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|
(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(local (include-book "../arithmetic/top"))
(local (in-theory (enable expt-minus)))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defthm bvecp-with-n-not-a-positive-integer
(implies (or (not (integerp k))
(<= k 0))
(equal (bvecp x k)
(equal 0 x)))
:hints (("Goal" :cases ((< x 1))
:in-theory (enable bvecp))))
(defthm bvecp-0
(bvecp 0 k)
:hints (("Goal" :in-theory (enable bvecp))))
;drop?
;just a special case of bvecp-with-n-not-a-positive-integer
(defthm bvecp-0-thm
(equal (bvecp x 0)
(equal x 0))
:hints (("Goal" :in-theory (enable bvecp))))
(defthm bvecp-ones
(implies (case-split (<= 0 k))
(bvecp (1- (expt 2 k)) k))
:hints (("goal" :in-theory (enable bvecp))))
(encapsulate
()
;k1 is a free var
(local (defthm bvecp-longer-aux
(implies (and (bvecp x k1)
(<= k1 k2)
(case-split (integerp k1))
(case-split (integerp k2))
)
(bvecp x k2))
:rule-classes ((:rewrite :match-free :all))
:hints (("Goal" :use (:instance expt-compare (lhs (expt 2 k1)) (rhs (expt 2 k2)))
:in-theory (set-difference-theories
(enable bvecp)
'(EXPT-COMPARE))
))))
;k1 is a free var
(defthm bvecp-longer
(implies (and (bvecp x k1)
(<= k1 k2)
(case-split (integerp k2))
)
(bvecp x k2))
:rule-classes ((:rewrite :match-free :all))
:hints (("Goal" :cases ((integerp k1)))))
)
;expensive and so disabled
;no free var
(defthmd bvecp-one-longer
(implies (and (integerp k)
(bvecp x (- k 1)))
(bvecp x k))
:hints (("Goal" :in-theory (disable bvecp-longer)
:use ((:instance bvecp-longer (k2 k) (k1 (- k 1))))))
:rule-classes ((:rewrite :backchain-limit-lst (nil 2))))
(defthm bvecp-of-non-integer
(implies (not (integerp x))
(not (bvecp x k)))
:hints (("Goal" :in-theory (enable bvecp))))
;gen (replace n+1 with an arbitrary integer > n)?
(defthm bvecp-expt-2-n
(implies (and (case-split (integerp n))
(case-split (<= 0 n))
)
(bvecp (expt 2 n) (+ 1 n)))
:hints (("Goal" :in-theory (set-difference-theories
(enable bvecp expt-split)
'(a15)))))
(defthm bvecp-if ; see comment in bvecp.lisp
(equal (bvecp (if test x y) k)
(if test (bvecp x k) (bvecp y k))))
; The following are analogous to mk-bvarr etc. in rtlarr.lisp.
;better name?
(defund mk-bvec (r k)
(declare (xargs :guard (integerp k)))
(if (bvecp r k) r 0))
(defthm mk-bvec-is-bvecp
(bvecp (mk-bvec r k) k)
:hints (("Goal" :in-theory (enable mk-bvec))))
(defthm mk-bvec-identity
(implies (bvecp r k)
(equal (mk-bvec r k) r))
:hints (("Goal" :in-theory (enable mk-bvec))))
;BOZO make a version to shift by a constant!
(defthm bvecp-shift
(implies (and (integerp x) ;note!
(<= 0 m)
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* x (expt 2 m)) n)
(bvecp x (- n m))))
:hints (("Goal" :in-theory (enable bvecp expt-split))))
(defthm bvecp-shift-alt
(implies (and (integerp x) ;note!
(<= 0 m)
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* (expt 2 m) x) n)
(bvecp x (- n m))))
:hints (("Goal" :in-theory (enable bvecp expt-split))))
;gen this!
;BOZO will this unify (* 2 x) with 0??
(defthm bvecp-shift-by-2
(implies (and (syntaxp (not (quotep x))) ;prevents loops...
(integerp x)
(<= 0 m) ;gen?
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* 2 x) n)
(bvecp x (- n 1))))
:hints (("Goal" :in-theory (enable bvecp expt-split))))
;gen?
;in general, rewrite (bvecp k n) where k is a constant to a fact about n
(defthm bvecp-1
(implies (and (<= 1 n)
(integerp n))
(bvecp 1 n))
:hints (("Goal" :in-theory (enable bvecp))))
;n is a free variable
;Disabled since may cause expensive backchaining.
(defthmd natp-bvecp
(implies (bvecp x n)
(natp x))
:hints (("Goal" :in-theory (enable bvecp)))
:rule-classes ((:rewrite :match-free :once)))
#|
;kill this rule?
;could just open bvecp instead of using this rule?
(defthmd bvecp<=
(implies (and (natp n) ;gen?
(bvecp x n))
(<= x (1- (expt 2 n))))
:hints (("Goal" :in-theory (enable bvecp))))
|#
(defthmd bvecp-forward
(implies (bvecp x k)
(and (integerp x)
(<= 0 x)
(< x (expt 2 k)))) ;tigher-bound?
:hints (("Goal" :in-theory (enable bvecp)))
:rule-classes :forward-chaining)
(defthm bvecp-product
(implies (and (bvecp x m)
(bvecp y n)
)
(bvecp (* x y) (+ m n)))
:rule-classes ()
:hints (("Goal" :cases ((and (integerp m) (equal 0 y) (integerp n))
(and (integerp m) (equal 0 y) (not (integerp n)))
(and (integerp m) (not (equal 0 y)) (integerp n))
(and (integerp m) (not (equal 0 y)) (not (integerp n)))
(and (not (integerp m))(equal 0 y) (integerp n))
(and (not (integerp m)) (equal 0 y) (not (integerp n)))
(and (not (integerp m)) (not (equal 0 y)) (integerp n))
(and (not (integerp m)) (not (equal 0 y)) (not (integerp n)))
)
:in-theory (enable bvecp natp expt-split))))
(defthmd bvecp-1-rewrite
(equal (bvecp x 1)
(or (equal x 0) (equal x 1)))
:hints (("Goal" :in-theory (enable bvecp))))
;make another for not-equal-0 implies equal-1?
(defthm bvecp-1-0
(implies (and (bvecp x 1)
(not (equal x 1)))
(equal x 0))
:rule-classes :forward-chaining
:hints (("Goal" :use bvecp-1-rewrite)))
(defthm bvecp+1
(implies (and (natp n)
(bvecp x n))
(bvecp x (+ 1 n)))
:hints (("Goal" :in-theory (enable bvecp expt-split))))
;same as bvecp-longer.decide which param names to use. j and k??
(defthmd bvecp-monotone
(implies (and (bvecp x n)
(<= n m)
(case-split (integerp m))
)
(bvecp x m)))
;This bounds the amount of carry out that we can have from the sum.
(defthm bvecp-sum-of-bvecps
(implies (and (bvecp x (1- k))
(bvecp y (1- k))
(case-split (integerp k)))
(bvecp (+ x y) k))
:hints (("Goal" :in-theory (enable bvecp expt-split)))
)
;add rule that (not (natp x)) implies (not (bvecp x k)) ??
;exported in lib/
(defthmd bvecp-0-1
(implies (and (bvecp x 1)
(not (equal x 0)))
(equal x 1))
:rule-classes :forward-chaining)
|
