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|
(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(include-book "cat-def")
(local (include-book "../arithmetic/top"))
(local (include-book "bits"))
(local (include-book "bitn"))
(local (include-book "bvecp"))
(local (include-book "cat"))
(defund bitn (x n)
(declare (xargs :guard (and (natp x)
(natp n))
:verify-guards nil))
(mbe :logic (bits x n n)
:exec (if (evenp (ash x (- n))) 0 1)))
#|
Currently we expect to leave setbits enabled so that it rewrites to cat, but there are some lemmas below which
might be useful if we choose to keep setbits disabled...
is this comment still valid? :
;it may happen that setbitn is called with an index which is a signal rather than a constant.
;in that case, we probably don't want it to expand to setbits.
;thus, we always expect the indices in setbits calls to be constants
;Set bits I down to J of the W-bit value X to Y.
(setbits x w i j y) is only well-defined when the following predicate is true:
(and (natp w)
(bvecp x w)
(integerp i)
(integerp j)
(<= 0 j)
(<= j i)
(< i w)
(bvecp y (+ 1 i (- j))))
|#
#| old:
(defund setbits (x w i j y)
(declare (xargs :guard (and (rationalp x) (rationalp y)
(acl2-numberp i) (acl2-numberp j) (acl2-numberp w))))
(if (not (natp w))
0
(cat (bits x (1- w) (+ 1 i))
(+ -1 w (- i))
(cat (bits y (+ i (- j)) 0)
(+ 1 i (- j))
(bits x (1- j) 0)
j)
(+ 1 i))))
|#
;Note: when j is 0, there is not lower part of x, but we have cat-with-n-0 to handle this case.
(defund setbits (x w i j y)
(declare (xargs :guard (and (natp x)
(natp y)
(integerp i)
(integerp j)
(<= 0 j)
(<= j i)
(integerp w)
(< i w))
:verify-guards nil))
(mbe :logic (cat (bits x (1- w) (1+ i))
(+ -1 w (- i))
(cat (bits y (+ i (- j)) 0)
(+ 1 i (- j))
(bits x (1- j) 0)
j)
(1+ i))
:exec (cond ((int= j 0)
(cond ((int= (1+ i) w)
(bits y (+ i (- j)) 0))
(t
(cat (bits x (1- w) (1+ i))
(+ -1 w (- i))
(bits y (+ i (- j)) 0)
(1+ i)))))
((int= (1+ i) w)
(cat (bits y (+ i (- j)) 0)
(+ 1 i (- j))
(bits x (1- j) 0)
j))
(t
(cat (bits x (1- w) (1+ i))
(+ -1 w (- i))
(cat (bits y (+ i (- j)) 0)
(+ 1 i (- j))
(bits x (1- j) 0)
j)
(1+ i))))))
#| old defn
(defun setbits (x i j y)
(ocat (ocat (ash x (- (1+ i)))
y
(1+ (- i j)))
(bits x (1- j) 0)
j))
|#
(defthm setbits-nonnegative-integer-type
(and (integerp (setbits x w i j y))
(<= 0 (setbits x w i j y)))
:hints (("Goal" :in-theory (enable setbits)))
:rule-classes (:type-prescription)
)
;this rule is no better than setbits-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription setbits)))
(defthm setbits-natp
(natp (setbits x w i j y)))
;BOZO r-c?
(defthm setbits-upper-bound
(< (setbits x w i j y) (expt 2 w))
:hints (("Goal" :in-theory (enable setbits cat-upper-bound))))
(defthm setbits-bvecp-simple
(bvecp (setbits x w i j y) w)
:hints (("goal" :in-theory (enable bvecp))))
(defthm setbits-bvecp
(implies (and (<= w k)
(case-split (integerp k))
)
(bvecp (setbits x w i j y) k))
:hints (("goal" :use setbits-bvecp-simple
:in-theory (disable setbits-bvecp-simple))))
(defthm setbits-does-nothing
(implies (and (case-split (< i w))
(case-split (<= j i))
(case-split (integerp i))
(case-split (integerp j))
(case-split (<= 0 j))
)
(equal (setbits x w i j (bits x i j))
(bits x (1- w) 0)))
:hints (("Goal" :in-theory (enable setbits))))
#| old, prove the two match for bvecps
(defun oldsetbits (x i j y)
(ocat (ocat (ash x (- (1+ i)))
y
(1+ (- i j)))
(bits x (1- j) 0)
j))
;we had this before
(defthm oldsetbits-rewrite-1
(implies (and (bvecp x n)
(natp n)
(> n 0)
(natp i)
(natp j)
(<= j i)
(bvecp y (1+ (- i j))))
(equal (oldsetbits x i j y)
(ocat (ocat (bits x (1- n) (1+ i))
y
(1+ (- i j)))
(bits x (1- j) 0)
j))))
(defthm setbits-match
(implies (and (bvecp x n)
(natp n)
(> n 0)
(natp w)
(<= n w)
(bvecp y (1+ (- i j)))
(natp i)
(natp j)
(<= j i))
(equal (oldsetbits x i j y)
(setbits x w i j y)))
:otf-flg t
:hints (("Goal" :in-theory (enable setbits oldsetbits bits-does-nothing
natp))))
|#
;taking bits from the lower third
;slow proof with may cases!
(defthm bits-setbits-1
(implies (and (< k j)
(case-split (<= 0 w))
(case-split (< i w))
(case-split (<= 0 l))
(case-split (<= j i)) ;drop?
(case-split (integerp w))
(case-split (integerp i))
(case-split (integerp j))
)
(equal (bits (setbits x w i j y) k l)
(bits x k l)))
:hints (("Goal" :in-theory (enable setbits))))
;taking bits from the middle third
;slow proof with may cases!
(defthm bits-setbits-2
(implies (and (<= k i)
(<= j l)
(case-split (integerp i))
(case-split (<= 0 j))
(case-split (integerp j))
(case-split (acl2-numberp k)); (case-split (integerp k))
(case-split (acl2-numberp l)) ; (case-split (integerp l))
(case-split (integerp w))
(case-split (<= 0 w))
(case-split (< i w))
)
(equal (bits (setbits x w i j y) k l)
(bits y (- k j) (- l j))))
:hints (("Goal" :in-theory (enable setbits natp))))
;taking bits from the upper third
(defthm bits-setbits-3
(implies (and (< i l)
(case-split (< i w))
(case-split (< k w)) ;handle this?
(case-split (<= j i))
(case-split (<= 0 l))
(case-split (<= 0 j))
(case-split (<= 0 w))
(case-split (integerp l))
(case-split (integerp w))
(case-split (integerp i))
(case-split (integerp j))
(case-split (integerp k))
)
(equal (bits (setbits x w i j y) k l)
(bits x k l)))
:hints (("Goal" :in-theory (enable setbits natp))))
(defthm setbits-with-0-width
(equal (setbits x 0 i j y)
0)
:hints (("Goal" :cases ((integerp j))
:in-theory (enable setbits))))
;add case-splits?
;why can't i prove this from bits-setbits?
(defthm bitn-setbits-1
(implies (and (< k j) ;case 1
(< i w)
(<= 0 i)
(<= 0 j)
(<= 0 k)
(<= j i)
(integerp k)
(integerp w)
(integerp i)
(integerp j)
)
(equal (bitn (setbits x w i j y) k)
(bitn x k)))
:hints (("Goal" :in-theory (enable setbits)))
)
(defthm bitn-setbits-2
(implies (and(<= k i) ;;case-2
(<= j k) ;;case-2
(<= 0 i)
(<= 0 j)
(< i w)
(integerp k)
(integerp w)
(integerp i)
(integerp j)
)
(equal (bitn (setbits x w i j y) k)
(bitn y (- k j))))
:hints (("Goal" :in-theory (enable setbits)))
)
(defthm bitn-setbits-3
(implies (and (< i k) ;;case-3
(< k w) ;;case-3
; (< i w)
(<= 0 i)
(<= 0 j)
(<= j i)
(integerp i)
(integerp j)
(integerp k)
(integerp w))
(equal (bitn (setbits x w i j y) k)
(bitn x k)))
:hints (("Goal" :in-theory (enable setbits)))
)
;taking a slice of each of the lower two thirds.
(defthm bits-setbits-4
(implies (and (<= k i) ;;case-4
(<= j k) ;;case-4
(< l j) ;;case-4
(< i w)
(<= 0 j)
(<= 0 l)
(integerp i)
(integerp j)
(integerp w)
(acl2-numberp l) ;(integerp l)
)
(equal (bits (setbits x w i j y) k l)
(cat (bits y (- k j) 0)
(+ 1 k (- j))
(bits x (1- j) l)
(- j l))))
:hints (("Goal" :in-theory (enable setbits))))
;taking a slice of each of the upper two thirds.
(defthm bits-setbits-5
(implies (and (< i k) ;case-5
(<= l i) ;case-5
(<= j l) ;case-5
(< k w) ;case-5 ;BOZO drop stuff like this?
(<= 0 j)
(integerp i)
(integerp j)
(integerp w)
(acl2-numberp l) ;(integerp l)
)
(equal (bits (setbits x w i j y) k l)
(cat (bits x k (1+ i))
(+ k (- i))
(bits y (- i j) (- l j))
(1+ (- i l)))))
:hints (("Goal" :in-theory (enable setbits))))
;taking a slice of each of the thirds.
;make one huge bits-setbits lemma?
(defthm bits-setbits-6
(implies (and (< i k) ;;case-6
(< l j) ;;case-6
(<= j i)
(< k w)
(<= 0 l)
(integerp i)
(integerp j)
(acl2-numberp l) ; (integerp l)
(integerp w)
)
(equal (bits (setbits x w i j y) k l)
(cat (bits x k (1+ i))
(+ k (- i))
(cat (bits y (+ i (- j)) 0)
(1+ (- i j))
(bits x (1- j) l)
(- j l))
(+ 1 i (- l)))))
:hints (("Goal" :in-theory (enable setbits))))
;prove that if (not (natp w)) setbits = 0 .
;combining these adjacent ranges [i..j][k..l]
(defthm setbits-combine
(implies (and (equal j (+ k 1))
(case-split (<= j i))
(case-split (<= l k))
(case-split (natp w))
(case-split (natp i))
(case-split (natp j))
(case-split (natp k))
(case-split (natp l))
)
(equal (setbits (setbits x w k l y1) w i j y2)
(setbits x w i l (cat y2
(+ 1 i (- j))
y1
(+ 1 k (- l))
))))
:hints (("goal" :in-theory (enable setbits))))
(defthm setbits-combine-2
(implies (and (equal j (+ k 1))
(case-split (< i w))
(case-split (<= j i))
(case-split (<= l k))
(case-split (natp w))
(case-split (natp i))
(case-split (natp j))
(case-split (natp k))
(case-split (natp l))
)
(equal (setbits (setbits x w i j y2) w k l y1)
(setbits x w i l (cat y2
(+ 1 i (- j))
y1
(+ 1 k (- l))
))))
:hints (("goal" :in-theory (enable setbits))))
(defthm setbits-combine-3
(implies (and (equal j (+ k 1))
(case-split (< i w))
(case-split (<= j i))
(case-split (<= l k))
(case-split (natp w))
(case-split (natp i))
(case-split (natp j))
(case-split (natp k))
(case-split (natp l)))
(equal (setbits (setbits x w i j y2) w k l y1)
(setbits x w i l
(cat y2 (+ 1 i (- j))
y1 (+ 1 k (- l)))))))
(defthm setbits-all
(implies (and (equal i (1- w))
(case-split (bvecp y w))
)
(equal (setbits x w i 0 y)
y))
:hints (("goal" :in-theory (enable setbits))))
|
