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|
(in-package "ACL2")
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
;what is this file??
;; 2. equality comparison
(defun log= (x y)
(declare (xargs :guard t))
(if (equal x y) 1 0))
(defun log<> (x y)
(declare (xargs :guard t))
(if (equal x y) 0 1))
;; 3. unsigned inequalities
(defun log< (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (< x y) 1 0))
(defun log<= (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (<= x y) 1 0))
(defun log> (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (> x y) 1 0))
(defun log>= (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (>= x y) 1 0))
;; 2. equality comparison
(defun log= (x y)
(declare (xargs :guard t))
(if (equal x y) 1 0))
(defun log<> (x y)
(declare (xargs :guard t))
(if (equal x y) 0 1))
;; 3. unsigned inequalities
(defun log< (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (< x y) 1 0))
(defun log<= (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (<= x y) 1 0))
(defun log> (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (> x y) 1 0))
(defun log>= (x y)
(declare (xargs :guard (and (rationalp x) (rationalp y))))
(if (>= x y) 1 0))
;; 4. signed inequalities
;; The following function is not generated by translate-rtl, it is only needed
;; for the definitions of comp2<, comp2<=, etc.
(defun comp2 (x n)
(declare (xargs :guard (and (rationalp x) (integerp n))))
(if (< x (expt 2 (1- n)))
x
(- (- (expt 2 n) x))))
(defun comp2< (x y n)
(declare (xargs :guard (and (rationalp x) (rationalp y) (integerp n))))
(log< (comp2 x n) (comp2 y n)))
(defun comp2<= (x y n)
(declare (xargs :guard (and (rationalp x) (rationalp y) (integerp n))))
(log<= (comp2 x n) (comp2 y n)))
(defun comp2> (x y n)
(declare (xargs :guard (and (rationalp x) (rationalp y) (integerp n))))
(log> (comp2 x n) (comp2 y n)))
(defun comp2>= (x y n)
(declare (xargs :guard (and (rationalp x) (rationalp y) (integerp n))))
(log>= (comp2 x n) (comp2 y n)))
;; 5. unary logical operations
;make separate books for these? logior1 has one?
(defun logand1 (x n)
(declare (xargs :guard (integerp n)))
(log= x (1- (expt 2 n))))
(defun logior1 (x)
(declare (xargs :guard t))
(if (equal x 0) 0 1))
(defun logxor1 (src)
(declare (xargs :guard (integerp src)))
(if (oddp (logcount src)) 1 0))
;should rtl.lisp disable these fns?
;; log<
(defthm log<-bvecp
(bvecp (log< x y) 1)
:hints (("Goal" :in-theory (enable log<))))
(defthm log<-nonnegative-integer-type
(and (integerp (log< x y))
(<= 0 (log< x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log<))))
;this rule is no better than log<-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log<)))
;just a rewrite rule
(defthm log<-natp
(natp (log< x y)))
;; log<=
(defthm log<=-bvecp
(bvecp (log<= x y) 1)
:hints (("Goal" :in-theory (enable log<=))))
(defthm log<=-nonnegative-integer-type
(and (integerp (log<= x y))
(<= 0 (log<= x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log<=))))
;this rule is no better than log<=-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log<=)))
;just a rewrite rule
(defthm log<=-natp
(natp (log<= x y)))
;; log>
(defthm log>-bvecp
(bvecp (log> x y) 1)
:hints (("Goal" :in-theory (enable log>))))
(defthm log>-nonnegative-integer-type
(and (integerp (log> x y))
(<= 0 (log> x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log>))))
;this rule is no better than log>-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log>)))
;just a rewrite rule
(defthm log>-natp
(natp (log> x y)))
;; log>=
(defthm log>=-bvecp
(bvecp (log>= x y) 1)
:hints (("Goal" :in-theory (enable log>=))))
(defthm log>=-nonnegative-integer-type
(and (integerp (log>= x y))
(<= 0 (log>= x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log>=))))
;this rule is no better than log>=-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log>=)))
;just a rewrite rule
(defthm log>=-natp
(natp (log>= x y)))
;; log=
(defthm log=-bvecp
(bvecp (log= x y) 1)
:hints (("Goal" :in-theory (enable log=))))
(defthm log=-nonnegative-integer-type
(and (integerp (log= x y))
(<= 0 (log= x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log=))))
(defthm log=-commutative
(equal (log= x y)
(log= y x)))
;this rule is no better than log=-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log=)))
;just a rewrite rule
(defthm log=-natp
(natp (log= x y)))
;; log<>
(defthm log<>-bvecp
(bvecp (log<> x y) 1)
:hints (("Goal" :in-theory (enable log<>))))
(defthm log<>-nonnegative-integer-type
(and (integerp (log<> x y))
(<= 0 (log<> x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable log<>))))
;this rule is no better than log<>-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription log<>)))
;just a rewrite rule
(defthm log<>-natp
(natp (log<> x y)))
(defthm log<>-commutative
(equal (log<> x y)
(log<> y x)))
;; logand1
(defthm logand1-bvecp
(bvecp (logand1 x y) 1)
:hints (("Goal" :in-theory (enable logand1))))
(defthm logand1-nonnegative-integer-type
(and (integerp (logand1 x y))
(<= 0 (logand1 x y)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable logand1))))
;this rule is no better than logand1-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription logand1)))
;just a rewrite rule
(defthm logand1-natp
(natp (logand1 x y)))
;; logior1
(defthm logior1-bvecp
(bvecp (logior1 x) 1)
:hints (("Goal" :in-theory (enable logior1))))
(defthm logior1-nonnegative-integer-type
(and (integerp (logior1 x))
(<= 0 (logior1 x)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable logior1))))
;this rule is no better than logior1-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription logior1)))
;just a rewrite rule
(defthm logior1-natp
(natp (logior1 x)))
;; logxor1
(defthm logxor1-bvecp
(bvecp (logxor1 x) 1)
:hints (("Goal" :in-theory (enable logxor1))))
(defthm logxor1-nonnegative-integer-type
(and (integerp (logxor1 x))
(<= 0 (logxor1 x)))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable logxor1))))
;this rule is no better than logxor1-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription logxor1)))
;just a rewrite rule
(defthm logxor1-natp
(natp (logxor1 x)))
|
