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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 "abs")
(include-book "eexp")
(include-book "phi-exists")
(include-book "phi-properties")
(include-book "computed-hints")
(include-book "tm-floor")
(in-theory (disable i-large))
(in-theory (disable standard-part-<=))
(defstub phi2 (tm) t)
(defaxiom phi2-type
(implies
(and
(realp tm))
(realp (phi2 tm)))
:rule-classes :type-prescription)
(defaxiom phi2-standard-thm
(implies
(and
(realp tm)
(standard-numberp tm))
(standard-numberp (phi2 tm)))
:rule-classes ((:type-prescription) (:rewrite)))
(defun resid2-tm (tm eps)
(+ (phi2 (+ tm eps))
(- (phi2 tm))
(- (* eps (f (phi2 tm))))))
(defthm resid2-tm-type
(implies
(and
(realp tm)
(realp eps))
(realp (resid2-tm tm eps)))
:rule-classes :type-prescription)
(defaxiom phi2-deriv
(implies
(and
(realp tm)
(i-limited tm)
(realp eps)
(not (equal eps 0))
(i-small eps))
(equal (standard-part (/ (- (phi2 (+ tm eps)) (phi2 tm)) eps))
(standard-part (f (phi2 tm))))))
(defthm phi2-equal-thm
(implies
(and
(realp tm)
(realp eps))
(equal (+ (phi2 tm)
(* eps (f (phi2 tm)))
(resid2-tm tm eps))
(phi2 (+ tm eps)))))
(defthm resid2/eps-small-thm
(implies
(and
(realp tm)
(i-limited tm)
(realp eps)
(not (equal eps 0))
(i-small eps))
(equal (standard-part (/ (resid2-tm tm eps) eps))
0))
:hints (("Goal" :use ((:instance phi2-deriv)))))
(defthm /eps-gt-1-thm
(implies
(and
(realp eps)
(not (equal eps 0))
(equal (standard-part eps) 0))
(< 1 (abs (/ eps))))
:hints (("Goal" :use ((:instance standard-part-<= (y 1) (x (/ (abs eps))))
(:instance i-large (x (abs (/ eps))))))))
(defthm resid2-small-thm
(implies
(and
(realp tm)
(i-limited tm)
(realp eps)
(not (equal eps 0))
(i-small eps))
(equal (standard-part (resid2-tm tm eps))
0))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs resid2-tm abs-*-thm)
:use ((:instance resid2/eps-small-thm)
(:instance pos-factor-<-thm
(x 1)
(y (abs (/ eps)))
(a (abs (resid2-tm tm eps))))
(:instance abs-*-thm
(x (/ eps))
(y (resid2-tm tm eps)))))))
(defthm phi2-tm-continuous-thm
(implies
(and
(realp tm)
(i-limited tm))
(equal (standard-part (phi2 tm))
(phi2 (standard-part tm))))
:hints (("Goal" :in-theory (disable resid2-tm standardp-standard-part)
:cases ((equal tm (standard-part tm))
(not (equal tm (standard-part tm)))))
("Subgoal 1" :use ((:instance phi2-equal-thm
(eps (- tm (standard-part tm)))
(tm (standard-part tm)))
(:instance resid2-small-thm
(eps (- tm (standard-part tm)))
(tm (standard-part tm)))
(:instance standardp-standard-part
(x tm))
(:instance standards-are-limited
(x (phi2 (standard-part tm))))))
("Subgoal 2" :use ((:instance standardp-standard-part
(x tm))))))
(defthm standardp-standard-part-limited
(implies
(and
(acl2-numberp x)
(standard-numberp (standard-part x)))
(i-limited x))
:hints (("Goal" :use ((:instance standard+small->i-limited
(x (standard-part x))
(eps (- x (standard-part x))))))))
(defthm phi2-limited-thm
(implies
(and
(realp tm)
(i-limited tm))
(i-limited (phi2 tm)))
:rule-classes ((:type-prescription) (:rewrite))
:hints (("Goal" :use ((:instance standardp-standard-part-limited
(x (phi2 tm)))))))
(defthm resid2-tm-limited
(implies
(and
(realp eps)
(realp tm)
(i-limited tm)
(i-small eps))
(i-limited (resid2-tm tm eps)))
:rule-classes ((:type-prescription) (:rewrite)))
(defun max-abs-resid2-tm (n eps)
(cond
((zp n) (abs (resid2-tm 0 eps)))
(t (max (abs (resid2-tm (* n eps) eps))
(max-abs-resid2-tm (- n 1) eps)))))
(defthm max-abs-resid2-is-bound
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(integerp m)
(<= 0 m)
(<= m n))
(<= (abs (resid2-tm (* eps m) eps))
(max-abs-resid2-tm n eps)))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs))))
(defun find-n (n eps)
(cond
((zp n) 0)
((equal (abs (resid2-tm (* n eps) eps))
(max-abs-resid2-tm n eps)) n)
(t (find-n (- n 1) eps))))
(defthm find-n-is-max
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(equal (abs (resid2-tm (* eps (find-n n eps)) eps))
(max-abs-resid2-tm n eps)))
:rule-classes nil)
(defthm find-n-range
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(and
(<= 0 (find-n n eps))
(<= (find-n n eps) n)))
:rule-classes :linear)
(defthm find-n-limited
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n)
(i-limited (* eps n)))
(i-limited (* eps (find-n n eps))))
:hints (("Goal" :in-theory (disable abs)
:do-not-induct t
:use ((:instance sandwich-limited-thm
(u 0)
(v (* eps n))
(x (* eps (find-n n eps))))
(:instance pos-factor-<=-thm
(x 0)
(y (find-n n eps))
(a eps))
(:instance pos-factor-<=-thm
(x (find-n n eps))
(y n)
(a eps))
(:instance find-n-range)))))
(defthm max-abs-resid2-tm-small
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n)
(i-small eps)
(i-limited (* eps n)))
(equal (standard-part (* (/ eps) (max-abs-resid2-tm n eps)))
0))
:hints (("Goal" :in-theory (disable abs
resid2-tm
abs-*-thm
ABS-POS-*-LEFT-THM
<-CANCEL-DIVISORS
EQUAL-CANCEL-DIVISORS)
:do-not-induct t
:use ((:instance find-n-is-max)
(:instance abs-pos-*-left-thm
(x (/ eps))
(y (RESID2-TM
(EPS-N-FUN EPS (FIND-N N EPS))
EPS)))
(:instance resid2/eps-small-thm
(tm (* eps (find-n n eps)))
)))))
(defthm max-abs-resid2-tm-limited
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n)
(i-small eps)
(i-limited (* eps n)))
(i-limited (* (/ eps) (max-abs-resid2-tm n eps))))
:rule-classes ((:rewrite) (:type-prescription)))
(defthm max-abs-resid2-tm-type
(implies
(and
(realp eps)
(integerp n))
(and
(realp (max-abs-resid2-tm n eps))
(<= 0 (max-abs-resid2-tm n eps))))
:rule-classes :type-prescription)
(defthm max-abs-resid2-tm-floor-small-hint
(implies
(and
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n)
(i-limited tm)
(i-small eps)
(i-limited (* eps n)))
(equal (standard-part (* tm (/ eps)
(max-abs-resid2-tm n eps)))
0))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs *-commut-3way)
:do-not-induct t
:use ((:instance max-abs-resid2-tm-small)))))
(defthm max-abs-resid2-tm-floor-small
(implies
(and
(realp tm)
(<= 0 tm)
(i-limited tm))
(equal (standard-part (* (floor1 (* tm (i-large-integer)))
(max-abs-resid2-tm
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))
0))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs resid2-tm
;; Cuong Chau: Proof failed if not
;; disable EPS-N-FUN-RW-3-THM.
EPS-N-FUN-RW-3-THM)
:do-not-induct t
:use ((:instance max-abs-resid2-tm-small
(eps (/ (i-large-integer)))
(n (floor1 (* tm (i-large-integer)))))
(:instance pos-factor-<-thm
(x (- (* tm (i-large-integer)) 1))
(y (floor1 (* tm (i-large-integer))))
(a (max-abs-resid2-tm
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))
(:instance pos-factor-<=-thm
(x (floor1 (* tm (i-large-integer))))
(y (* tm (i-large-integer)))
(a (max-abs-resid2-tm
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))))
("Subgoal 1" :in-theory (disable abs resid2-tm)
:use ((:instance max-abs-resid2-tm-floor-small-hint
(n (floor1 (* tm (i-large-integer))))
(eps (/ (i-large-integer))))))))
(defun n-induct-scheme (n)
(cond
((zp n) 0)
(t (n-induct-scheme (- n 1)))))
(defthm phi-run-step-diff
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(< 0 n)
(integerp m)
(<= n m))
(<= (abs (- (step1 x eps)
(phi2 (* n eps))))
(+ (* (+ 1 (* eps (L))) (abs (- x (phi2 (* (- n 1) eps)))))
(max-abs-resid2-tm m eps))))
:hints (("Goal" :in-theory (disable abs resid2-tm)
:do-not '(generalize)
:do-not-induct t
:use ((:instance f-lim-thm
(x1 x)
(x2 (phi2 (* (- n 1) eps))))
(:instance max-abs-resid2-is-bound
(m (- n 1))
(n m))
(:instance pos-factor-<=-thm
(x (abs (- (f x) (f (phi2 (* (- n 1) eps))))))
(y (* (L) (abs (- x (phi2 (* (- n 1) eps))))))
(a eps))
(:instance phi2-equal-thm
(tm (* (- n 1) eps)))
(:instance abs-pos-*-left-thm
(x eps)
(y (+ (F X)
(* -1
(f (PHI2 (+ (* -1 EPS) (* EPS N))))))))
(:instance abs-triangular-inequality-3way-thm
(x (- x (phi2 (* (- n 1) eps))))
(y (- (* eps (f x))
(* eps (f (phi2 (* (- n 1) eps))))))
(z (- (resid2-tm (* (- n 1) eps) eps))))))))
(defthm eexp-unite-exponents-thm
(implies
(and
(realp x)
(realp y)
(realp z))
(equal (* (eexp x) y (eexp z))
(* y (eexp (+ x z))))))
(defthm phi2-run-eq-thm
(implies
(and
(integerp m)
(integerp n)
(<= 0 n)
(<= n m)
(< 0 eps)
(realp eps))
(<= (abs (- (run (phi2 0) n eps)
(phi2 (* eps n))))
(* (eexp (* eps n (L)))
n
(max-abs-resid2-tm m eps))))
:rule-classes nil
:hints (("Goal" :do-not '(generalize)
:induct (n-induct-scheme n)
:do-not-induct t
:in-theory (disable abs step1 resid2-tm MAX-ABS-RESID2-TM))
("Subgoal *1/2" :use ((:instance phi-run-step-diff
(x (run (phi2 0) (- n 1) eps)))
(:instance pos-factor-<=-thm
(x (ABS (+ (RUN (PHI2 0) (+ -1 N) EPS)
(* -1 (PHI2 (+ (* -1 EPS)
(* EPS N)))))))
(y (+ (* -1 (MAX-ABS-RESID2-TM M EPS)
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS N))))
(* N (MAX-ABS-RESID2-TM M EPS)
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS N))))))
(a (eexp (* (L) eps))))
(:instance pos-factor-<=-thm
(x 1)
(y (EEXP (* (L) EPS N)))
(a (MAX-ABS-RESID2-TM M EPS)))
(:instance 1+x-<=eexp-thm
(x (* eps (L))))
(:instance pos-factor-<=-thm
(x (+ 1 (* eps (L))))
(y (eexp (* eps (L))))
(a (ABS (+ (RUN (PHI2 0) (+ -1 N) EPS)
(* -1 (PHI2 (+ (* -1 EPS)
(* EPS N)))))))))))
:otf-flg t)
(defthm phi2-st-run-eq-hint
(implies
(and
(realp tm)
(i-limited y)
(<= 0 tm)
(i-limited tm))
(equal (standard-part (* y
(floor1 (* tm (i-large-integer)))
(max-abs-resid2-tm
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))
0))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs resid2-tm)
:do-not-induct t
:use ((:instance max-abs-resid2-tm-floor-small)))))
(defthm phi2-st-run-eq-thm
(implies
(and
(realp tm)
(standard-numberp tm)
(<= 0 tm))
(equal (standard-part (phi2 tm))
(standard-part (phi (phi2 0) tm))))
:rule-classes nil
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs
;; Cuong Chau: Proof failed if not
;; disable the following two lemmas.
EPS-N-FUN-RW-1-THM
EPS-N-FUN-RW-2-THM)
:use ((:instance phi2-run-eq-thm
(eps (/ (i-large-integer)))
(n (floor1 (* tm (i-large-integer))))
(m (floor1 (* tm (i-large-integer)))))
(:instance phi2-st-run-eq-hint
(y (EEXP (* (L) (TM-FUN TM)))))))))
;; --------------------------------------------
;; The following is the theorem which states
;; that, for any phi2 satisfying the
;; differential equation, phi2 is equal to
;; phi evaluated at initial condition phi2(0).
;; --------------------------------------------
(defthm-std phi2-st-run-eq-std-thm
(implies
(and
(realp tm)
(<= 0 tm))
(equal (phi2 tm)
(phi (phi2 0) tm)))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:use ((:instance phi2-st-run-eq-thm)))))
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