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(in-package "ACL2")
(include-book "cat-def")
(local (include-book "setbitn-proofs"))
(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))))
(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)))
(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))))))
(defund setbitn (x w n y)
(declare (xargs :guard (and (natp x)
(natp y)
(integerp n)
(<= 0 n)
(integerp w)
(< n w))
:verify-guards nil))
(setbits x w n n y))
(defthm setbitn-nonnegative-integer-type
(and (integerp (setbitn x w n y))
(<= 0 (setbitn x w n y)))
:rule-classes (:type-prescription))
;this rule is no better than setbits-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription setbitn)))
(defthm setbitn-natp
(natp (setbitn x w n y)))
;add setbitn-bvecp-simple?
(defthm setbitn-bvecp
(implies (and (<= w k)
(case-split (integerp k)))
(bvecp (setbitn x w n y) k)))
(defthm setbitn-rewrite
(implies (syntaxp (quotep n))
(equal (setbitn x w n y)
(setbits x w n n y))))
;gen?
(defthm bitn-setbitn
(implies (and (case-split (bvecp y 1))
(case-split (< 0 w))
(case-split (< n w))
(case-split (< k w))
(case-split (<= 0 k))
(case-split (integerp w))
(case-split (integerp n))
(<= 0 n)
(case-split (integerp k))
)
(equal (bitn (setbitn x w n y) k)
(if (equal n k)
y
(bitn x k)))))
(defthm setbitn-setbitn
(implies (and (case-split (<= 0 n))
(case-split (< n w))
(case-split (integerp w))
(case-split (integerp n))
)
(equal (setbitn (setbitn x w n y) w n y2)
(setbitn x w n y2))))
(defthm setbitn-does-nothing
(implies (and (case-split (<= 0 n))
(case-split (< n w))
(case-split (integerp w))
(case-split (integerp n))
)
(equal (setbitn x w n (bitn x n))
(bits x (1- w) 0))))
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