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|
; Copyright (C) 2007 by Shant Harutunian
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
(include-book "arith-nsa4")
(include-book "nsa")
(include-book "eexp")
(defstub f (x) t)
(defstub L () t)
(defstub h (x1 x2 eps) t)
(defaxiom L-type
(and
(realp (L))
(< 0 (L)))
:rule-classes :type-prescription)
(defaxiom L-standard-thm
(standard-numberp (L))
:rule-classes :type-prescription)
(defaxiom L-i-limited-thm
(i-limited (L))
:rule-classes ((:type-prescription) (:rewrite)))
(defaxiom f-type
(realp (f x))
:rule-classes :type-prescription)
(defaxiom f-standard-thm
(implies
(and
(realp x)
(standard-numberp x))
(standard-numberp (f x)))
:rule-classes :type-prescription)
(defaxiom f-i-limited-thm
(implies
(and
(realp x)
(i-limited x))
(i-limited (f x)))
:rule-classes ((:type-prescription) (:rewrite)))
(defaxiom f-lim-thm
(implies
(and
(realp x1)
(realp x2))
(and
(<= (abs (- (f x1) (f x2)))
(* (L) (abs (- x1 x2)))))))
(defun step1 (x eps)
(+ x (* (f x) eps)))
(defun run (x n eps)
(cond
((zp n) x)
(t (run (step1 x eps) (- n 1) eps))))
(defun run-limit (delta n eps)
(cond
((zp n) delta)
(t (* (+ 1 (* eps (L))) (run-limit delta (- n 1) eps)))))
(defthm run-limit-realp
(implies
(and
(realp delta)
(realp eps))
(realp (run-limit delta n eps)))
:rule-classes :type-prescription)
(defthm run-limit-n+1-thm
(implies
(and
(realp eps)
(realp delta)
(<= 0 delta)
(< 0 eps)
(integerp n)
(<= 0 n))
(<= (run-limit delta n eps)
(run-limit delta (+ n 1) eps)))
:hints (("Goal" :in-theory (disable distributivity-left))))
; Added by Matt K.: Avoids rewriter loop in abs-step-thm.
(local (in-theory (disable commutativity-2-of-+)))
(defthm abs-step-thm
(implies
(and
(realp x1)
(realp x2)
(realp eps)
(< 0 eps))
(<= (abs (- (step1 x1 eps)
(step1 x2 eps)))
(+ (abs (- x1 x2))
(abs (* eps (- (f x1) (f x2)))))))
:rule-classes :linear)
(defthm step-1+leps-thm-1
(implies
(and
(realp x1)
(realp x2)
(realp eps)
(< 0 eps))
(<= (abs (- (step1 x1 (/ eps))
(step1 x2 (/ eps))))
(* (+ 1 (* (L) (/ eps))) (abs (- x1 x2)))))
:hints (("Goal"
:use ((:instance f-lim-thm)
(:instance abs-step-thm (eps (/ eps))))))
:rule-classes nil)
(defthm step-1+leps-thm
(implies
(and
(realp x1)
(realp x2)
(realp eps)
(< 0 eps))
(<= (abs (- (step1 x1 eps)
(step1 x2 eps)))
(* (+ 1 (* (L) eps)) (abs (- x1 x2)))))
:hints (("Goal"
:use ((:instance step-1+leps-thm-1 (eps (/ eps)))))))
(defthm run-realp
(implies
(and
(realp x)
(realp eps))
(realp (run x n eps)))
:rule-classes :type-prescription)
(defun n-scheme (n)
(cond
((zp n) 0)
(t (n-scheme (- n 1)))))
(defthm step-run-thm
(implies
(and
(integerp n)
(<= 0 n)
(realp x)
(realp eps))
(equal (step1 (run x n eps) eps)
(run x (+ n 1) eps))))
(defthm run-limit-thm
(implies
(and
(realp x1)
(realp x2)
(realp eps)
(< 0 eps))
(<= (abs (- (run x1 n eps) (run x2 n eps)))
(run-limit (abs (- x1 x2)) n eps)))
:hints (("Goal" :do-not '(generalize)
:induct (n-scheme n)
:in-theory (disable abs)
:nonlinearp t)
("Subgoal *1/2" :in-theory (disable abs <-*-LEFT-CANCEL)
:use ((:instance run-limit-n+1-thm
(n (- n 1))
(delta (abs (- (step1 x1 eps)
(step1 x2 eps)))))
(:instance step-1+leps-thm
(x1 (run x1 (- n 1) eps))
(x2 (run x2 (- n 1) eps)))
(:instance pos-factor-<=-thm
(x (ABS (+ (RUN X1 (+ -1 N) eps)
(* -1 (RUN X2
(+ -1 N)
eps)))))
(y (RUN-LIMIT (ABS (+ X1 (* -1 X2)))
(+ -1 N) eps))
(a (+ 1 (* (L) EPS))))))))
(defthm eexp-1+epsl-thm
(implies
(and
(realp eps)
(integerp n)
(< 0 eps)
(realp delta)
(<= 0 delta)
(<= 0 n))
(<= (* delta (eexp (* (- n 1) eps (L)))
(+ 1 (* eps (L))))
(* delta (eexp (* n eps (L))))))
:hints (("Goal" :in-theory (disable 1+x-<=eexp-thm)
:use ((:instance 1+x-<=eexp-thm (x (* eps (L))))
(:instance pos-factor-<=-thm
(x (+ 1 (* eps (L))))
(y (eexp (* eps (L))))
(a (* delta
(eexp (+ (* n eps (L))
(* -1 eps (L))))))))))
:rule-classes nil)
(defthm run-limit-eexp-thm-1
(implies
(and
(realp eps)
(realp delta)
(<= 0 delta)
(< 0 eps)
(integerp n)
(<= 0 n))
(<= (run-limit delta n eps)
(* delta (eexp (* eps (L) n)))))
:hints (("Goal" :do-not '(generalize))
("Subgoal *1/4" :use ((:instance pos-factor-<=-thm
(x (RUN-LIMIT DELTA (+ -1 N) eps))
(y (* DELTA
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS N)))))
(a (+ 1 (* eps (L)))))
(:instance eexp-1+epsl-thm))))
:rule-classes nil)
(defthm run-diff-limit-eexp-thm
(implies
(and
(realp eps)
(< 0 eps)
(realp x1)
(realp x2)
(integerp n)
(<= 0 n))
(<= (abs (- (run x1 n eps) (run x2 n eps)))
(* (abs (- x1 x2)) (eexp (* eps (L) n)))))
:hints (("Goal" :do-not '(generalize))
("Subgoal *1/2" :in-theory (disable abs)
:use ((:instance run-limit-thm)
(:instance run-limit-eexp-thm-1
(delta (abs (- x1 x2)))))))
:rule-classes nil)
(defthm run-plus-thm
(implies
(and
(integerp m)
(integerp n)
(<= 0 m)
(<= 0 n))
(equal (run (run x n eps) m eps)
(run x (+ m n) eps))))
(defun run-n-limit (x n eps)
(+ (abs x)
(* (eexp (* (L) n eps)) (abs (f x)) n eps)))
(defthm f-step-thm-hint1
(implies
(and
(realp eps)
(< 0 eps)
(realp x))
(<= (abs (f (step1 x eps)))
(+ (abs (f x)) (abs (- (f (step1 x eps)) (f x))))))
:rule-classes nil)
(defthm f-step-thm-hint2
(implies
(and
(realp eps)
(< 0 eps)
(realp x))
(<= (abs (f (step1 x eps)))
(* (eexp (* (L) eps)) (abs (f x)))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:use ((:instance f-step-thm-hint1)
(:instance f-lim-thm (x1 (+ x (* (f x) eps)))
(x2 x))
(:instance 1+x-<=eexp-thm (x (* (L) eps)))
(:instance pos-factor-<=-thm (x (+ 1 (* eps (L))))
(y (eexp (* eps (L) )))
(a (abs (f x))))))))
(defthm arith-3
(implies
(and
(integerp n)
(< 0 n))
(<= 0 (- n 1)))
:rule-classes :type-prescription)
(defthm f-step-thm-hint3
(implies
(and
(integerp n)
(< 0 n)
(realp x)
(realp eps)
(< 0 eps))
(<= 0
(* (EEXP (* -1 EPS (L)))
(EEXP (* EPS (L) N))
(+ -1 N)
EPS)))
:rule-classes nil)
(defthm step1-type-thm
(implies
(and
(realp x)
(realp eps))
(realp (step1 x eps)))
:rule-classes :type-prescription)
(defthm abs-step1-thm
(implies
(and
(realp eps)
(< 0 eps)
(realp x))
(<= (abs (step1 x eps))
(+ (abs x)
(* (abs (f x)) eps))))
:rule-classes nil)
(defthm run-limit-eexp-step-thm
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(< 0 n)
(realp x))
(<= (run-n-limit (step1 x eps)
(- n 1)
eps)
(run-n-limit x n eps)))
:rule-classes :linear
:hints (("Goal" :in-theory (disable abs <-*-LEFT-CANCEL)
:use ((:instance abs-step1-thm)
(:instance f-step-thm-hint3)
(:instance f-step-thm-hint2)
(:instance pos-factor-<=-thm (x 1)
(y (eexp (* eps (L) n)))
(a (* (abs (f x)) eps)))
(:instance pos-factor-<=-thm
(x (abs (f (step1 x eps))))
(y (* (eexp (* (L) eps)) (abs (f x))))
(a (* (eexp (* -1 eps (L)))
(eexp (* eps (L) n))
(- n 1) eps)))))))
(defthm run-limit-eexpt-thm
(implies
(and
(realp eps)
(< 0 eps)
(realp x)
(integerp n)
(<= 0 n))
(<= (abs (run x n eps))
(run-n-limit x n eps)))
:rule-classes nil
:hints (("Goal" :in-theory (disable step1 run-n-limit abs))
("Subgoal *1/1" :in-theory (e/d (step1 run-n-limit) (abs)))))
(defun run-tm-limit (x tm)
(+ (abs x)
(* (eexp (* (L) tm)) (abs (f x)) tm)))
;; Cuong Chau: Add the :use hint to the following theorem.
(defthm run-tm-limit-standard-thm
(implies
(and
(realp tm)
(standard-numberp tm)
(realp x)
(standard-numberp x))
(standard-numberp (run-tm-limit x tm)))
:hints (("Goal" :use ((:instance EEXP-STANDARD-THM
(x (* (L) tm)))
(:instance L-STANDARD-THM)
(:instance F-STANDARD-THM))))
:rule-classes nil)
(defthm run-n-limit-standard-thm
(implies
(and
(realp eps)
(integerp n)
(standard-numberp (* eps n))
(realp x)
(standard-numberp x))
(standard-numberp (run-n-limit x n eps)))
:rule-classes nil
:hints (("Goal" :use ((:instance run-tm-limit-standard-thm
(tm (* eps n)))))))
(defthm run-tm-limit-limited-thm
(implies
(and
(realp tm)
(i-limited tm)
(realp x)
(i-limited x))
(i-limited (run-tm-limit x tm)))
:rule-classes nil
:hints (("Goal" :in-theory (disable i-large)
:use ((:instance standards-are-limited)))))
(defthm run-n-limit-limited-thm
(implies
(and
(realp eps)
(integerp n)
(i-limited (* eps n))
(realp x)
(i-limited x))
(i-limited (run-n-limit x n eps)))
:rule-classes :type-prescription
:hints (("Goal" :use ((:instance run-tm-limit-limited-thm
(tm (* eps n)))))))
(defthm run-n-limit-type-thm
(implies
(and
(realp x)
(realp eps)
(integerp n))
(realp (run-n-limit x n eps)))
:rule-classes :type-prescription)
(defthm run-standard-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n)
(i-limited (* eps n))
(standard-numberp x))
(standard-numberp (standard-part (run x n eps))))
:hints (("Goal" :in-theory (disable run
run-n-limit
plus-limited
times-limited
divide-limited)
:use ((:instance run-limit-eexpt-thm)
(:instance run-n-limit-limited-thm)
(:instance limited-bound-x-implies-limited-x-thm
(y (run-n-limit x n eps))
(x (run x n eps)))
(:instance standardp-standard-part
(x (run x n eps)))))))
(defthm floor-limited-thm-hint-1
(implies (and (i-small (/ (i-large-integer)))
(realp tm)
(i-limited tm))
(i-limited (+ (* -1 (/ (i-large-integer))) tm)))
:rule-classes nil
:hints (("goal" :in-theory (disable i-large
i-small
/-large-integer-is-ismall-thm))))
(defthm floor-limited-thm
(implies
(and
(realp tm)
(i-limited tm))
(i-limited (* (/ (i-large-integer)) (floor1 (* (i-large-integer) tm)) )))
:hints (("goal" :in-theory (disable i-large)
:use ((:instance sandwich-limited-thm
(u (/ (- (* tm (i-large-integer)) 1)
(i-large-integer)))
(v (/ (* tm (i-large-integer))
(i-large-integer)))
(x (* (/ (i-large-integer))
(floor1 (* (i-large-integer) tm)) )))))
("subgoal 2"
:use ((:instance /-large-integer-is-ismall-thm)
(:instance floor-limited-thm-hint-1)))))
(defthm phi-thm
(implies
(and (standard-numberp x)
(standard-numberp tm)
(realp x)
(realp tm))
(standard-numberp (standard-part (run x (floor1 (* (i-large-integer) tm))
(/ (i-large-integer))))))
:hints (("goal" :in-theory (disable i-large)
:use ((:instance run-standard-thm
(n (floor1 (* tm (i-large-integer))))
(eps (/ (i-large-integer))))
(:instance floor-limited-thm)))))
; The following is the definition of the
; standard function
(defun-std phi (x tm)
(cond
((not (and
(realp x)
(realp tm))) 0)
(t (standard-part (run x
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))))
|
