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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))))
(local (include-book "../arithmetic/expt"))
(local (include-book "../arithmetic/expo"))
(local (include-book "../arithmetic/arith2"))
(local (include-book "../arithmetic/fp2"))
(local (include-book "../arithmetic/integerp"))
(local (in-theory (enable expt-minus)))
(defund ocat (x y n)
(declare (xargs :guard t))
(+ (* (expt 2 (nfix n)) (nfix x)) (nfix y)))
(defthm ocat-nonnegative-integer-type
(and (integerp (OCAT X Y N))
(<= 0 (OCAT X Y N)))
:rule-classes (:type-prescription)
)
;this rule is no better than ocat-nonnegative-integer-type and might be worse
(in-theory (disable (:type-prescription ocat)))
;just a rewrite rule
(defthm ocat-natp
(natp (ocat x y n)))
;became less general when we made ocat nfix its args
(defthm ocat-0
(implies (and (case-split (<= 0 y))
(case-split (integerp y)))
(equal (ocat 0 y n) y))
:hints (("Goal" :in-theory (enable ocat))))
;became less general when we made ocat nfix its args
(defthm ocat-x-0-0
(implies (and (case-split (<= 0 x))
(case-split (integerp x)))
(equal (ocat x 0 0)
x))
:hints (("Goal" :in-theory (enable ocat))))
#|
;old form:
(defthm ocat-upper-bound
(implies (and (integerp x)
(bvecp y n) ;expensive?
(integerp n))
(< (ocat x y n)
(+ (* (expt 2 n) x) (expt 2 n))))
:hints (("Goal" :in-theory (enable ocat bvecp)))
:rule-classes (:rewrite (:linear :trigger-terms ((ocat x y n))))
)
|#
;this can be really expensive
;old form:
(defthm ocat-upper-bound
(implies (and (< y (expt 2 n))
(<= 0 x)
(integerp x)
(<= 0 y)
(integerp n)
(<= 0 n)
)
(< (ocat x y n)
(+ (* (expt 2 n) x) (expt 2 n))))
:hints (("Goal" :in-theory (enable ocat )))
:rule-classes (:rewrite (:linear :trigger-terms ((ocat x y n))))
)
(encapsulate
()
(local (defthm ocat-bvecp-rewrite-fw
(implies (and (integerp n)
(<= 0 n)
(integerp k)
(<= 0 k)
(integerp x)
(<= 0 x)
(>= k n) ;drop?
(force (bvecp y n))
)
(implies (bvecp (ocat x y n) k)
(bvecp x (- k n))))
:rule-classes nil
:hints (("goal" :in-theory (enable bvecp expt-split ocat)))
))
(local (defthm hack-hack
(implies (and (integerp x)
(integerp y)
(integerp m)
(<= 0 m)
(integerp n)
(<= 0 n)
(< x (expt 2 m))
(< y (expt 2 n))
)
(< (+ (/ y (expt 2 n)) x)
(expt 2 m)))
:hints (("goal" :in-theory (set-difference-theories
(enable expt-split)
'())))))
(local (defthm hack-ocat
(implies (and (integerp x)
(integerp y)
(integerp m)
(<= 0 m)
(integerp n)
(<= 0 n)
(< x (expt 2 m))
(< y (expt 2 n))
)
(< (+ y (* x (expt 2 n)))
(expt 2 (+ m n))))
:hints (("goal" :in-theory (set-difference-theories
(enable expt-split)
'( hack-hack))
:use (hack-hack
(:instance mult-both-sides-of-<-by-positive (a (+ x (* y (/ (expt 2 n)))))
(b (expt 2 m))
(c (expt 2 n))))))))
(local (in-theory (enable bvecp)))
(local (defthm ocat-bvecp-rewrite-bk
(implies (and (integerp n)
(<= 0 n)
(integerp k)
(<= 0 k)
(integerp x)
(<= 0 x)
(>= k n) ;drop?
(force (bvecp y n))
)
(implies
(bvecp x (- k n))
(bvecp (ocat x y n) k)))
:rule-classes nil
:hints (("goal" :in-theory (set-difference-theories
(enable ocat)
'(hack-ocat))
:use (:instance hack-ocat (n n) (m (- k n)))))))
(defthm ocat-bvecp-rewrite
(implies (and (>= k n) ;handle the other case?
(case-split (integerp x))
(case-split (<= 0 x))
(case-split (bvecp y n))
(case-split (natp n))
(case-split (natp k))
)
(equal (bvecp (ocat x y n) k)
(bvecp x (- k n))))
:hints (("goal"
:use (ocat-bvecp-rewrite-fw ocat-bvecp-rewrite-bk))
))
(local (defthm hack-4
(implies (and (integerp x)
(<= 0 x)
(not (equal x 0)))
(>= x 1))))
;expensive? handle this better somehow?
(local (defthm hack-3
(implies (and (integerp x)
(<= 0 x)
(not (equal x 0))
(rationalp a)
(<= 0 a)
)
(>= (* x a) a))
:rule-classes :linear
:hints (("goal" :in-theory (disable
hack-4
; cancel-in-prods-<
; cancel-times-<-eric-1
)
:use (hack-4
(:instance mult-both-sides-of-<-by-positive
(b a) (a (* a x)) (c (/ a))))))))
;better names?
(defthm ocat-bvecp-other-case
(implies (and (< p n)
(integerp n)
(<= 0 n)
(integerp p)
(<= 0 p)
(integerp x)
(<= 0 x)
(integerp y)
(<= 0 y)
)
(equal (bvecp (ocat x y n) p)
(and (bvecp y p)
(equal 0 x)))
)
:hints (("goal" :in-theory (set-difference-theories
(enable power2p ocat)
'(expt-compare
))
:use (:instance expt-compare (lhs (expt 2 p)) (rhs (expt 2 n)))))
:otf-flg t
)
)
#|
;make more general
;also make ncat ver
(defthm highbits-ocat
(implies (and (integerp x)
(<= 0 x)
(force (bvecp y n))
(integerp n)
(<= 0 n)
)
(equal (highbits (OCAT x y n) n)
x))
:hints (("Goal" :in-theory (enable expt-split
ocat
highbits))))
|#
(local (defthm hack-10
(implies (and (integerp x)
(integerp y)
(< x y))
(<= x (1- y)))
:rule-classes ()))
(local (defthm ocat-bvecp-simple
(implies (and (natp n)
(natp k)
(bvecp x m)
(natp m)
(bvecp y n)
(>= k (+ m n)))
(bvecp (ocat x y n) k))
:rule-classes ()
:hints (("Goal" :in-theory (set-difference-theories
(enable natp bvecp ocat)
'(expt-compare EXPT-COMPARE-EQUAL))
:use ((:instance expt-split (i m) (j n) (r 2))
(:instance hack-10 (y (expt 2 m)))
(:instance expt-weak-monotone (m p))
(:instance expt-weak-monotone (m k) (n (+ m n))))))))
(defthm ocat-bvecp
(implies (and (>= k n) ;handle other case?
(bvecp x (- k n))
(case-split (natp n))
(case-split (natp k))
(case-split (bvecp y n))
)
(bvecp (ocat x y n) k))
:hints (("Goal" :in-theory (enable natp bvecp)
:use ((:instance ocat-bvecp-simple (m (- k n)))))))
(defthm ocat-0-rewrite
(implies (and (case-split (integerp x))
(case-split (<= 0 x)))
(equal (ocat 0 x n) x))
:hints (("Goal" :in-theory (enable ocat))))
(defthm ocat-with-x-not-a-natural
(implies (or (not (integerp x))
(< x 0))
(equal (ocat x y n)
(nfix y)))
:hints (("Goal" :in-theory (enable ocat))))
(defthm ocat-with-y-not-a-natural
(implies (or (not (integerp y))
(< y 0))
(equal (ocat x y n)
(* (nfix x) (expt 2 (nfix n)))))
:hints (("Goal" :in-theory (enable ocat))))
(defthm ocat-with-n-not-a-natural
(implies (or (not (integerp n))
(< n 0))
(equal (ocat x y n)
(+ (nfix x) (nfix y))))
:hints (("Goal" :in-theory (enable ocat))))
;might be able to generalize this more
;this look like it will fire as often as we'd like
(defthm ocat-upper-bound-2
(implies (and (< x (expt 2 k)) ; k is a free var. huh?
(case-split (< y (expt 2 n)))
(case-split (integerp k))
(case-split (<= 0 k))
(case-split (integerp n))
(case-split (<= 0 n))
)
(< (ocat x y n)
(expt 2 (+ n k))))
:hints (("Goal" :in-theory (set-difference-theories
(enable bvecp expt-split)
'(OCAT-BVECP-REWRITE OCAT-BVECP ))
:use ((:instance <-TRANSITIVE (a y) (b (expt 2 n)) (c (* (EXPT 2 K) (EXPT 2 N))))
(:instance ocat-bvecp (k (+ n k)))))))
(defthm ocat-associative
(implies (and (case-split (<= 0 m)) ;new now that ocat fixes its args
(case-split (<= 0 n)) ;new now that ocat fixes its args
(case-split (integerp m))
(case-split (integerp n))
)
(equal (ocat (ocat x y m) z n)
(ocat x (ocat y z n) (+ m n))))
:hints (("Goal" :in-theory (enable ocat))))
(defthm ocat-equal-0
(implies (and (case-split (<= 0 x))
(case-split (<= 0 y))
(case-split (integerp x))
(case-split (integerp y))
)
(equal (equal (ocat x y n) 0)
(and (equal x 0)
(equal y 0))))
:hints (("Goal" :in-theory (enable ocat)))
)
(defthm ocat-combine-constants
(implies (and (syntaxp (and (quotep x)
(quotep y)
(quotep n)
(quotep m)))
(case-split (<= 0 m))
(case-split (<= 0 n))
(case-split (<= m n))
(case-split (integerp m))
(case-split (integerp n)))
(equal (ocat x (ocat y z m) n)
(ocat (ocat x y (- n m)) z m)))
:otf-flg t
:hints
(("goal" :in-theory (enable ocat))))
|
