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(in-package "ACL2")
(local (include-book "arithmetic-5/top" :dir :system))
(include-book "integrable-functions")
(include-book "continuous-function")
(include-book "equivalence-integrals")
(include-book "nonstd/nsa/equivalence-continuity" :dir :system)
(defun map-rcfn-hyper (p)
(if (consp p)
(cons (rcfn-hyper (car p))
(map-rcfn-hyper (cdr p)))
nil))
(defthm real-listp-map-rcfn-hyper
(implies (partitionp p)
(real-listp (map-rcfn-hyper p))))
(defun riemann-rcfn-hyper (p)
(dotprod (deltas p)
(map-rcfn-hyper (cdr p))))
(defthm real-riemann-rcfn-hyper
(implies (partitionp p)
(realp (riemann-rcfn-hyper p))))
(defthm limited-riemann-rcfn-hyper-small-partition
(implies (and (realp a) (standardp a)
(realp b) (standardp b)
(inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(< a b))
(i-limited (riemann-rcfn-hyper (make-small-partition a b))))
:hints (("Goal"
:by (:functional-instance limited-riemann-rcfn-small-partition
(rcfn rcfn-hyper)
(rcfn-domain rcfn-hyper-domain)
(map-rcfn map-rcfn-hyper)
(riemann-rcfn riemann-rcfn-hyper)))
("Subgoal 3"
:use ((:instance rcfn-hyper-is-continuous-using-nonstandard-criterion
(x0 x)
(x y))))
("Subgoal 2"
:use ((:instance rcfn-hyper-domain-non-trivial)))
))
(encapsulate
nil
(local (in-theory (disable riemann-rcfn-hyper)))
(defun-std strict-int-rcfn-hyper (a b)
(if (and (realp a)
(realp b)
(inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(< a b))
(standard-part (riemann-rcfn-hyper (make-small-partition a b)))
0))
)
(defthm-std realp-strict-int-rcfn-hyper
(implies (and (realp a)
(realp b))
(realp (strict-int-rcfn-hyper a b)))
:hints (("Goal"
:use ((:instance real-riemann-rcfn-hyper
(p (make-small-partition a b))))
:in-theory (disable riemann-rcfn-hyper
real-riemann-rcfn-hyper))))
(defun int-rcfn-hyper (a b)
(if (<= a b)
(strict-int-rcfn-hyper a b)
(- (strict-int-rcfn-hyper b a))))
(defthm realp-int-rcfn-nyper
(implies (and (realp a)
(realp b))
(realp (int-rcfn-hyper a b))))
(defthm strict-int-rcfn-hyper-is-integral-of-rcfn-hyper
(implies (and (standardp a)
(standardp b)
(<= a b)
(inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(partitionp p)
(equal (car p) a)
(equal (car (last p)) b)
(i-small (mesh p)))
(i-close (riemann-rcfn-hyper p)
(strict-int-rcfn-hyper a b)))
:hints (("Goal"
:do-not-induct t
:by (:functional-instance strict-int-rcfn-is-integral-of-rcfn
(rcfn rcfn-hyper)
(rcfn-domain rcfn-hyper-domain)
(map-rcfn map-rcfn-hyper)
(riemann-rcfn riemann-rcfn-hyper)
(strict-int-rcfn strict-int-rcfn-hyper)))
))
(defun-sk forall-partitions-riemann-sum-is-close-to-int-rcfn-hyper (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-rcfn-hyper p)
(strict-int-rcfn-hyper a b)))
eps)))
:rewrite :direct)
(defun-sk exists-delta-so-that-riemann-sum-is-close-to-int-rcfn-hyper (a b eps)
(exists (delta)
(implies (and (inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(<= a b)
(realp eps)
(< 0 eps))
(and (realp delta)
(< 0 delta)
(forall-partitions-riemann-sum-is-close-to-int-rcfn-hyper a b eps delta)))))
(defthm rcfn-hyper-is-integrable-hyperreal
(implies (and (inside-interval-p a (rcfn-hyper-domain))
(inside-interval-p b (rcfn-hyper-domain))
(<= a b)
(realp eps)
(< 0 eps))
(exists-delta-so-that-riemann-sum-is-close-to-int-rcfn-hyper a b eps))
:hints (("Goal"
:by (:functional-instance rifn-is-integrable-hyperreal
(rifn rcfn-hyper)
(domain-rifn rcfn-hyper-domain)
(map-rifn map-rcfn-hyper)
(riemann-rifn riemann-rcfn-hyper)
(strict-int-rifn strict-int-rcfn-hyper)
(exists-delta-so-that-riemann-sum-is-close-to-int-rifn
exists-delta-so-that-riemann-sum-is-close-to-int-rcfn-hyper)
(exists-delta-so-that-riemann-sum-is-close-to-int-rifn-witness
exists-delta-so-that-riemann-sum-is-close-to-int-rcfn-hyper-witness)
(forall-partitions-riemann-sum-is-close-to-int-rifn
forall-partitions-riemann-sum-is-close-to-int-rcfn-hyper)
(forall-partitions-riemann-sum-is-close-to-int-rifn-witness
forall-partitions-riemann-sum-is-close-to-int-rcfn-hyper-witness)
))
("Subgoal 8"
:use ((:instance EXISTS-DELTA-SO-THAT-RIEMANN-SUM-IS-CLOSE-TO-INT-RCFN-HYPER-suff)))
("Subgoal 6"
:use ((:instance FORALL-PARTITIONS-RIEMANN-SUM-IS-CLOSE-TO-INT-RCFN-HYPER-necc)))
("Subgoal 4"
:use ((:instance strict-int-rcfn-hyper-is-integral-of-rcfn-hyper)))
("Subgoal 3"
:use ((:instance rcfn-hyper-domain-non-trivial)))
))
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