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(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(include-book "ground-zero")
(local (include-book "logeqv"))
(local (include-book "logorc1"))
(local (include-book "lognot"))
(local (include-book "../arithmetic/top"))
(defthm logxor-integerp-type
(integerp (logxor i j))
:rule-classes :type-prescription)
(defthm logxor-0
(implies (case-split (integerp i))
(equal (logxor 0 i)
i))
:hints (("goal" :in-theory (enable logxor))))
(defthm logxor-non-negative-integer-type-prescription
(implies (and (<= 0 i)
(<= 0 j))
(and (<= 0 (logxor i j))
(integerp (logxor i j))))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (enable logxor))))
(defthm logxor-non-negative
(implies (and (<= 0 i)
(<= 0 j)
)
(<= 0 (logxor i j)))
:rule-classes (:rewrite :type-prescription)
:hints (("goal" :in-theory (enable logxor))))
(defthm logxor-even
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (integerp (* 1/2 (logxor i j)))
(or (and (integerp (* 1/2 i))
(integerp (* 1/2 j)))
(and (not (integerp (* 1/2 j)))
(not (integerp (* 1/2 i)))))))
:hints (("goal" :in-theory (enable logxor))))
(defthm logxor-commutative
(equal (logxor j i)
(logxor i j))
:hints (("goal" :in-theory (enable logxor))))
(defthm logxor-with-non-integer-arg
(implies (not (integerp i))
(and (equal (logxor i j)
(ifix j))
(equal (logxor j i)
(ifix j))))
:hints (("goal" :in-theory (enable logxor))))
;do we really want to go to lognot?
(defthm logxor-with-an-arg-of-minus-one
(implies (case-split (integerp i))
(equal (logxor -1 i)
(lognot i)))
:hints (("goal" :in-theory (enable logxor))))
(defthmd floor-logxor-by-2
(implies (and (case-split (integerp i))
(case-split (integerp j)))
(equal (floor (logxor i j) 2)
(logxor (floor i 2) (floor j 2))))
:hints (("Goal" :in-theory (set-difference-theories
(enable logxor)
'(lognot logeqv floor)))))
(defthm fl-logxor-by-2
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (fl (* 1/2 (logxor i j)))
(logxor (fl (* 1/2 i)) (fl (* 1/2 j)))))
:hints (("goal" :in-theory (enable logxor))))
(defthm mod-logxor-by-2
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (mod (logxor i j) 2)
(logxor (mod i 2) (mod j 2))))
:hints (("Goal" :in-theory (enable mod-by-2))))
(defthmd logxor-def
(implies (and (case-split (integerp i))
(case-split (integerp j))
)
(equal (logxor i j)
(+ (* 2 (logxor (fl (* 1/2 i)) (fl (* 1/2 j))))
(logxor (mod i 2) (mod j 2)))))
:rule-classes ((:definition :controller-alist ((binary-logxor t t))))
:hints (("goal" :use (:instance fl-mod-equal
(x (logxor i j))
(y (+ (* 2 (logxor (fl (* 1/2 i)) (fl (* 1/2 j))))
(logxor (mod i 2) (mod j 2)))))
:in-theory (enable mod-by-2))))
;i'm not sure which way this rule should go but note that both parts of this rule rewrite to the same rhs
(defthm lognot-logxor
(and (equal (logxor (lognot i) j)
(lognot (logxor i j)))
(equal (logxor j (lognot i))
(lognot (logxor i j))))
:hints (("goal" :in-theory (enable logxor ))))
(defthm logxor-associative
(equal (logxor (logxor i j) k)
(logxor i (logxor j k)))
:hints (("subgoal *1/2" :use ( ;(:instance logxor-assoc-1)
(:instance fl-mod-equal
(x (logxor (logxor i j) k))
(y (logxor i (logxor j k))))))
("goal" :in-theory (enable logxor-def mod-by-2)
:induct ( logand-three-args-induct i j k))
))
(defthm logxor-commutative-2
(equal (logxor j i k)
(logxor i j k))
:hints (("Goal" :in-theory (disable LOGXOR-ASSOCIATIVE
logxor-commutative)
:use (LOGXOR-ASSOCIATIVE
logxor-commutative
(:instance LOGXOR-ASSOCIATIVE (j i) (i j))))))
(defthm logxor-combine-constants
(implies (syntaxp (and (quotep i)
(quotep j)))
(equal (logxor i j k)
(logxor (logxor i j) k))))
(defthm logxor-self
(equal (logxor i i) 0)
:hints (("goal" :in-theory (enable logxor))))
(defthmd logxor-def-rewrite
(implies (and (case-split (integerp x))
(case-split (integerp y))
)
(equal (logxor x y)
(+ (* 2 (logxor (fl (/ x 2)) (fl (/ y 2))))
(logxor (mod x 2) (mod y 2)))))
:hints (("Goal" :in-theory (enable logxor-def)))
:rule-classes ((:definition :controller-alist ((binary-logxor t t)))))
;gen?
(defthm logxor-upper-bound-tight
(implies (and (< i (expt 2 n))
(< j (expt 2 n))
(integerp i) (>= i 0)
(integerp j) (>= j 0)
(integerp n) (>= n 0)
)
(<= (logxor i j) (1- (expt 2 n))))
:hints (("Goal" :induct (op-dist-induct i j n))
("Subgoal *1/2" :in-theory (set-difference-theories
(enable expt-split
)
'(a15))
:use ((:instance logxor-def)
(:instance mod012 (x i))
(:instance mod012 (x j))))))
;change var names
(defthm logxor-upper-bound
(implies (and (< i (expt 2 n))
(< j (expt 2 n))
(integerp i) (>= i 0)
(integerp j) (>= j 0)
(integerp n) (>= n 0)
)
(< (logxor i j) (expt 2 n)))
:hints (("Goal" :in-theory (disable logxor-upper-bound-tight)
:use (:instance logxor-upper-bound-tight))))
(defthm logxor-bvecp
(implies (and (bvecp x n)
(bvecp y n)
(natp n) ;gen?
)
(bvecp (logxor x y) n))
:hints (("Goal" :in-theory (enable bvecp))))
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