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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 "computed-hints")
(in-theory (disable i-large))
(in-theory (disable standard-part-<=))
(defthm tm-floor-thm-hint-1
(implies
(and
(realp tm)
(realp eps)
(< 0 eps)
(i-small eps))
(equal (standard-part (* eps (floor1 (/ tm eps))))
(standard-part tm)))
:rule-classes nil
:hints ((standard-part-hint stable-under-simplificationp clause)
("Goal" :in-theory (disable <-CANCEL-DIVISORS)
:use ((:instance pos-factor-<-thm (x (- (/ tm eps) 1))
(y (floor1 (/ tm eps))) (a eps))
(:instance pos-factor-<=-thm (x (floor1 (/ tm eps)))
(y (/ tm eps)) (a eps))))))
(defthm tm-floor-thm
(implies
(and
(realp tm)
(standard-numberp tm))
(equal (STANDARD-PART (* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM))))
tm))
:hints (("Goal" :use ((:instance tm-floor-thm-hint-1
(eps (/ (i-large-integer))))))))
(defthm tm-floor-limited-thm
(implies
(and
(realp tm)
(standard-numberp tm))
(i-limited (* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM)))))
:hints (("Goal" :use ((:instance standard+small->i-limited
(x (standard-part
(* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM)))))
(eps (- (* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM)))
(standard-part
(* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM)))
))))))))
(defthm tm-floor-*a-hint-1
(implies
(and
(realp tm)
(realp a)
(i-limited a)
(standard-numberp tm))
(equal (STANDARD-PART (* a (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM))))
(standard-part (* a tm))))
:rule-classes nil
:hints (("Goal" :in-theory (disable *-COMMUT-3WAY))))
(defthm tm-floor-*a-thm
(implies
(and
(realp tm)
(realp a)
(i-limited a)
(standard-numberp tm))
(equal (STANDARD-PART (* (/ (I-LARGE-INTEGER)) a
(FLOOR1 (* (I-LARGE-INTEGER) TM))))
(standard-part (* a tm))))
:hints (("Goal" :in-theory (disable i-large)
:use ((:instance tm-floor-*a-hint-1)))))
(defun tm-fun (tm)
(* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM))))
(defthm tm-fun-limited-thm
(implies
(and
(realp a)
(realp tm)
(standard-numberp tm))
(i-limited (tm-fun tm))))
(defthm tm-fun-standard-part-thm
(implies
(and
(realp a)
(realp tm)
(standard-numberp tm))
(equal (standard-part (tm-fun tm))
tm)))
(defthm tm-fun-rw-1-thm
(implies
(and
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM)))
(tm-fun tm))))
(defthm tm-fun-rw-2-thm
(implies
(and
(realp a)
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER)) a
(FLOOR1 (* (I-LARGE-INTEGER) TM)))
(* (tm-fun tm) a)))
:hints (("Goal" :in-theory (disable tm-fun-rw-1-thm))))
(defthm tm-fun-rw-3-thm
(implies
(and
(realp a)
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER))
(FLOOR1 (* (I-LARGE-INTEGER) TM))
a)
(* (tm-fun tm) a)))
:hints (("Goal" :in-theory (disable tm-fun-rw-1-thm tm-fun-rw-2-thm))))
(defthm tm-fun-rw-4-thm
(implies
(and
(realp a)
(realp b)
(realp tm)
(standard-numberp tm))
(equal (* (/ (I-LARGE-INTEGER))
a
(FLOOR1 (* (I-LARGE-INTEGER) TM))
b)
(* (tm-fun tm) a b)))
:hints (("Goal" :in-theory (disable tm-fun-rw-1-thm
tm-fun-rw-2-thm
tm-fun-rw-3-thm))))
;; tm-fun should be disabled. Enabling it may result in rewrite loops.
(in-theory (disable tm-fun))
(defun eps-n-fun (eps n)
(* eps n))
(defthm eps-n-fun-type-thm
(implies
(and
(realp eps)
(realp n))
(realp (eps-n-fun eps n)))
:rule-classes :type-prescription)
(defthm eps-n-fun-0-thm
(equal (eps-n-fun eps 0) 0))
(defthm eps-n-fun-limited-thm
(implies
(and
(realp eps)
(integerp n)
(i-limited (* eps n)))
(i-limited (eps-n-fun eps n))))
(defthm eps-n-fun-rw-1-thm
(implies
(and
(integerp n)
(realp eps)
(syntaxp (eq eps 'eps))
(i-limited (* eps n)))
(equal (* eps n)
(eps-n-fun eps n))))
(defthm eps-n-fun-rw-2-thm
(implies
(and
(realp a)
(syntaxp (eq eps 'eps))
(integerp n)
(realp eps)
(i-limited (* eps n)))
(equal (* eps a n)
(* (eps-n-fun eps n) a)))
:hints (("Goal" :in-theory (disable eps-n-fun-rw-1-thm))))
(defthm eps-n-fun-rw-3-thm
(implies
(and
(realp a)
(syntaxp (eq eps 'eps))
(integerp n)
(realp eps)
(i-limited (* eps n)))
(equal (* eps n a)
(* (eps-n-fun eps n) a)))
:hints (("Goal" :in-theory (disable eps-n-fun-rw-1-thm
eps-n-fun-rw-2-thm))))
(defthm eps-n-fun-rw-4-thm
(implies
(and
(realp a)
(realp b)
(syntaxp (eq eps 'eps))
(integerp n)
(realp eps)
(i-limited (* eps n)))
(equal (* eps a n b)
(* (eps-n-fun eps n) a b)))
:hints (("Goal" :in-theory (disable eps-n-fun-rw-1-thm
eps-n-fun-rw-2-thm
eps-n-fun-rw-3-thm))))
;; eps-n-fun should be disabled. Enabling it may result in rewrite loops.
(in-theory (disable eps-n-fun))
(defun tm-eps-fun (tm eps)
(* eps
(FLOOR1 (/ TM eps))))
(defthm tm-eps-type
(implies
(realp eps)
(realp (tm-eps-fun tm eps)))
:rule-classes :type-prescription
:hints (("Goal" :in-theory (enable tm-eps-fun))))
(defthm tm-eps-pos-thm
(implies
(and
(realp tm)
(realp eps)
(<= 0 tm)
(< 0 eps))
(<= 0 (tm-eps-fun tm eps)))
:rule-classes :type-prescription
:hints (("Goal" :in-theory (enable tm-eps-fun))))
(defthm tm-eps-fun-standard-part-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp tm)
(standard-numberp tm))
(equal (standard-part (tm-eps-fun tm eps))
tm))
:hints (("Goal" :use ((:instance tm-floor-thm-hint-1)))))
(defthm tm-eps-fun-limited-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp tm)
(standard-numberp tm))
(i-limited (tm-eps-fun tm eps)))
:hints (("Goal" :in-theory (disable tm-eps-fun)
:use ((:instance standard+small->i-limited
(x (standard-part (tm-eps-fun tm eps)))
(eps (- (tm-eps-fun tm eps)
(standard-part
(tm-eps-fun tm eps)))))))))
(defthm tm-eps-fun-rw-1-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp tm)
(standard-numberp tm))
(equal (* eps
(FLOOR1 (* (/ EPS) TM)))
(tm-eps-fun tm eps))))
(defthm tm-eps-fun-rw-2-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp a)
(realp tm)
(standard-numberp tm))
(equal (* eps a
(FLOOR1 (* (/ EPS) TM)))
(* (tm-eps-fun tm eps) a)))
:hints (("Goal" :in-theory (disable tm-eps-fun-rw-1-thm))))
(defthm tm-eps-fun-rw-3-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp a)
(realp tm)
(standard-numberp tm))
(equal (* eps
(FLOOR1 (* (/ EPS) TM))
a)
(* (tm-eps-fun tm eps) a)))
:hints (("Goal" :in-theory (disable tm-eps-fun-rw-1-thm
tm-eps-fun-rw-2-thm))))
(defthm tm-eps-fun-rw-4-thm
(implies
(and
(realp eps)
(< 0 eps)
(i-small eps)
(realp a)
(realp b)
(realp tm)
(standard-numberp tm))
(equal (* eps
a
(FLOOR1 (* (/ EPS) TM))
b)
(* (tm-eps-fun tm eps) a b)))
:hints (("Goal" :in-theory (disable tm-eps-fun-rw-1-thm
tm-eps-fun-rw-2-thm
tm-eps-fun-rw-3-thm))))
;; tm-eps-fun should be disabled. Enabling it may result in rewrite loops.
(in-theory (disable tm-eps-fun))
|
