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|
(in-package "ACL2")
(include-book "ground-zero")
(include-book "../arithmetic/power2p")
(include-book "../arithmetic/negative-syntaxp")
(local (include-book "bitn-proofs"))
(set-inhibit-warnings "theory") ; avoid warning in the next event
(local (in-theory nil))
;; Necessary defuns:
(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 fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund bits (x i j)
(declare (xargs :guard (and (natp x)
(natp i)
(natp j))
:verify-guards nil))
(mbe :logic (if (or (not (integerp i))
(not (integerp j)))
0
(fl (/ (mod x (expt 2 (1+ i))) (expt 2 j))))
:exec (if (< i j)
0
(logand (ash x (- j)) (1- (ash 1 (1+ (- i j))))))))
(defun expo-measure (x)
; (declare (xargs :guard (and (real/rationalp x) (not (equal x 0)))))
(cond ((not (rationalp x)) 0)
((< x 0) '(2 . 0))
((< x 1) (cons 1 (fl (/ x))))
(t (fl x))))
(defund expo (x)
(declare (xargs :guard t
:measure (expo-measure x)))
(cond ((or (not (rationalp x)) (equal x 0)) 0)
((< x 0) (expo (- x)))
((< x 1) (1- (expo (* 2 x))))
((< x 2) 0)
(t (1+ (expo (/ x 2))))))
;;
;; Begin bitn stuff...
;;
(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)))
(defthm bitn-with-n-not-an-integer
(implies (not (integerp n))
(equal (bitn x n)
0)))
(defthm bitn-of-non-rational
(implies (not (rationalp x))
(equal (bitn x n)
0)))
(defthm bitn-nonnegative-integer
(and (integerp (bitn x n))
(<= 0 (bitn x n)))
:rule-classes (:type-prescription))
;this rule is no better than bitn-nonnegative-integer and might be worse:
(in-theory (disable (:type-prescription bitn)))
(defthm bitn-natp
(natp (bitn x n)))
(defthm bitn-upper-bound
(<= (bitn x n) 1))
(defthm bitn-upper-bound-linear
(<= (bitn x n) 1)
:rule-classes ((:LINEAR :TRIGGER-TERMS ((bitn x n)))))
;include separate cases?
;BOZO one of the branches simplifies to 0 - see bits-minus
(defthm bitn-minus
(implies (and (syntaxp (negative-syntaxp x))
(case-split (rationalp x)) ;gen?
(case-split (integerp n))
)
(equal (bitn x n)
(if (integerp (/ x (expt 2 (+ 1 n))))
(- (bitn (- x) n))
(if (integerp (/ x (expt 2 n)))
(- 2 (bitn (- x) n))
(- 1 (bitn (- x) n)))))))
;1 rewrite to odd?
;trying disabled
(defthmd bitn-0-rewrite-to-even
(implies (integerp x)
(equal (equal (bitn x 0) 0)
(integerp (* 1/2 x)))))
;we probably want this enabled in lib/ but not in support/
(defthmd bits-n-n-rewrite
(equal (bits x n n)
(bitn x n)))
(theory-invariant (incompatible (:rewrite bits-n-n-rewrite)
(:definition bitn)
)
:key bitn-and-bits-n-n-can-loop)
(defthm bitn-0-1
(or (equal (bitn x n) 0)
(equal (bitn x n) 1))
:rule-classes nil)
;my strategy with the rules below is to prefer (not (equal (bitn x n) 0)) over (equal (bitn x n) 1)
;this allows subsumption to ...
;but maybe this is a bad idea!
;BOZO if we have f-w chaining rule to handle this issue, perhaps drop these rules?
;bad to have both?
(defthm bitn-not-0-means-1
(equal (not (equal (bitn x n) 0))
(equal (bitn x n) 1)))
(defthm bitn-not-1-means-0
(equal (not (equal (bitn x n) 1))
(equal (bitn x n) 0)))
;these are bad rules?
(in-theory (disable bitn-not-1-means-0 bitn-not-0-means-1))
(defthm bitn-bitn
(equal (bitn (bitn x n) 0)
(bitn x n)))
(defthm bitn-known-not-0-replace-with-1
(implies (not (equal (bitn x n) 0)) ; backchain-limit?
(equal (bitn x n)
1))
:rule-classes ((:rewrite :backchain-limit-lst (1)))
)
;needed?
(defthm bitn->-0
(equal (< 0 (bitn x n))
(not (equal 0 (bitn x n)))))
(defthm bitn-<-1
(equal (< (BITN X n) 1)
(equal (BITN X n) 0)))
;useful if bitn-upper-bound and bitn-upper-bound-2 are disabled
(defthm bitn-not->-1
(implies (and (syntaxp (quotep k))
(<= 1 k))
(equal (< k (bitn x n))
nil)))
;useful if bitn-upper-bound and bitn-upper-bound-2 are disabled
(defthm bitn-<=-1
(implies (and (syntaxp (quotep k))
(< 1 k))
(equal (< (bitn x n) k)
t)))
(defthmd bitn-rec-0
(implies (integerp x)
(equal (bitn x 0)
(mod x 2))))
;rename?
;is there a bits analog of this theorem?
;BOZO change formal k to n
(defthmd bitn-rec-pos
(implies (< 0 k) ;k cannot be 0 or negative
(equal (bitn x k)
(bitn (fl (/ x 2)) (1- k))))
:rule-classes ((:definition :controller-alist ((bitn t t)))))
;BOZO change k param to n
(defthmd bitn-def
(implies (case-split (integerp k))
(equal (bitn x k)
(mod (fl (/ x (expt 2 k)))
2))))
;make bit-not, bit-and, etc. ?
;BOZO or remove this function?
(defun not-eric (x)
(if (equal x 0)
1
0))
(defthm bitn-drop-crucial-bit-and-flip-result
(implies (and (case-split (rationalp x))
(case-split (integerp n)) ;drop?
)
(and (equal (bitn (+ (expt 2 n) x) n)
(not-eric (bitn x n)))
(equal (bitn (+ x (expt 2 n)) n)
(not-eric (bitn x n))))))
;BOZO this looped!
(defthmd bitn-drop-crucial-bit-and-flip-result-alt-gen
(implies (and (syntaxp (and (quotep j)
(< (cadr j) (expt 2 (+ 1 (cadr n)))) ;bitn-sum-lowbits does most of the work
(>= (cadr j) (expt 2 (cadr n)))))
(rationalp j)
(rationalp x)
(integerp n)
)
(equal (bitn (+ j x) n)
(not-eric (bitn (+ (- j (expt 2 n)) x) n)))))
;for negative constants j
;might be slow if the negative constant has a large absolute value
;make a negative version of bitn-sum-lowbits
(defthm bitn-add-crucial-bit-and-flip-result
(implies (and (syntaxp (and (quotep j)
(quotep n)
(< (cadr j) 0)))
(rationalp j)
(rationalp x)
(integerp n)
)
(equal (bitn (+ j x) n)
(not-eric (bitn (+ (+ j (expt 2 n)) x) n)))))
(defthm bitn-equal-to-silly-value
(implies (and (syntaxp (quotep k))
(not (or (equal 0 k) (equal 1 k)))
)
(equal (equal k (bitn x n))
nil)))
(defthm bitn-split-around-zero
(implies (and (<= (- (expt 2 n)) x)
(< x (expt 2 n))
(rationalp x)
(integerp n)
)
(equal (equal (bitn x n) 0)
(<= 0 x))))
;drop silly hyps like: (<= -128 (bitn x 24))
(defthm bitn-drop-silly-bound
(implies (and (syntaxp (quotep k))
(<= k 0)
)
(equal (< (bitn x n) k)
nil)))
(defthm bitn-drop-silly-bound-2
(implies (and (syntaxp (quotep k))
(< k 0)
)
(equal (< k (bitn x n))
t)))
;there are many other ways to say that something is even (include those?)
(defthm bitn-even-means-0
(equal (integerp (* 1/2 (bitn x n)))
(equal (bitn x n) 0)))
;new - export disabled?
(defthm bitn-too-small
(implies (and (< x (expt 2 n))
(<= 0 x) ;case-split?
)
(equal (bitn x n)
0))
:rule-classes ((:rewrite :backchain-limit-lst (1 nil))))
;not sure how to handle this.
(defthmd bitn-normal-form
(equal (equal (bitn x n) 1)
(not (equal (bitn x n) 0))))
(defthm bitn-bvecp
(implies (and (<= 1 k)
(case-split (integerp k)))
(bvecp (bitn x n) k)))
(defthm bitn-times-fraction-integerp
(implies (and (not (integerp k))
(case-split (acl2-numberp k))
)
(equal (INTEGERP (* k (BITN x n)))
(equal (BITN x n) 0))))
(defthm bitn-in-product-split-cases
(and (implies (case-split (acl2-numberp k))
(equal (* (bitn x n) k)
(if (equal (bitn x n) 0)
0
k)))
(implies (case-split (acl2-numberp k))
(equal (* k (bitn x n))
(if (equal (bitn x n) 0)
0
k)))))
(defthm bitn-in-sum-split-cases
(and (implies (case-split (acl2-numberp k))
(equal (+ k (bitn x n))
(if (equal (bitn x n) 0)
k
(+ k 1))))
(implies (case-split (acl2-numberp k))
(equal (+ (bitn x n) k)
(if (equal (bitn x n) 0)
k
(+ k 1))))))
;BOZO change params
(defthm bitn-0
(equal (bitn 0 k)
0))
(defthmd bitn-fw-1
(implies (not (equal (bitn x n) 0))
(equal (bitn x n) 1)
)
:rule-classes (:forward-chaining))
(defthmd bitn-fw-2
(implies (not (equal (bitn x n) 1))
(equal (bitn x n) 0)
)
:rule-classes (:forward-chaining))
;may cause case splits (maybe that's good?)
(defthm bitn-expt-gen
(implies (case-split (integerp i))
(equal (bitn (expt 2 i) n)
(if (equal i n)
1
0))))
;BOZO consider having only the rule above?
(defthmd bitn-expt
(implies (case-split (integerp n))
(equal (bitn (expt 2 n) n) 1)))
;These are intended for the (perhaps weird) case when x in (bitn x n) is a constant but n is not a constant.
;I actually had this term in a proof: (EQUAL (BITN 128 (BITS <signal-name> 8 6)) 0)
(defthm bitn-of-expt-equal-0
(implies (and (syntaxp (quotep x))
(equal x (expt 2 (expo x))) ;means x is a power of 2
)
(equal (equal (bitn x n) 0)
(not (equal n (expo x))))));note that (expo x) will be a constant since x is
(defthm bitn-of-expt-equal-1
(implies (and (syntaxp (quotep x))
(equal x (expt 2 (expo x))) ;means x is a power of 2
)
(equal (equal (bitn x n) 1)
(equal n (expo x))))) ;note that (expo x) will be a constant since x is
(defthmd bitn-expt-0
(implies (and (not (equal i n))
(case-split (integerp i)))
(equal (bitn (expt 2 i) n) 0)))
(defthm bitn-0-1
(or (equal (bitn x n) 0)
(equal (bitn x n) 1))
:rule-classes ())
;BOZO enable?
(defthmd bitn-shift-eric
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* x (expt 2 k)) n)
(bitn x (+ n (- k))))))
(defthmd bitn-shift-eric-2
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* (expt 2 k) x) n)
(bitn x (+ n (- k))))))
;BOZO replace with bitn-shift-eric ??
(defthmd bitn-shift
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* x (expt 2 k)) (+ n k)) ;BOZO rewrite the (+ n k) to match better
(bitn x n))))
;dammit, ACL2 unifies 0 with (* 2 x), so this rule can loop!
(defthm bitn-shift-by-2
(implies (and (syntaxp (not (quotep x)))
(acl2-numberp n))
(equal (BITN (* 2 x) n)
(bitn x (1- n)))))
(defthmd bitn-plus-mult
(implies (and (< n m)
(integerp m)
(integerp k)
)
(equal (bitn (+ x (* k (expt 2 m))) n)
(bitn x n))))
(defthmd bitn-plus-mult-rewrite
(implies (and (syntaxp (quotep c))
(equal (mod c (expt 2 (1+ n))) 0))
(equal (bitn (+ c x) n)
(bitn x n))))
;we almost always want to leave this disabled!
(defthmd bitn-plus-bits
(implies (and (<= m n)
(integerp m)
(integerp n)
)
(= (bits x n m)
(+ (* (bitn x n) (expt 2 (- n m)))
(bits x (1- n) m)))))
;BOZO it's in r-c nil. we almost always want to leave this disabled!
(defthm bits-plus-bitn
(implies (and (<= m n)
(integerp m)
(integerp n)
)
(= (bits x n m)
(+ (bitn x m)
(* 2 (bits x n (1+ m))))))
:rule-classes ())
;drop?
(defthm bits-0-bitn-0
(implies (and (<= 0 n)
(integerp n)
)
(iff (= (bits x n 0) 0)
(and (= (bitn x n) 0)
(= (bits x (1- n) 0) 0))))
:rule-classes ())
;Follows from bits-shift-down
(defthmd bitn-shift-2
(implies (and (<= 0 k)
(integerp k)
(integerp r)
)
(equal (bitn (fl (/ x (expt 2 r))) k)
(bitn x (+ k r)))))
(defthm bitn-shift-by-constant-power-of-2
(implies (and (syntaxp (quotep k))
(power2p k)
(case-split (integerp n))
)
(equal (bitn (* k x) n)
(bitn x (- n (expo k))))))
;generalize to bits-mod?
(defthmd bitn-mod
(implies (and (< k n)
(integerp n)
(integerp k)
)
(equal (bitn (mod x (expt 2 n)) k)
(bitn x k))))
;dup?
(defthm BIT-EXPO-A
(implies (and (< x (expt 2 n))
(>= x 0)
(integerp n)
)
(equal (bitn x n) 0))
:rule-classes ())
;special case of bit-expo-c?
(defthm BIT-EXPO-B
(implies (and (<= (expt 2 n) x)
(< x (expt 2 (1+ n)))
(rationalp x)
(integerp n)
;(>= x 0)
;(>= n 0)
)
(equal (bitn x n) 1))
:rule-classes ())
;bozo. combine these next 2?
;bozo. dup?
(defthm bitn-plus-expt-1
(implies (and (rationalp x)
(integerp n)
)
(not (equal (bitn (+ x (expt 2 n)) n)
(bitn x n))))
:rule-classes ()
)
;bozo. dup?
;prove from bitn-plus-mult?
(defthm bitn-plus-expt-2
(implies (and (< n m)
(integerp n)
(integerp m)
)
(equal (bitn (+ x (expt 2 m)) n)
(bitn x n))))
;this is the most interesting case. perhaps add the other cases for k<0 and k>i-j
(defthm bitn-bits
(implies (and (<= k (- i j))
(case-split (<= 0 k))
(case-split (integerp i))
(case-split (integerp j))
(case-split (integerp k))
)
(equal (bitn (bits x i j) k)
(bitn x (+ j k)))))
;The following trivial corollary of bitn-bits is worth keeping enabled.
(defthm bitn-bits-constants
(implies (and (syntaxp (quotep i))
(syntaxp (quotep j))
(syntaxp (quotep k))
(<= k (- i j))
(<= 0 k)
(integerp i)
(integerp j)
(integerp k))
(equal (bitn (bits x i j) k)
(bitn x (+ j k)))))
(defthmd bitn-shift-3
(implies (and (bvecp x m)
(<= m n)
(integerp k)
(case-split (integerp n))
(case-split (integerp m))
)
(equal (bitn (+ x (* k (expt 2 m))) n)
(bitn k (- n m)))))
;reconcile param names with bits version?
;like bitn-shift-3
;rename!
(defthmd bit+*k-2
(implies (and (< x (expt 2 m))
(<= 0 x)
(rationalp x)
(<= m n)
(integerp k)
(case-split (integerp n))
(case-split (integerp m))
)
(equal (bitn (+ x (* k (expt 2 m))) n)
(bitn k (- n m)))))
(defthm bit-expo-c
(implies (and (<= (- (expt 2 n) (expt 2 k)) x)
(< x (expt 2 n))
(< k n)
(rationalp x);(integerp x) ;gen more!
(integerp n)
(integerp k)
)
(equal (bitn x k) 1))
:rule-classes ())
;Follows from bit-expo-c
;requires x to be an integer, unlike bit-expo-c.
(defthmd bvecp-bitn-2
(implies (and (bvecp x n) ; bind free var n here
(< k n)
(<= (- (expt 2 n) (expt 2 k)) x)
(integerp n)
(integerp k)
)
(equal (bitn x k) 1))
:rule-classes ((:rewrite :match-free :all))
:hints (("Goal" :in-theory (enable bvecp)
:use (bit-expo-c))))
(defthm bitn-bvecp-forward
(bvecp (bitn x n) 1)
:rule-classes ((:forward-chaining :trigger-terms ((bitn x n)))))
;could combine these next two?
;BOZO enable?
(defthmd bvecp-bitn-0
(implies (bvecp x n)
(equal (bitn x n) 0)))
;make an alt version?
;trying disabled..
(defthmd bitn-bvecp-0
(implies (and (bvecp x n)
(<= 0 m)
)
(equal (bitn x (+ m n)) 0)))
;k is a free var
;do we need this, if we have bvecp-longer?
(defthm bitn-bvecp-0-eric
(implies (and (bvecp x k)
(<= k n))
(equal (bitn x n) 0))
:rule-classes ((:rewrite :match-free :all)))
;sort of a "bitn-tail" like bits-tail?
(defthm bitn-bvecp-1
(implies (bvecp x 1)
(equal (bitn x 0) x)))
;rename
(defthmd bvecp-bitn-1
(implies (and (bvecp x (1+ n))
(<= (expt 2 n) x)
(natp n))
(equal (bitn x n) 1)))
;handle the case where we don't go down to 0?
(defthm bits-bitn
(implies (and (case-split (integerp i))
(case-split (<= 0 i))
)
(equal (bits (bitn x n) i 0)
(bitn x n))))
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