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|
(in-package "ACL2")
(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))))))))
(local (include-book "../arithmetic/top"))
(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))))))
(include-book "../arithmetic/negative-syntaxp")
(include-book "../arithmetic/power2p")
(local (include-book "ground-zero"))
(local (include-book "bits"))
(local (include-book "bvecp")) ;to get bvecp-longer
;(in-theory (disable expt-inverse))
(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-nonnegative-integer
(and (integerp (bitn x n))
(<= 0 (bitn x n)))
:hints (("Goal" :in-theory (enable bitn)))
:rule-classes (:type-prescription))
(defthm bitn-with-n-not-an-integer
(implies (not (integerp n))
(equal (bitn x n)
0))
:hints (("Goal" :in-theory (enable bitn))))
(encapsulate
()
;gen
(local (defthm bitn-upper-bound-case-1
(implies (integerp n)
(<= (bitn x n) 1))
:otf-flg t
:hints (("Goal" :use (:instance fl-def-linear-part-2 (x (* 1/2 X (/ (EXPT 2 N)))))
:in-theory (set-difference-theories
(enable mod bitn bits expt-split)
'( fl-def-linear-part-2
a10
; REARRANGE-ERIC-4
; REARRANGE-FRACTIONAL-COEFS-<
))))
:rule-classes (:rewrite :linear)))
;separate out the linear rule?
(local (defthm bitn-upper-bound-case-2
(implies (not (integerp n))
(<= (bitn x n) 1))
:otf-flg t
:hints (("Goal" :cases ((integerp (+ n 1)))
:in-theory (set-difference-theories
(enable mod bitn bits expt-split)
'(A10
fl-def-linear-part-2
; REARRANGE-FRACTIONAL-COEFS-<
))))
:rule-classes (:rewrite :linear)))
(defthm bitn-upper-bound
(<= (bitn x n) 1)
:hints (("Goal" :cases ((integerp n)))))
)
(defthm bitn-upper-bound-linear
(<= (bitn x n) 1)
:rule-classes ((:LINEAR :TRIGGER-TERMS ((bitn x n)))))
;!! was looping with expt-compare
; look into this more
(local (in-theory (disable EXPO-COMPARISON-REWRITE-TO-BOUND)))
(encapsulate
()
;derive from bits-minus?
(local (defthm bitn-minus-case-1
(implies (and (rationalp x)
(integerp n)
(integerp (/ x (expt 2 (+ 1 n))))
)
(equal (bitn (* -1 x) n)
(- (bitn x n))
))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn
bits
mod-cancel
expt-minus
expt-split)
'( ;expt-inverse
))))))
(local (defthm bitn-minus-case-2
(implies (and (rationalp x)
(integerp n)
(not (integerp (/ x (expt 2 n))))
)
(equal (bitn (* -1 x) n)
(- 1 (bitn x n))
))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn
mod
mod-cancel
bits
even-int-implies-int
expt-minus
expt-split)
'( ;expt-inverse
))))))
(local (defthm bitn-minus-case-3
(implies (and (rationalp x)
(integerp n)
(not (integerp (/ x (expt 2 (+ 1 n)))))
(integerp (/ x (expt 2 n)))
)
(equal (bitn (* -1 x) n)
(- 2 (bitn x n))
))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn
mod
mod-cancel
bits
expt-minus
expt-split)
'( ;expt-inverse
))))))
(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))))))))
;(in-theory (disable FL-EQUAL-0))
;1 rewrite to odd?
(defthm bitn-0-rewrite-to-even
(implies (integerp x)
(equal (equal (bitn x 0) 0)
(integerp (* 1/2 x))))
:hints (("Goal" :in-theory (enable bitn bits mod-by-2-rewrite-to-even)))
)
;...
;(in-theory (disable bitn-sum-lowbits)) ;was causing loops
;this one should remain last? <-- huh?
(theory-invariant (incompatible (:rewrite bits-n-n-rewrite)
(:definition bitn)
)
:key bitn-and-bits-n-n-shouldnt-alternate)
(defthmd bits-n-n-rewrite
(equal (BITS X n n)
(bitn x n))
:hints (("Goal" :in-theory (enable bitn)))
)
#|
;should only fire if it really does simplify x, that is, if x really has bits to be dropped
(defthm bitn-sum-simplify-first-term
(implies (and (>= (abs x) (expt 2 (+ n 1))) ;prevents loop
(rationalp x)
(rationalp y)
(integerp n))
(equal (bitn (+ x y) n)
(bitn (+ (lowbits x n) y) n)))
:hints (("Goal" :in-theory (set-difference-theories
(enable
lowbits
bitn bits)
'()))))
;should only fire if it really does simplify y, that is, if y really has bits to be dropped
(defthm bitn-sum-simplify-second-term
(implies (and (>= (abs y) (expt 2 (+ n 1))) ;prevents loop
(rationalp x)
(rationalp y)
(integerp n))
(equal (bitn (+ x y) n)
(bitn (+ x (lowbits y n)) n)))
:hints (("Goal" :in-theory (set-difference-theories
(enable lowbits
bitn bits)
'()))))
(defthm bitn-sum-simplify-third-term
(implies (and (>= (abs z) (expt 2 (+ n 1))) ;prevents loop
(rationalp x)
(rationalp y)
(rationalp z)
(integerp n))
(equal (bitn (+ x y z) n)
(bitn (+ x y (lowbits z n)) n)))
:hints (("Goal" :in-theory (disable bitn-sum-simplify-first-term
bitn-sum-simplify-second-term)
:use (:instance bitn-sum-simplify-first-term (x z) (y (+ x y))))))
|#
(defthm bitn-upper-bound-2
(< (bitn x n) 2)
:hints (("Goal" :in-theory (disable bitn-upper-bound)
:use bitn-upper-bound)))
(defthm bitn-0-1
(or (equal (bitn x n) 0)
(equal (bitn x n) 1))
:hints (("Goal" :in-theory (disable bitn)))
:rule-classes nil)
;my strategey with the rules below is to rewrite prefer (not (equal (bitn x n) 0)) over (equal (bitn x n) 1)
;this allows subsumption to ...
;bad to have both?
(defthm bitn-not-0-means-1
(equal (not (equal (bitn x n) 0))
(equal (bitn x n) 1))
:hints (("Goal" :use bitn-0-1)))
(defthm bitn-not-1-means-0
(equal (not (equal (bitn x n) 1))
(equal (bitn x n) 0))
:hints (("Goal" :use bitn-0-1)))
;these are bad rules?
(in-theory (disable bitn-not-1-means-0 bitn-not-0-means-1))
;add matt's forward chaining rules for dealing with single bits (maybe they should go in bvecp.lisp)
(encapsulate
()
(local (defthm bitn-bitn-case-1
(implies (case-split (integerp n))
(equal (bitn (bitn x n) 0)
(bitn x n)))
:hints (("Goal"
:in-theory (set-difference-theories
(enable bitn bits)
'())))))
(local (defthm bitn-bitn-case-2
(implies (not (integerp n))
(equal (bitn (bitn x n) 0)
(bitn x n)))
:hints (("Goal" :cases ((acl2-numberp n))
:in-theory (set-difference-theories
(enable bitn bits mod)
'())))))
(defthm bitn-bitn
(equal (bitn (bitn x n) 0)
(bitn x n))))
;bb
(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)))
:hints (("Goal" :use (:instance bitn-0-1)))
)
;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))
:hints (("Goal"
:use bitn-0-1)))
;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))
:hints (("Goal" :in-theory (disable bitn-upper-bound bitn-upper-bound-2)
:use bitn-upper-bound)))
;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))
:hints (("Goal" :in-theory (disable BITN-NOT->-1 bitn-upper-bound bitn-upper-bound-2)
:use bitn-upper-bound)))
#|
;cc
(defthm bitn-shift-alt
(implies (and (syntaxp (should-have-a-2-factor-divided-out x))
(> n 0) ;restricts application
(rationalp x)
(integerp n)
)
(equal (bitn x n)
(bitn (/ x 2) (- n 1))))
:hints (("Goal" :in-theory (set-difference-theories
(enable bits bitn)
'(bits-shift-alt
))
:use (:instance bits-shift-alt (i n) (j n)))))
|#
(defthmd bitn-def
(implies (case-split (integerp k))
(equal (bitn x k)
(mod (fl (/ x (expt 2 k)))
2)))
:hints (("Goal" :in-theory (enable bits bitn expt-split))))
(defun not-eric (x)
(if (equal x 0)
1
0))
#|
;this does most of the work (i.e., it gets the constant below 2^i+1
(defthm bitn-sum-lowbits
(implies (and (syntaxp (and (quotep x) (>= (cadr x) (expt 2 (+ 1 (cadr n)))))) ;dropped negative case
(rationalp x)
(rationalp y)
(integerp n)
)
(equal (bitn (+ x y) n)
(bitn (+ (lowbits x n) y) n)))
:hints (("Goal" :in-theory (enable bitn)
:use (:instance bits-sum-lowbits (i n) (j n) ))))
|#
(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)))))
:otf-flg t
:hints (("Goal" :in-theory (set-difference-theories
(enable bits bitn-def
expt-split
)
'(
MOD-PULL-INSIDE-FL-SHIFT-ALT-ALT-ALT-ALT
floor-fl)))))
(defthm 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))))
:otf-flg t
:hints (("Goal" :in-theory (disable bitn-drop-crucial-bit-and-flip-result)
:use (:instance bitn-drop-crucial-bit-and-flip-result (x (+ j (- (expt 2 n)) x))))))
;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))))
:otf-flg t
:hints (("Goal" :in-theory (disable bitn-drop-crucial-bit-and-flip-result)
:use (:instance bitn-drop-crucial-bit-and-flip-result (x (+ j x))))))
(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)))
:otf-flg t
:hints (("Goal" :cases ((<= 0 x))
:in-theory (enable bitn bits expt-split mod-force-chosen-a-neg)))
)
;drop silly hyps like: (<= -128 (BITN FOO 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)))
(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))
:hints (("Goal" :cases ((rationalp x)) ;why needed?
:in-theory (enable bitn bits expt-split)))
:rule-classes ((:rewrite :backchain-limit-lst (1 nil)))
)
(defthm bitn-normal-form
(equal (equal (bitn x n) 1)
(not (equal (bitn x n) 0))))
(defthm bitn-of-non-rational
(implies (not (rationalp x))
(equal (bitn x n)
0))
:hints (("Goal" :in-theory (enable bitn)))
)
(encapsulate
()
(local (defthm bitn-bvecp-simple
(bvecp (bitn x n) 1)
:hints (("Goal" :use bitn-0-1
:in-theory (set-difference-theories
(enable bvecp)
'()
)))))
(defthm bitn-bvecp
(implies (and (<= 1 k)
(case-split (integerp k)))
(bvecp (bitn x n) k))
:hints (("Goal" :use bitn-bvecp-simple
:in-theory (disable bitn-bvecp-simple
))))
)
(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)))))
;(in-theory (disable bitn-in-product-split-cases))
(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))))))
;(in-theory (disable bitn-in-sum-split-cases))
#|
(defthm bitn-shift-better
(implies (and (bind-free (can-take-out-numeric-power-of-2 x) (c))
(force (power2p c))
(case-split (integerp n))
)
(equal (bitn x n)
(bitn (/ x c) (- n (expo c)))))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn)
'(bits-shift-better)
)
:use (:instance bits-shift-better (i n) (j n)))))
|#
(defthm bitn-0
(equal (bitn 0 k)
0)
:hints (("goal" :in-theory (enable bitn))))
;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)))
:hints (("Goal" :in-theory (enable bitn))))
(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
#|
(defthm bitn-of-expt-constant
(implies (and (syntaxp (quotep x))
(equal e (expo x)) ;having E means we don't have to evaluate (expo x) in the conclusion
(equal x (expt 2 e)) ;means x is a power of 2
)
(equal (bitn x n)
(log= n e)))) ;note that e will be a constant
|#
;This is the rule Doc is allowing in lib/, since it doesn't cause as many case-splits are bitn-expt-gen?
(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 ()
:hints (("goal" :in-theory (enable bitn))))
(defthmd bitn-shift-eric
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* x (expt 2 k)) n)
(bitn x (+ n (- k)))))
:hints (("Goal" :in-theory (enable bitn))))
;BOZO replace with bitn-shift-eric ??
(defthmd bitn-shift
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* x (expt 2 k)) (+ n k))
(bitn x n)))
:hints (("Goal" :in-theory (enable bitn))))
;gen!
;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))))
:hints (("Goal" :use (:instance bitn-shift-eric (k 1))))
)
(defthmd bitn-plus-mult
(implies (and (< n m)
(integerp m)
(integerp k)
)
(equal (bitn (+ x (* k (expt 2 m))) n)
(bitn x n)))
:hints (("Goal" :in-theory (enable bitn bits-plus-mult-2))))
(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)))
:hints (("Goal" :use ((:instance bitn-plus-mult
(x x)
(k (/ c (expt 2 (1+ n))))
(m (1+ n))
(n n)))
:in-theory (enable mod))))
;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))))
:hints (("goal" :in-theory (enable bitn)
:use ((:instance bits-plus-bits (n n) (p n) (m m)))
)))
;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 ()
:hints (("goal" :in-theory (enable bitn)
:use ((:instance bits-plus-bits (n n) (m m) (p (+ m 1)))))))
;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 ()
:hints (("Goal" :use (:instance bitn-plus-bits (m 0)))))
(defthmd bitn-shift-2
(implies (and (<= 0 k)
(integerp k)
(integerp r)
)
(equal (bitn (fl (/ x (expt 2 r))) k)
(bitn x (+ k r))))
:hints (("goal" :in-theory (e/d (bits-shift-down bitn) (BITS-FL)))))
(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)))))
:hints (("Goal" :use (:instance bits-shift-by-constant-power-of-2 (i n) (j n))
:in-theory (enable bitn))))
(defthmd bitn-shift-eric-2
(implies (and (integerp n)
(integerp k)
)
(equal (bitn (* (expt 2 k) x) n) ;BOZO rewrite the (+ n k) to match better
(bitn x (+ n (- k)))))
:hints (("Goal" :in-theory (enable bitn))))
(defthmd bitn-rec-0
(implies (integerp x)
(equal (bitn x 0)
(mod x 2)))
:hints (("goal" :use ((:instance bitn-def (k 0))))))
;rename?
;is there a bits analog of this theorem?
;move or copy to bitn?
;change k to n
;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))))
:hints (("goal" :in-theory (set-difference-theories
(enable bitn-def expt-split)
'( ; bitn-def
fl/int-rewrite
fl-shift-fl
mod-pull-inside-fl-shift-alt-alt-alt
mod-pull-inside-fl-shift-alt-alt-alt-alt))
:use ((:instance fl/int-rewrite (x (/ x 2)) (n (expt 2 (1- 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)))
:hints (("Goal"; :cases ((integerp n))
:in-theory (enable bitn bits))))
;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 ()
:hints (("Goal" :in-theory (enable expt-split bitn-def)
:use ((:instance fl-unique (x (/ x (expt 2 n))) (n 1))))))
(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)))
:hints (("Goal" :in-theory (enable bitn))))
;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))))
:hints (("Goal" :in-theory (e/d ( bitn) (BITS-FL)))))
;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 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))))
:hints (("Goal" :in-theory (enable bitn bits+2**k-2))))
(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))))
:hints (("Goal" :in-theory (enable bvecp)
:use (bit+*k-2))))
;try
(local
(defthm bit-expo-c-4
(implies (and (rationalp x)
(integerp n)
(integerp k)
(<= k n)
(< x (expt 2 n))
(<= (- (expt 2 n) (expt 2 k)) x))
(= (fl (/ x (expt 2 k)))
(1+ (* 2 (1- (expt 2 (1- (- n k))))))))
:rule-classes ()
:hints (("goal" :in-theory (set-difference-theories
(enable expt-split expt-minus )
'())
:use ((:instance fl-unique (x (/ x (expt 2 k))) (n (1- (expt 2 (- n k))))))))))
(local
(defthm bit-expo-c-6
(implies (and (rationalp x)
(integerp n)
(integerp k)
(< k n)
(< x (expt 2 n))
(<= (- (expt 2 n) (expt 2 k)) x))
(= (mod (fl (/ x (expt 2 k))) 2)
1))
:rule-classes ()
:hints (("goal" :in-theory (disable expt-split
)
:use ( ;(:instance bit-expo-c-5)
(:instance bit-expo-c-4)
(:instance mod-mult-eric (x 1) (y 2) (a (1- (expt 2 (1- (- n k))))))
)))))
;prove this from a more general result about bits??
;BOZO bad name. doesn't mention expo !
(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 ()
:hints (("goal" :use ((:instance bitn-def)
(:instance bit-expo-c-6)))))
(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)))))
#| old:
(defun BITN (x n)
(if (logbitp n x) 1 0))
|#
(defthm bitn-natp
(natp (bitn x n)))
;BOZO do we want these?
(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))
(defthmd bvecp-bitn-0
(implies (bvecp x n)
(equal (bitn x n) 0))
:hints (("Goal" :in-theory (enable bitn bvecp-bits-0))))
;make an alt version?
(defthm bitn-bvecp-0
(implies (and (bvecp x n)
(<= 0 m)
)
(equal (bitn x (+ m n)) 0))
:hints (("Goal" :in-theory (disable bvecp-bitn-0)
:use ((:instance bvecp-bitn-0 (n (+ m n)))))))
;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))
:hints (("Goal" :in-theory (enable bvecp-bitn-0))))
;sort of a "bitn-tail" like bits-tail?
(defthm bitn-bvecp-1
(implies (bvecp x 1)
(equal (bitn x 0) x))
:hints (("Goal" :in-theory (enable bvecp-1-rewrite))))
(defthmd bvecp-bitn-1
(implies (and (bvecp x (1+ n))
(<= (expt 2 n) x)
(natp n))
(equal (bitn x n) 1))
:hints (("Goal" :in-theory (enable bvecp)
:use (bit-expo-b))))
;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)))
:hints (("Goal" :in-theory (set-difference-theories
(enable bitn)
'()))))
|
