-rcfn-classical-domain
(interval-p (rcfn-classical-domain))
:rule-classes (:type-prescription :rewrite))
(defthm rcfn-classical-domain-real
(implies (inside-interval-p x (rcfn-classical-domain))
(realp x))
:rule-classes (:forward-chaining))
(defthm rcfn-classical-domain-non-trivial
(or (null (interval-left-endpoint (rcfn-classical-domain)))
(null (interval-right-endpoint (rcfn-classical-domain)))
(< (interval-left-endpoint (rcfn-classical-domain))
(interval-right-endpoint (rcfn-classical-domain))))
:rule-classes nil)
(defthm rcfn-classical-real
(realp (rcfn-classical x))
:rule-classes (:rewrite :type-prescription))
(defun-sk forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f (x0 eps delta)
(forall (x)
(implies (and (inside-interval-p x (rcfn-classical-domain))
(inside-interval-p x0 (rcfn-classical-domain))
(realp delta)
(< 0 delta)
(realp eps)
(< 0 eps)
(< (abs (- x x0)) delta)
(not (equal x x0)))
(< (abs (- (rcfn-classical x) (rcfn-classical x0))) eps)))
:rewrite :direct)
(defun-sk exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcfn-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-classical-f x0 eps delta))))
:classicalp nil)
(defthmd rcfn-classic-is-continuous
(implies (and (inside-interval-p x0 (rcfn-classical-domain))
(standardp x0)
(realp eps)
(standardp eps)
(< 0 eps))
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f x0 eps))
:hints (("Goal"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f-suff
(delta 1))
(:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f
(x0 x0)
(eps eps)
(delta 1)))
:in-theory (disable abs))))
)
(defthm rcfn-classical-is-continuous-using-nonstandard-criterion
(implies (and (standardp x0)
(inside-interval-p x0 (rcfn-classical-domain))
(i-close x x0)
(inside-interval-p x (rcfn-classical-domain))
(not (equal x x0)))
(i-close (rcfn-classical x) (rcfn-classical x0)))
:hints (("Goal"
:by (:functional-instance rlfn-classical-has-a-limit-using-nonstandard-criterion
(rlfn-classical rcfn-classical)
(rlfn-classical-limit rcfn-classical)
(rlfn-classical-domain rcfn-classical-domain)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-limit forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f)
(forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-limit-witness forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f-witness)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-limit exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f)
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-limit-witness exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f-witness)
)
)
("Subgoal 7"
:by (:instance rcfn-classic-is-continuous))
("Subgoal 5"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-classical-f-suff)))
("Subgoal 3"
:use ((:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-classical-f-necc)))
("Subgoal 2"
:use ((:instance rcfn-classical-domain-non-trivial)))
))
;; Corollaries: Show the intermediate value theorem and extreme value theorems hold
;; for the classical definition
(defun-sk exists-intermediate-point-classical (a b z)
(exists (x)
(and (realp x)
(< a x)
(< x b)
(equal (rcfn-classical x) z))))
(defthm intermediate-value-theorem-classical-sk
(implies (and (inside-interval-p a (rcfn-classical-domain))
(inside-interval-p b (rcfn-classical-domain))
(realp z)
(< a b)
(or (and (< (rcfn-classical a) z) (< z (rcfn-classical b)))
(and (< z (rcfn-classical a)) (< (rcfn-classical b) z))))
(exists-intermediate-point-classical a b z))
:hints (("Goal"
:by (:functional-instance intermediate-value-theorem-sk
(rcfn rcfn-classical)
(rcfn-domain rcfn-classical-domain)
(exists-intermediate-point exists-intermediate-point-classical)
(exists-intermediate-point-witness exists-intermediate-point-classical-witness)))
("Subgoal 3"
:use ((:instance rcfn-classical-is-continuous-using-nonstandard-criterion
(x0 x)
(x y))))
("Subgoal 2"
:use ((:instance rcfn-classical-domain-non-trivial)))))
(defun-sk is-maximum-point-of-rcfn-classical (a b max)
(forall (x)
(implies (and (realp x)
(<= a x)
(<= x b))
(<= (rcfn-classical x) (rcfn-classical max)))))
(defun-sk rcfn-classical-achieves-maximum-point (a b)
(exists (max)
(implies (and (realp a)
(realp b)
(<= a b))
(and (realp max)
(<= a max)
(<= max b)
(is-maximum-point-of-rcfn-classical a b max)))))
(defthm maximum-point-theorem-classical-sk
(implies (and (inside-interval-p a (rcfn-classical-domain))
(inside-interval-p b (rcfn-classical-domain))
(< a b))
(rcfn-classical-achieves-maximum-point a b))
:hints (("Goal"
:by (:functional-instance maximum-point-theorem-sk
(rcfn rcfn-classical)
(rcfn-domain rcfn-classical-domain)
(is-maximum-point is-maximum-point-of-rcfn-classical)
(is-maximum-point-witness is-maximum-point-of-rcfn-classical-witness)
(achieves-maximum-point rcfn-classical-achieves-maximum-point)
(achieves-maximum-point-witness rcfn-classical-achieves-maximum-point-witness)
)
)
("Subgoal 4"
:use ((:instance rcfn-classical-achieves-maximum-point-suff)))
("Subgoal 2"
:use ((:instance is-maximum-point-of-rcfn-classical-necc)))))
(defun-sk is-minimum-point-of-rcfn-classical (a b min)
(forall (x)
(implies (and (realp x)
(<= a x)
(<= x b))
(<= (rcfn-classical min) (rcfn-classical x)))))
(defun-sk rcfn-classical-achieves-minimum-point (a b)
(exists (min)
(implies (and (realp a)
(realp b)
(<= a b))
(and (realp min)
(<= a min)
(<= min b)
(is-minimum-point-of-rcfn-classical a b min)))))
(defthm minimum-point-theorem-classical-sk
(implies (and (inside-interval-p a (rcfn-classical-domain))
(inside-interval-p b (rcfn-classical-domain))
(< a b))
(rcfn-classical-achieves-minimum-point a b))
:hints (("Goal"
:by (:functional-instance minimum-point-theorem-sk
(rcfn rcfn-classical)
(rcfn-domain rcfn-classical-domain)
(is-minimum-point is-minimum-point-of-rcfn-classical)
(is-minimum-point-witness is-minimum-point-of-rcfn-classical-witness)
(achieves-minimum-point rcfn-classical-achieves-minimum-point)
(achieves-minimum-point-witness rcfn-classical-achieves-minimum-point-witness)
)
)
("Subgoal 4"
:use ((:instance rcfn-classical-achieves-minimum-point-suff)))
("Subgoal 2"
:use ((:instance is-minimum-point-of-rcfn-classical-necc)))))
;; Part 3: The hyperreal definition of continuity implies the classical definition
(encapsulate
((rcfn-hyper (x) t)
(rcfn-hyper-domain () t))
(local (defun rcfn-hyper (x) (declare (ignore x)) 0))
(local (defun rcfn-hyper-domain () (interval nil nil)))
(defthm intervalp-rcfn-hyper-domain
(interval-p (rcfn-hyper-domain))
:rule-classes (:type-prescription :rewrite))
(defthm rcfn-hyper-domain-real
(implies (inside-interval-p x (rcfn-hyper-domain))
(realp x))
:rule-classes (:forward-chaining))
(defthm rcfn-hyper-domain-non-trivial
(or (null (interval-left-endpoint (rcfn-hyper-domain)))
(null (interval-right-endpoint (rcfn-hyper-domain)))
(< (interval-left-endpoint (rcfn-hyper-domain))
(interval-right-endpoint (rcfn-hyper-domain))))
:rule-classes nil)
(defthm rcfn-hyper-real
(realp (rcfn-hyper x))
:rule-classes (:rewrite :type-prescription))
(defun-sk forall-x-within-delta-of-x0-f-x-within-epsilon-of-hyper-f (x0 eps delta)
(forall (x)
(implies (and (inside-interval-p x (rcfn-hyper-domain))
(inside-interval-p x0 (rcfn-hyper-domain))
(realp delta)
(< 0 delta)
(realp eps)
(< 0 eps)
(< (abs (- x x0)) delta)
(not (equal x x0)))
(< (abs (- (rcfn-hyper x) (rcfn-hyper x0))) eps)))
:rewrite :direct)
(defun-sk exists-delta-for-x0-to-make-x-within-epsilon-of-hyper-f (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcfn-hyper-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-hyper-f x0 eps delta)))))
(defthmd rcfn-hyper-is-continuous
(implies (and (inside-interval-p x0 (rcfn-hyper-domain))
;(standardp x0)
(realp eps)
;(standardp eps)
(< 0 eps))
(exists-delta-for-x0-to-make-x-within-epsilon-of-hyper-f x0 eps))
:hints (("Goal"
:use ((:instance exists-delta-for-x0-to-make-x-within-epsilon-of-hyper-f-suff
(delta 1))
(:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-hyper-f
(x0 x0)
(eps eps)
(delta 1)))
:in-theory (disable abs))))
)
(defun-sk exists-standard-delta-for-x0-to-make-x-within-epsilon-of-hyper-f (x0 eps)
(exists (delta)
(implies (and (inside-interval-p x0 (rcfn-hyper-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-hyper-f x0 eps delta))))
:classicalp nil)
(defthm-std standard-exists-delta-for-x0-to-make-x-within-epsilon-of-hyper-f-witness
(implies (and (standardp x0)
(standardp eps))
(standardp (EXISTS-DELTA-FOR-X0-TO-MAKE-X-WITHIN-EPSILON-OF-HYPER-F-WITNESS
X0 EPS))))
(defthmd rcfn-hyper-is-continuous-using-standard-criterion
(implies (and (inside-interval-p x0 (rcfn-hyper-domain))
(standardp x0)
(realp eps)
(standardp eps)
(< 0 eps))
(exists-standard-delta-for-x0-to-make-x-within-epsilon-of-hyper-f x0 eps))
:hints (("Goal"
:use ((:instance rcfn-hyper-is-continuous)
(:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-hyper-f-suff
(delta (EXISTS-DELTA-FOR-X0-TO-MAKE-X-WITHIN-EPSILON-OF-HYPER-F-WITNESS
X0 EPS))))
:in-theory (disable rcfn-hyper-is-continuous
forall-x-within-delta-of-x0-f-x-within-epsilon-of-hyper-f)
)))
(defthm rcfn-hyper-is-continuous-using-nonstandard-criterion
(implies (and (standardp x0)
(inside-interval-p x0 (rcfn-hyper-domain))
(i-close x x0)
(inside-interval-p x (rcfn-hyper-domain))
(not (equal x x0)))
(i-close (rcfn-hyper x) (rcfn-hyper x0)))
:hints (("Goal"
:by (:functional-instance rcfn-classical-is-continuous-using-nonstandard-criterion
(rcfn-classical rcfn-hyper)
(rcfn-classical-domain rcfn-hyper-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-hyper-f)
(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-hyper-f-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-hyper-F)
(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-hyper-F-witness)))
("Subgoal 7"
:use ((:instance rcfn-hyper-is-continuous-using-standard-criterion)))
("Subgoal 5"
:use ((:instance exists-standard-delta-for-x0-to-make-x-within-epsilon-of-hyper-f-suff)))
("Subgoal 3"
:use ((:instance forall-x-within-delta-of-x0-f-x-within-epsilon-of-hyper-f-necc)))
("Subgoal 2"
:use ((:instance rcfn-hyper-domain-non-trivial)))
))
;; Corollaries: Show the intermediate value theorem and extreme value theorems hold
;; for the hyperreal definition
(defun-sk exists-intermediate-point-hyper (a b z)
(exists (x)
(and (realp x)
(< a x)
(< x b)
(equal (rcfn-hyper x) z))))
(defthm intermediate-value-theorem-hyper-sk
(implies (and (inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(realp z)
(< a b)
(or (and (< (rcfn-hyper a) z) (< z (rcfn-hyper b)))
(and (< z (rcfn-hyper a)) (< (rcfn-hyper b) z))))
(exists-intermediate-point-hyper a b z))
:hints (("Goal"
:by (:functional-instance intermediate-value-theorem-classical-sk
(rcfn-classical rcfn-hyper)
(rcfn-classical-domain rcfn-hyper-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-hyper-f)
(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-hyper-f-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-hyper-F)
(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-hyper-F-witness)
(exists-intermediate-point-classical exists-intermediate-point-hyper)
(exists-intermediate-point-classical-witness exists-intermediate-point-hyper-witness)))
))
(defun-sk is-maximum-point-of-rcfn-hyper (a b max)
(forall (x)
(implies (and (realp x)
(<= a x)
(<= x b))
(<= (rcfn-hyper x) (rcfn-hyper max)))))
(defun-sk rcfn-hyper-achieves-maximum-point (a b)
(exists (max)
(implies (and (realp a)
(realp b)
(<= a b))
(and (realp max)
(<= a max)
(<= max b)
(is-maximum-point-of-rcfn-hyper a b max)))))
(defthm maximum-point-theorem-hyper-sk
(implies (and (inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(< a b))
(rcfn-hyper-achieves-maximum-point a b))
:hints (("Goal"
:by (:functional-instance maximum-point-theorem-classical-sk
(rcfn-classical rcfn-hyper)
(rcfn-classical-domain rcfn-hyper-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-hyper-f)
(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-hyper-f-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-hyper-F)
(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-hyper-F-witness)
(is-maximum-point-of-rcfn-classical is-maximum-point-of-rcfn-hyper)
(is-maximum-point-of-rcfn-classical-witness is-maximum-point-of-rcfn-hyper-witness)
(rcfn-classical-achieves-maximum-point rcfn-hyper-achieves-maximum-point)
(rcfn-classical-achieves-maximum-point-witness rcfn-hyper-achieves-maximum-point-witness)
))
("Subgoal 4"
:use ((:instance rcfn-hyper-achieves-maximum-point-suff)))
("Subgoal 2"
:use ((:instance is-maximum-point-of-rcfn-hyper-necc)))
))
(defun-sk is-minimum-point-of-rcfn-hyper (a b min)
(forall (x)
(implies (and (realp x)
(<= a x)
(<= x b))
(<= (rcfn-hyper min) (rcfn-hyper x)))))
(defun-sk rcfn-hyper-achieves-minimum-point (a b)
(exists (min)
(implies (and (realp a)
(realp b)
(<= a b))
(and (realp min)
(<= a min)
(<= min b)
(is-minimum-point-of-rcfn-hyper a b min)))))
(defthm minimum-point-theorem-hyper-sk
(implies (and (inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(< a b))
(rcfn-hyper-achieves-minimum-point a b))
:hints (("Goal"
:by (:functional-instance minimum-point-theorem-classical-sk
(rcfn-classical rcfn-hyper)
(rcfn-classical-domain rcfn-hyper-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-hyper-f)
(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-hyper-f-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-hyper-F)
(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-hyper-F-witness)
(is-minimum-point-of-rcfn-classical is-minimum-point-of-rcfn-hyper)
(is-minimum-point-of-rcfn-classical-witness is-minimum-point-of-rcfn-hyper-witness)
(rcfn-classical-achieves-minimum-point rcfn-hyper-achieves-minimum-point)
(rcfn-classical-achieves-minimum-point-witness rcfn-hyper-achieves-minimum-point-witness)
))
("Subgoal 4"
:use ((:instance rcfn-hyper-achieves-minimum-point-suff)))
("Subgoal 2"
:use ((:instance is-minimum-point-of-rcfn-hyper-necc)))))
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