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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 "computed-hints")
(include-book "tm-floor")
(in-theory (disable i-large))
(in-theory (disable standard-part-<=))
;; Cuong Chau: Some proofs failed when I left the following lemmas enable.
(local
(in-theory (disable EPS-N-FUN-RW-1-THM
EPS-N-FUN-RW-2-THM
EPS-N-FUN-RW-3-THM
EPS-N-FUN-RW-4-THM)))
(defun f-sum (x n eps)
(cond
((zp n) 0)
(t (+ (* eps (f (run x (- n 1) eps)))
(f-sum x (- n 1) eps)))))
(defthm run-f-sum-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps))
(equal (run x n eps)
(+ x (f-sum x n eps))))
:rule-classes nil
:hints (("Goal" :do-not '(generalize))
("Subgoal *1/2" :in-theory (disable step-run-thm)
:use ((:instance
step-run-thm (n (- n 1)))))))
(defun resid (x n eps)
(- (f-sum x n eps) (* n eps (f x))))
(defthm f-sum-type
(implies
(and
(realp x)
(realp eps))
(realp (f-sum x n eps)))
:rule-classes :type-prescription)
(defthm run-type
(implies
(and
(realp x)
(realp eps))
(realp (run x n eps)))
:rule-classes :type-prescription)
(defthm run-limited-thm
(implies
(and
(realp x)
(i-limited x)
(realp eps)
(integerp n)
(< 0 eps)
(<= 0 n)
(i-limited (* eps n)))
(i-limited (run x n eps)))
:rule-classes ((:type-prescription) (:rewrite))
:hints (("Goal" :in-theory (disable abs
run
run-n-limit
i-large
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)))))))
(defthm resid-limited-thm
(implies
(and
(realp x)
(i-limited x)
(integerp n)
(<= 0 n)
(realp eps)
(< 0 eps)
(i-limited (* eps n)))
(i-limited (resid x n eps)))
:rule-classes :type-prescription
:hints (("Goal" :in-theory (disable i-large)
:use ((:instance run-f-sum-thm)
(:instance run-limited-thm)))
("Goal'''" :use ((:instance times-limited
(x (* eps n))
(y (f x)))
(:instance plus-limited
(x (+ X (F-SUM X N EPS)))
(y (- x)))))))
(defthm resid-standard-thm
(implies
(and
(realp x)
(i-limited x)
(integerp n)
(<= 0 n)
(realp eps)
(< 0 eps)
(i-limited (* eps n)))
(standard-numberp (standard-part (resid x n eps))))
:hints (("Goal" :in-theory (disable i-large resid)
:use ((:instance resid-limited-thm)))))
(defthm resid-std-thm
(IMPLIES
(AND (STANDARD-NUMBERP X)
(STANDARD-NUMBERP TM)
(REALP X)
(REALP TM)
(<= 0 TM))
(STANDARD-NUMBERP (STANDARD-PART
(resid X
(FLOOR1 (* (I-LARGE-INTEGER) TM))
(/ (I-LARGE-INTEGER))))))
:hints (("Goal" :in-theory (disable i-large)
:use ((:instance resid-standard-thm
(n (floor1 (* tm (i-large-integer))))
(eps (/ (i-large-integer))))))))
(in-theory (disable resid))
(defun-std resid-tm (x tm)
(cond
((not (and (realp x)
(realp tm)
(<= 0 tm))) 0)
(t (standard-part (resid x
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer)))))))
(in-theory (enable resid))
(defthm f-run-bound-hint-1
(implies
(and
(realp x)
(realp eps)
(< 0 eps))
(equal (f (step1 x eps))
(+ (f x) (- (f (step1 x eps)) (f x)))))
:rule-classes nil)
(defthm f-run-bound-hint-2
(implies
(and
(realp x)
(realp eps)
(< 0 eps))
(<= (abs (f (step1 x eps)))
(* (+ 1 (* eps (L))) (abs (f x)))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:use ((:instance f-run-bound-hint-1)
(:instance abs-triangular-inequality-thm
(x (f x))
(y (- (f (step1 x eps)) (f x))))
(:instance f-lim-thm
(x1 (step1 x eps))
(x2 x))))))
(defthm f-run-bound-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(<= (abs (f (run x n eps)))
(* (eexp (* n eps (L))) (abs (f x)))))
:hints (("Goal" :in-theory (disable abs step1)
:do-not '(generalize))
("Subgoal *1/5" :use ((:instance pos-factor-<=-thm
(x (abs (f (step1 x eps))))
(y (* (+ 1 (* eps (L))) (abs (f x))))
(a (EEXP (+ (* -1 (L) EPS)
(* (L) EPS N)))))
(:instance pos-factor-<=-thm
(x (+ 1 (* eps (L))))
(y (eexp (* eps (L))))
(a (* (EEXP (+ (* -1 (L) EPS)
(* (L) EPS N)))
(abs (f x)))))
(:instance f-run-bound-hint-2)
(:instance 1+x-<=eexp-thm
(x (* eps (L))))))))
(defthm f-sum-exp-bound-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(<= (abs (f-sum x n eps))
(* n eps (eexp (* eps n (L))) (abs (f x)))))
:rule-classes nil
:hints (("Goal" :induct (f-sum x n eps)
:do-not-induct t
:in-theory (disable abs <-*-LEFT-CANCEL)
:do-not '(generalize))
("Subgoal *1/2" :use ((:instance f-run-bound-thm (n (- n 1)))
(:instance pos-factor-<=-thm
(x (abs (f (run x (- n 1) eps))))
(y (* (eexp (* (- n 1) eps (L)))
(abs (f x))))
(a eps))
(:instance abs-triangular-inequality-thm
(x (F-SUM X (+ -1 N) EPS))
(y (* EPS (F (RUN X (+ -1 N) EPS)))))
(:instance pos-factor-<=-thm
(x 1)
(y (EEXP (* (L) EPS)))
(a (* EPS N (ABS (F X))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS N)))))
)))))
(defthm f-sum-diff-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(integerp m)
(<= 0 n)
(<= n m))
(equal (- (f-sum x m eps)
(f-sum x n eps))
(f-sum (run x n eps) (- m n) eps))))
(defthm f-run-diff-eq-f-sum-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(integerp m)
(<= 0 n)
(<= n m))
(equal (- (run x m eps)
(run x n eps))
(f-sum (run x n eps) (- m n) eps)))
:hints (("Goal" :do-not-induct t
:use ((:instance run-f-sum-thm (n m))
(:instance run-f-sum-thm)
(:instance f-sum-diff-thm)))))
(defthm pos-*-<=-thm
(implies
(and
(realp a)
(realp b)
(realp x)
(realp y)
(<= 0 a)
(<= 0 b)
(<= a x)
(<= b y))
(<= (* a b) (* x y)))
:hints (("Goal" :use ((:instance pos-factor-<=-thm
(x a)
(y x)
(a b))
(:instance pos-factor-<=-thm
(x b)
(y y)
(a x))))))
(defthm f-run-diff-tm-thm-hint
(implies
(and
(realp eps)
(< 0 eps)
(integerp m)
(integerp n)
(<= n m))
(<= 0 (* (- m n) eps (eexp (* (- m n) (L) eps)))))
:rule-classes nil
:hints (("Goal" :in-theory (disable distributivity
distributivity-left)
:use ((:instance pos-factor-<=-thm
(x 0)
(y (- m n))
(a (* eps
(eexp (* (- m n)
(L)
eps))))
)))))
(defthm f-run-diff-tm-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(integerp m)
(<= 0 n)
(<= n m))
(<= (abs (- (run x m eps)
(run x n eps)))
(* (- m n) eps (eexp (* eps m (L))) (abs (f x)))))
:hints (("Goal" :do-not-induct t
:in-theory (disable abs)
:use ((:instance f-run-diff-tm-thm-hint)
(:instance f-run-diff-eq-f-sum-thm)
(:instance f-sum-exp-bound-thm
(x (run x n eps))
(n (- m n)))
(:instance f-run-bound-thm)
(:instance pos-factor-<=-thm
(x (abs (f (run x n eps))))
(y (* (eexp (* n eps (L))) (abs (f x))))
(a (* (- m n)
eps
(eexp (* eps
(- m n)
(L))))))))))
;; Cuong Chau: I need this lemma to prove some theorems in this book.
(defthm standard-number-is-limited
(implies (standard-numberp r)
(i-limited r)))
(defthm run-diff-tm-standard-part-thm
(implies
(and
(realp x)
(standard-numberp x)
(integerp n)
(integerp m)
(realp eps)
(< 0 eps)
(<= 0 n)
(<= 0 m)
(i-limited (* eps n))
(i-limited (* eps m))
(equal (standard-part (* m eps))
(standard-part (* n eps))))
(equal (- (standard-part (run x m eps))
(standard-part (run x n eps)))
0))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (e/d (EPS-N-FUN-RW-1-THM
EPS-N-FUN-RW-3-THM)
(abs))
:cases ((<= n m) (< m n))
:do-not-induct t)
("Subgoal 2" :use ((:instance f-run-diff-tm-thm)))
("Subgoal 1" :use ((:instance f-run-diff-tm-thm
(m n)
(n m))))))
(defthm floor1-large-1<=
(implies
(and
(realp x)
(< 0 x)
(standard-numberp x))
(i-large (* (i-large-integer) x)))
:hints (("Goal" :in-theory (enable i-large))))
(defthm large-gt-1
(implies
(and
(realp x)
(< 0 x)
(i-large x))
(< 1 x))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (enable i-large)
:use ((:instance standard-part-<=
(x 1)
(y (/ x)))))))
(defthm standard-diff-*-large-thm
(implies
(and
(realp x)
(realp y)
(standard-numberp x)
(standard-numberp y)
(< x y))
(<= (+ (* (i-large-integer) x) 1)
(* (i-large-integer) y)))
:hints (("Goal" :use ((:instance floor1-large-1<= (x (- y x)))
(:instance large-gt-1
(x (+ (* -1 (I-LARGE-INTEGER) X)
(* (I-LARGE-INTEGER) Y))))
(:instance pos-factor-<-thm
(x 0)
(y (- y x))
(a (i-large-integer)))))))
(defthm abs-standard-numberp
(implies
(and
(realp x)
(standard-numberp x))
(standard-numberp (abs x))))
(defthm-std phi-diff-tm-thm
(implies
(and
(realp x)
(realp tm1)
(realp tm2)
(<= 0 tm1)
(<= tm1 tm2))
(<= (abs (- (phi x tm2)
(phi x tm1)))
(* (- tm2 tm1) (eexp (* tm2 (L))) (abs (f x)))))
:rule-classes nil
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs <-CANCEL-DIVISORS)
:use ((:instance f-run-diff-tm-thm
(eps (/ (i-large-integer)))
(n (floor1 (* tm1 (i-large-integer))))
(m (floor1 (* tm2 (i-large-integer)))))
(:instance L-STANDARD-THM)
(:instance F-STANDARD-THM)
(:instance standard-diff-*-large-thm
(x TM1)
(y TM2))))))
;; ----------------------------------------------
;; The following is the theorem which states
;; that phi is continuous with respect to time
;; ----------------------------------------------
(defthm phi-tm-continuous-thm
(implies
(and
(realp x)
(standard-numberp x)
(realp tm1)
(<= 0 tm1)
(i-limited tm1))
(equal (standard-part (phi x tm1))
(phi x (standard-part tm1))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs)
:cases ((<= tm1 (standard-part tm1))
(< (standard-part tm1) tm1)))
("Subgoal 2" :use ((:instance phi-diff-tm-thm
(tm1 tm1)
(tm2 (standard-part tm1)))))
("Subgoal 1" :use ((:instance phi-diff-tm-thm
(tm2 tm1)
(tm1 (standard-part tm1)))))))
(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))))
(defthm phi-diff-hint-1
(implies
(and
(standard-numberp x1)
(standard-numberp x2)
(standard-numberp tm)
(realp x1)
(realp x2)
(realp tm)
(<= 0 tm))
(equal (STANDARD-PART (* (EEXP (* (L) (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM))))
(ABS (+ X1 (* -1 X2)))))
(* (EEXP (* (L) TM))
(ABS (+ X1 (* -1 X2))))))
:rule-classes nil
:hints (("Goal" :in-theory (disable *-commut-3way)
:use (:instance L-STANDARD-THM))))
(defthm-std phi-x-diff-thm
(implies
(and
(realp x1)
(realp x2)
(realp tm)
(<= 0 tm))
(<= (abs (- (phi x1 tm) (phi x2 tm)))
(* (abs (- x1 x2)) (eexp (* tm (L))))))
:rule-classes nil
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs i-large )
:use ((:instance run-diff-limit-eexp-thm
(n (floor1 (* tm (i-large-integer))))
(eps (/ (i-large-integer))))
(:instance phi-diff-hint-1)))))
(defthm run-x-continuous-thm
(implies
(and
(realp eps)
(< 0 eps)
(realp x)
(i-limited x)
(integerp n)
(i-limited (* eps n))
(<= 0 n))
(equal (standard-part (run (standard-part x) n eps))
(standard-part (run x n eps))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs)
:do-not-induct t
:use ((:instance run-diff-limit-eexp-thm
(x1 x)
(x2 (standard-part x)))))))
;; -------------------------------------------
;; The following is the theorem which states
;; that phi is continuous with respect to x
;; -------------------------------------------
(defthm phi-x-continuous-thm
(implies
(and
(realp x)
(i-limited x)
(realp tm)
(standard-numberp tm)
(<= 0 tm))
(equal (standard-part (phi x tm))
(phi (standard-part x) tm)))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs)
:use ((:instance phi-x-diff-thm
(x1 x)
(x2 (standard-part x)))))))
(defthm phi-plus-hint1
(implies
(and
(realp tm1)
(realp tm2))
(equal (standard-part (* (+ (FLOOR1 (* (I-LARGE-INTEGER) TM1))
(FLOOR1 (* (I-LARGE-INTEGER) TM2)))
(/ (I-LARGE-INTEGER))))
(standard-part (* (floor1 (* (i-large-integer) (+ tm1 tm2)))
(/ (i-large-integer))))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable <-CANCEL-DIVISORS)
:use ((:instance pos-factor-<-thm
(x (+ (* (I-LARGE-INTEGER) TM1)
(* (I-LARGE-INTEGER) TM2)
-2))
(y (+ (FLOOR1 (* (I-LARGE-INTEGER) TM1))
(FLOOR1 (* (I-LARGE-INTEGER) TM2))))
(a (/ (i-large-integer))))
(:instance pos-factor-<=-thm
(x (+ (FLOOR1 (* (I-LARGE-INTEGER) TM1))
(FLOOR1 (* (I-LARGE-INTEGER) TM2))))
(y (+ (* (I-LARGE-INTEGER) TM1)
(* (I-LARGE-INTEGER) TM2)))
(a (/ (i-large-integer))))
(:instance pos-factor-<-thm
(x (+ (* (i-large-integer) (+ tm1 tm2)) -1))
(y (floor1 (* (i-large-integer) (+ tm1 tm2))))
(a (/ (i-large-integer))))
(:instance pos-factor-<=-thm
(x (floor1 (* (i-large-integer) (+ tm1 tm2))))
(y (* (i-large-integer) (+ tm1 tm2)))
(a (/ (i-large-integer))))))))
(defthm tm1+tm2-limited-thm
(implies
(and
(standard-numberp tm1)
(standard-numberp tm2)
(realp tm1)
(realp tm2))
(i-limited (* (/ (i-large-integer))
(floor1 (* (i-large-integer) (+ tm1 tm2))))))
:rule-classes nil
:hints (("Goal" :use ((:instance phi-plus-hint1)
(:instance standard+small->i-limited
(x (standard-part
(* (/ (i-large-integer))
(floor1 (* (i-large-integer)
(+ tm1 tm2))))))
(eps (- (* (/ (i-large-integer))
(floor1 (* (i-large-integer)
(+ tm1 tm2))))
(standard-part
(* (/ (i-large-integer))
(floor1 (* (i-large-integer)
(+ tm1 tm2)))))))
)))))
;; -------------------------------------------
;; The following is the theorem which states
;; that phi is time invariant
;; -------------------------------------------
(defthm-std phi-plus-thm
(implies
(and
(realp x)
(realp tm1)
(realp tm2)
(<= 0 tm1)
(<= 0 tm2))
(equal (phi (phi x tm1) tm2)
(phi x (+ tm1 tm2))))
:hints (("Goal" :in-theory (disable i-large run-plus-thm)
:use ((:instance phi-plus-hint1)
(:instance tm1+tm2-limited-thm)
(:instance run-plus-thm (n (floor1 (* tm1
(i-large-integer))))
(m (floor1 (* tm2
(i-large-integer))))
(eps (/ (i-large-integer))))
(:instance run-diff-tm-standard-part-thm
(eps (/ (i-large-integer)))
(m (+ (floor1 (* (i-large-integer) tm1))
(floor1 (* (i-large-integer) tm2))))
(n (floor1 (* (i-large-integer) (+ tm1 tm2)))))
(:instance phi-x-continuous-thm
(x (RUN X
(FLOOR1 (* (I-LARGE-INTEGER) TM1))
(/ (I-LARGE-INTEGER))))
(tm tm2))))))
;; For some reason, enabling the following rule as a
;; linear lemma may cause ACL2 to stall during simplification
(defthm floor1-1-thm
(implies
(and (realp x)
(<= 1 x))
(< 0 (floor1 x)))
:rule-classes nil)
;; For some reason, enabling the following rewrite rule
;; may cause ACL2 to stall during simplification
(defthm floor1-0-thm
(implies
(and (realp x)
(<= 0 x)
(< x 1))
(equal (floor1 x) 0))
:rule-classes nil)
(defun tm-m (tm eps)
(cond
((not (and
(realp eps)
(< 0 eps)
(realp tm)
(<= eps tm))) 0)
(t (floor1 (/ tm eps)))))
(defun phi-run (x tm eps)
(declare (xargs :measure (tm-m tm eps)
:hints (("Subgoal 1.2"
:use ((:instance floor1-1-thm
(x (* (/ eps) tm))))))))
(cond
((not (and (realp eps)
(< 0 eps)
(<= eps tm)
(realp tm))) (phi x tm))
(t (phi-run (phi x eps) (- tm eps) eps))))
(defthm phi-run-eq-phi-thm
(implies
(and
(realp x)
(realp tm)
(<= 0 tm)
(< 0 eps)
(realp eps))
(equal (phi-run x tm eps)
(phi x tm)))
:rule-classes nil)
(defthm run-equal-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(equal (+ x (* eps n (f x)) (- (f-sum x n eps) (* n eps (f x))))
(run x n eps)))
:hints (("Goal" :use ((:instance run-f-sum-thm)))))
(defthm-std phi-equal-thm
(implies
(and
(realp x)
(realp tm)
(<= 0 tm))
(equal (+ x (* tm (f x)) (resid-tm x tm))
(phi x tm)))
:hints (("Goal" :use ((:instance F-STANDARD-THM)
(:instance run-f-sum-thm
(eps (/ (i-large-integer)))
(n (floor1 (* tm (i-large-integer))))
)))))
(defthm step-resid-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(equal (+ (resid x n eps)
(- (* eps (f (run x n eps)))
(* eps (f x))))
(resid x (+ n 1) eps))))
(defthm resid-type
(implies
(and
(realp x)
(realp eps)
(integerp n))
(realp (resid x n eps)))
:rule-classes :type-prescription)
(defthm-std resid-tm-type
(implies
(and
(realp x)
(realp tm))
(realp (resid-tm x tm)))
:rule-classes :type-prescription)
(defthm step-resid-bound-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(< 0 n))
(<= (abs (resid x n eps))
(+ (abs (resid x (- n 1) eps))
(* eps eps (L) (- n 1)
(eexp (* eps (- n 1) (L)))
(abs (f x))))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs step-resid-thm resid)
:use ((:instance f-sum-exp-bound-thm (n (- n 1)))
(:instance abs-triangular-inequality-thm
(x (RESID X (+ -1 N) EPS))
(y (- (* eps (f (run x (- n 1) eps)))
(* eps (f x)))))
(:instance abs-pos-*-left-thm
(x eps)
(y (+ (* -1 (F X))
(* (F (RUN X (+ -1 N) EPS))))))
(:instance f-lim-thm (x1 (run x (- n 1) eps))
(x2 x))
(:instance pos-factor-<=-thm
(x (ABS (+ (* -1 (F X))
(* (F (RUN X (+ -1 N) EPS))))))
(y (* (L)
(ABS (+ (* -1 X)
(RUN X (+ -1 N) EPS)))))
(a eps))
(:instance run-f-sum-thm (n (- n 1)))
(:instance step-resid-thm (n (- n 1)))
(:instance pos-factor-<=-thm
(x (abs (f-sum x (- n 1) eps)))
(y (* (- n 1)
eps
(eexp (* eps (- n 1) (L)))
(abs (f x))))
(a (* eps (L))))))))
(defthm resid-bound-thm
(implies
(and
(realp x)
(realp eps)
(< 0 eps)
(integerp n)
(<= 0 n))
(<= (abs (resid x n eps))
(* n n eps eps (L) (eexp (* n eps (L))) (abs (f x)))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:do-not '(generalize)
:do-not-induct t
:induct (f-sum x n eps))
("Subgoal *1/2" :in-theory (disable abs resid
<-*-LEFT-CANCEL)
:use ((:instance step-resid-bound-thm)
(:instance pos-factor-<=-thm
(x 0)
(y (- n 1))
(a (* (L) eps)))
(:instance pos-factor-<=-thm
(x 1)
(y (eexp (* eps (L))))
(a (* (L) EPS EPS N N
(ABS (F X))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS N)))))
)))))
(defthm tm-fun-rw-5-thm
(implies
(and
(realp a)
(realp b)
(realp c)
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER)) a b
(FLOOR1 (* (I-LARGE-INTEGER) TM)) c)
(* (tm-fun tm) a b c)))
:hints (("Goal" :in-theory (e/d (tm-fun) (tm-fun-rw-1-thm
tm-fun-rw-2-thm
tm-fun-rw-3-thm
tm-fun-rw-4-thm)))))
(defthm tm-fun-rw-6-thm
(implies
(and
(realp a)
(realp b)
(realp c)
(realp d)
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER)) a b c
(FLOOR1 (* (I-LARGE-INTEGER) TM)) d)
(* (tm-fun tm) a b c d)))
:hints (("Goal" :in-theory (e/d (tm-fun) (tm-fun-rw-1-thm
tm-fun-rw-2-thm
tm-fun-rw-3-thm
tm-fun-rw-4-thm
tm-fun-rw-5-thm)))))
(defthm-std resid-tm-bound-thm
(implies
(and
(realp x)
(realp tm)
(<= 0 tm))
(<= (abs (resid-tm x tm))
(* tm tm (L) (eexp (* tm (L))) (abs (f x)))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs <-CANCEL-DIVISORS)
:use ((:instance F-STANDARD-THM)
(:instance L-STANDARD-THM)
(:instance resid-bound-thm
(eps (/ (i-large-integer)))
(n (floor1 (* tm (i-large-integer)))))))))
(defthm tm-floor-0-thm
(implies
(and
(realp tm)
(realp eps)
(<= 0 tm)
(< 0 eps)
(< tm eps))
(equal (FLOOR1 (* (/ EPS) TM)) 0))
:hints (("Goal" :use ((:instance floor1-0-thm (x (/ tm eps)))))))
(defthm-std phi-0-thm
(implies
(realp x)
(equal (phi x 0)
x)))
(defun tm-induct (tm eps)
(declare (xargs :measure (tm-m tm eps)
:hints (("Subgoal 1.2"
:use ((:instance floor1-1-thm
(x (* (/ eps) tm))))))))
(cond
((not (and
(realp eps)
(< 0 eps)
(realp tm)
(<= eps tm))) tm)
(t (tm-induct (- tm eps)
eps))))
(defthm phi-eps-step-thm
(implies
(and
(realp x1)
(realp x2)
(realp eps)
(< 0 eps))
(<= (abs (- (phi x1 eps) (step1 x2 eps)))
(+ (* (abs (- x1 x2)) (+ 1 (* eps (L))))
(* eps eps (L) (eexp (* eps (L))) (abs (f x1))))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:do-not '(generalize)
:do-not-induct t
:use ((:instance phi-equal-thm (x x1) (tm eps))
(:instance abs-triangular-inequality-thm
(x (+ X1 (* -1 X2)))
(y (+(RESID-TM X1 EPS)
(* EPS (F X1))
(* -1 EPS (F X2)))))
(:instance abs-triangular-inequality-thm
(x (RESID-TM X1 EPS))
(y (+ (* EPS (F X1))
(* -1 EPS (F X2)))))
(:instance resid-tm-bound-thm
(tm eps)
(x x1))
(:instance abs-pos-*-left-thm
(x eps)
(y (- (F X1)
(F X2))))
(:instance f-lim-thm)
(:instance pos-factor-<=-thm
(x (abs (- (f x1) (f x2))))
(y (* (L) (abs (- x1 x2))))
(a eps))))))
(defthm phi-phi-run-thm
(implies
(and
(realp eps)
(realp x)
(realp tm)
(<= 0 tm)
(<= eps tm)
(< 0 eps))
(equal (phi (phi-run x
(+ (* -1 EPS)
(* EPS
(FLOOR1 (* (/ EPS) TM))))
eps) eps)
(phi-run x (* eps (floor1 (/ tm eps))) eps)))
:hints (("Goal" :induct (phi-run x tm eps)
:do-not '(generalize))
("Subgoal *1/4" :use ((:instance floor1-1-thm
(x (/ tm eps)))
(:instance pos-factor-<=-thm
(x 1)
(y (floor1 (/ tm eps)))
(a eps))))
("Subgoal *1/2" :use ((:instance floor1-1-thm (x (/ tm eps)))
(:instance pos-factor-<=-thm
(x 1)
(y (floor1 (/ tm eps)))
(a eps))))
("Subgoal *1/1" :use ((:instance tm-floor-0-thm
(tm (- tm eps)))
(:instance distributivity
(y (FLOOR1 (* (/ EPS) TM)))
(z -1)
(x eps))))))
(defthm f-standard-part-thm
(implies
(and
(realp x)
(i-limited x))
(equal (standard-part (f x))
(f (standard-part x))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs STANDARDP-STANDARD-PART)
:use ((:instance F-STANDARD-THM
(x (standard-part x)))
(:instance f-lim-thm
(x1 (standard-part x))
(x2 x))
(:instance standardp-standard-part)))))
(defthm-std f-phi-bound-thm
(implies
(and
(realp x)
(realp tm)
(<= 0 tm))
(<= (abs (f (phi x tm)))
(* (eexp (* tm (L))) (abs (f x)))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs)
:use ((:instance L-STANDARD-THM)
(:instance f-run-bound-thm
(eps (/ (i-large-integer)))
(n (floor1 (* tm (i-large-integer))))
)))))
(defthm phi-eps-arith-hint
(implies
(and
(realp tm)
(< 0 eps)
(realp eps))
(equal
(* (EEXP (* (L) EPS))
(FLOOR1 (* (/ EPS) TM))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS (FLOOR1 (* (/ EPS) TM))))))
(* (FLOOR1 (* (/ EPS) TM))
(EEXP (* (L) EPS (FLOOR1 (* (/ EPS) TM)))))))
:hints (("Goal" :in-theory (disable *-commut-3way)
:use ((:instance *-commut-3way
(x (EEXP (* (L) EPS)))
(y (FLOOR1 (* (/ EPS) TM)))
(z (EEXP (+ (* -1 (L) EPS)
(* (L) EPS
(FLOOR1 (* (/ EPS) TM))))))
)))))
(defthm phi-eps-thm
(implies
(and
(realp x)
(realp tm)
(realp eps)
(<= 0 tm)
(< 0 eps))
(<= (abs (- (phi-run x
(* eps (floor1 (/ tm eps)))
eps)
(run x
(floor1 (/ tm eps))
eps)))
(* eps eps
(floor1 (/ tm eps))
(L)
(abs (f x))
(eexp (* eps (floor1 (/ tm eps)) (L))))))
:rule-classes nil
:hints (("Goal" :in-theory (disable abs)
:induct (tm-induct tm eps)
:do-not '(generalize))
("Subgoal *1/2" :in-theory (disable abs run step1)
:use ((:instance floor1-1-thm (x (/ tm eps)))
(:instance pos-factor-<=-thm
(x 1)
(y (floor1 (/ tm eps)))
(a eps))))
("Subgoal *1/2.1" :in-theory (disable abs run step1
<-*-left-cancel
<-cancel-divisors)
:use ((:instance phi-run-eq-phi-thm
(tm (+ (* -1 EPS)
(* EPS (FLOOR1 (* (/ EPS) TM))))))
(:instance phi-run-eq-phi-thm
(tm (* EPS (FLOOR1 (* (/ EPS) TM)))))
(:instance 1+x-<=eexp-thm (x (* eps (L))))
(:instance phi-eps-arith-hint)
(:instance pos-factor-<=-thm
(x (+ 1 (* eps (L))))
(y (eexp (* eps (L))))
(a (ABS (+ (* -1
(RUN X
(+ -1
(FLOOR1
(* (/ EPS) TM)))
EPS))
(PHI-RUN X
(+ (* -1 EPS)
(* EPS
(FLOOR1
(* (/ EPS) TM))))
EPS)))))
(:instance pos-factor-<=-thm
(x (ABS (+ (* -1
(RUN X
(+ -1 (FLOOR1
(* (/ EPS) TM)))
EPS))
(PHI-RUN X
(+ (* -1 EPS)
(* EPS
(FLOOR1
(* (/ EPS) TM))))
EPS))))
(y (+ (* -1 (L)
EPS EPS (ABS (F X))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS
(FLOOR1
(* (/ EPS) TM))))))
(* (L)
EPS EPS (ABS (F X))
(FLOOR1 (* (/ EPS) TM))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS
(FLOOR1
(* (/ EPS) TM)))))
)))
(a (EEXP (* (L) EPS))))
(:instance pos-factor-<=-thm
(x (ABS (F (PHI X
(+ (* -1 EPS)
(* EPS
(FLOOR1
(* (/ EPS) TM))))
))))
(y (* (ABS (F X))
(EEXP (+ (* -1 (L) EPS)
(* (L) EPS
(FLOOR1
(* (/ EPS) TM))
)))))
(a (* (L) EPS EPS (EEXP (* (L) EPS)))))
(:instance phi-eps-step-thm
(x1 (phi-run x
(* eps
(- (floor1 (/ tm eps)) 1))
eps))
(x2 (run x (- (floor1
(/ tm eps)) 1) eps)))
(:instance f-phi-bound-thm
(tm (+ (* -1 EPS)
(* EPS (FLOOR1 (* (/ EPS) TM)))))
)))))
(defthm phi-any-small-eps-thm
(implies
(and
(realp x)
(realp tm)
(standard-numberp x)
(standard-numberp tm)
(realp eps)
(<= 0 tm)
(< 0 eps)
(i-small eps))
(equal (standard-part (phi x tm))
(standard-part (run x (floor1 (/ tm eps)) eps))))
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable abs phi)
:use ((:instance small-are-limited (x eps))
(:instance phi-eps-thm)
(:instance phi-run-eq-phi-thm
(tm (* EPS (FLOOR1 (* (/ EPS) TM)))))))))
;; ----------------------------------------------
;; The following is the theorem which states
;; that run, hence phi, is independent of the
;; choice of eps.
;; ----------------------------------------------
(defthm run-any-small-eps-thm
(implies
(and
(realp x)
(realp tm)
(standard-numberp x)
(standard-numberp tm)
(realp eps)
(<= 0 tm)
(< 0 eps)
(i-small eps))
(equal (standard-part (run x
(floor1 (* tm (i-large-integer)))
(/ (i-large-integer))))
(standard-part (run x
(floor1 (/ tm eps))
eps))))
:hints (("Goal" :use ((:instance phi-any-small-eps-thm)))))
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