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;;;***************************************************************
;;;An ACL2 Library of Floating Point Arithmetic
;;;David M. Russinoff
;;;Advanced Micro Devices, Inc.
;;;February, 1998
;;;***************************************************************
(in-package "ACL2")
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(include-book "fp2")
(local (include-book "even-odd"))
;;; natp ;;;
;Currently, we plan to leave natp enabled...
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defthm natp-compound-recognizer
(equal (natp x)
(and (integerp x)
(<= 0 x)))
:rule-classes :compound-recognizer)
; The fpaf3a proof of far-exp-low-lemma-1 in far.lisp requires the
; following to be a :rewrite rule, not just a :type-prescription rule.
; Let's make most or all of our :type-prescription rules into :rewrite
; rules as well.
(defthmd natp+
(implies (and (natp x) (natp y))
(natp (+ x y))))
;move
(defthmd natp*
(implies (and (natp x) (natp y))
(natp (* x y))))
;abs
;Currently, we plan to leave abs enabled, but here are some rules about it:
(defthm abs-nonnegative-acl2-numberp-type
(implies (case-split (acl2-numberp x))
(and (<= 0 (abs x))
(acl2-numberp (abs x))))
:rule-classes (:TYPE-PRESCRIPTION))
(defthm abs-nonnegative-rationalp-type
(implies (case-split (rationalp x))
(and (<= 0 (abs x))
(rationalp (abs x))))
:rule-classes (:TYPE-PRESCRIPTION))
(defthm abs-nonnegative-integerp-type
(implies (integerp x)
(and (<= 0 (abs x))
(rationalp (abs x))))
:rule-classes (:TYPE-PRESCRIPTION))
(defthm abs-nonnegative
(<= 0 (abs x)))
(local (include-book "fl"))
(defthm fl-def-linear
(implies (case-split (rationalp x))
(and (<= (fl x) x)
(< x (1+ (fl x)))))
:rule-classes :linear)
;(in-theory (disable a13)) ;the same rule as fl-def-linear!
;bad? free var.
(defthm fl-monotone-linear
(implies (and (<= x y)
(rationalp x)
(rationalp y))
(<= (fl x) (fl y)))
:rule-classes :linear)
(defthm n<=fl-linear
(implies (and (<= n x)
(rationalp x)
(integerp n))
(<= n (fl x)))
:rule-classes :linear)
;may need to disable? <-- why did I write that? expensive backchaining?
(defthm fl+int-rewrite
(implies (and (integerp n)
(rationalp x))
(equal (fl (+ x n)) (+ (fl x) n))))
;from fl.lisp
(defthm fl/int-rewrite
(implies (and (integerp n)
(<= 0 n) ;can't gen?
(rationalp x))
(equal (fl (/ (fl x) n))
(fl (/ x n))))
:hints (("Goal" :use ((:instance fl/int-1)
(:instance fl/int-2)))))
;needed?
(defthm fl-integer-type
(integerp (fl x))
:rule-classes (:type-prescription))
;this rule is no better than fl-integer-type and might be worse:
(in-theory (disable (:type-prescription fl)))
(defthm fl-int
(implies (integerp x)
(equal (fl x) x)))
(encapsulate
()
(local (include-book "fl"))
(defthm fl-integerp
(equal (equal (fl x) x)
(integerp x))))
(defthm fl-unique
(implies (and (rationalp x)
(integerp n)
(<= n x)
(< x (1+ n)))
(equal (fl x) n))
:rule-classes ())
(encapsulate
()
(local (include-book "expt"))
(defthm expt-2-positive-rational-type
(and (rationalp (expt 2 i))
(< 0 (expt 2 i)))
:rule-classes (:rewrite (:type-prescription :typed-term (expt 2 i))))
(defthm expt-2-positive-integer-type
(implies (<= 0 i)
(and (integerp (expt 2 i))
(< 0 (expt 2 i))))
:rule-classes (:type-prescription))
;the rewrite rule counterpart to expt-2-positive-integer-type
(defthm expt-2-integerp
(implies (<= 0 i)
(integerp (expt 2 i))))
; (in-theory (disable a14)) ;the rules above are better than this one for (expt 2 i)
(defthm expt-2-type-linear
(implies (<= 0 i)
(<= 1 (expt 2 i)))
:rule-classes ((:linear :trigger-terms ((expt 2 i)))))
(defthmd expt-split
(implies (and (integerp i)
(integerp j)
(case-split (acl2-numberp r)) ;(integerp r)
(case-split (not (equal r 0)))
)
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(theory-invariant (incompatible (:rewrite expt-split)
(:definition a15))
:key expt-split-invariant)
(defthmd expt-weak-monotone
(implies (and (integerp n)
(integerp m))
(equal (<= (expt 2 n) (expt 2 m))
(<= n m))))
(defthmd expt-weak-monotone-linear
(implies (and (<= n m)
(case-split (integerp n))
(case-split (integerp m)))
(<= (expt 2 n) (expt 2 m)))
:rule-classes ((:linear :match-free :all)))
(defthmd expt-strong-monotone
(implies (and (integerp n)
(integerp m))
(equal (< (expt 2 n) (expt 2 m))
(< n m))))
(defthmd expt-strong-monotone-linear
(implies (and (< n m)
(case-split (integerp n))
(case-split (integerp m))
)
(< (expt 2 n) (expt 2 m)))
:rule-classes ((:linear :match-free :all)))
(defthmd a15
(implies (and (rationalp i)
(not (equal i 0))
(integerp j1)
(integerp j2))
(and (equal (* (expt i j1) (expt i j2))
(expt i (+ j1 j2)))
(equal (* (expt i j1) (* (expt i j2) x))
(* (expt i (+ j1 j2)) x))))
)
)
; The next two events were added by Matt K. June 2004: Some proofs require
; calls of expt to be evaluated, but some calls are just too large (2^2^n for
; large n). So we use the following hack, which allows calls of 2^n for n<130
; to be evaluated even when the executable-counterpart of expt is disabled.
; The use of 130 is somewhat arbitrary, chosen in the hope that it suffices for
; relieving of hyps related to widths of bit vectors
(defun expt-exec (r i)
(declare (xargs :guard
(and (acl2-numberp r)
(integerp i)
(not (and (eql r 0) (< i 0))))
:guard-hints (("Goal" :expand (hide (expt r i))))))
(mbe :logic (hide (expt r i)) ; hide may avoid potential loop
:exec (expt r i)))
(defthm expt-2-evaluator
(implies (syntaxp (and (quotep n)
(natp (cadr n))
(< (cadr n) 130)
))
(equal (expt 2 n)
(expt-exec 2 n)))
:hints (("Goal" :expand ((hide (expt 2 n))))))
;weakly?
;cases for other signs?
(defthm *-doubly-monotonic
(implies (and (rationalp x)
(rationalp y)
(rationalp a)
(rationalp b)
(<= 0 x)
(<= 0 y)
(<= 0 a)
(<= 0 b)
(<= x y)
(<= a b))
(<= (* x a) (* y b)))
:rule-classes ())
(defund fl-half (x)
; (declare (xargs :guard (real/rationalp x)))
(1- (fl (/ (1+ x) 2))))
(defthm fl-half-lemma
(implies (and (integerp x)
(not (integerp (/ x 2)))) ;if x is odd, ...
(= x (1+ (* 2 (fl-half x)))))
:rule-classes ()
:hints (("goal" :in-theory (e/d (fl-half) (fl-int))
:use ((:instance x-or-x/2)
(:instance fl-int (x (/ (1+ x) 2)))))))
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