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|
(in-package "ACL2")
; The lemmas below were initially provd after including ../lib/top (or at least
; they could have been; actually we used ../lib/ books bits, arith, and util).
; But we want to prove them in this support/ directory, so we include a
; book, top1, whose events are a superset of those that were in the
; aforementioned lib/ books, and then we put ourselves in the same theory as
; though we had included ../lib/top.
(include-book "rtl")
(include-book "fadd")
(local (include-book "top1"))
(local (in-theory (theory 'lib-top1)))
; Proof of bits-sum-swallow:
; Proof: Since y < 2^(k+1), y[i:k+1] = 0.
;
; Since x[k] = 0, x[k:0] = x[k-1:0] < 2^k.
;
; Hence,
;
; x[k:0] + y[k:0] < 2^k + 2^k = 2^(k+1)
;
; and
;
; (x[k:0] + y[k:0])[k+1] = 0.
;
; By BITS-SUM,
;
; (x+y)[i:k+1] = (x[i:k+1] + y[i:k+1] + (x[k:0] + y[k:0])[k+1])[i-k-1:0]
; = (x[i:k+1])[i-k-1:0]
; = x[i:k+1] {by BITS-BITS}.
;
; By BITS-BITS,
;
; (x+y)[i:j] = (x+y)[i:k+1][i-k-1:j-k-1]
; = x[i:k+1][i-k-1:j-k-1]
; = x[i:j].
(local-defthm bits<expt
(implies (and (natp x)
(integerp i)
(natp j)
(<= j i)
(equal n (1+ (- i j))))
(< (bits x i j)
(expt 2 n)))
:hints (("Goal" :use ((:instance bits-bvecp
(k n)))
:in-theory (e/d (bvecp) (bits-bvecp bits-bvecp-simple))))
:rule-classes :linear)
(local-defthm bits-sum-swallow-1-1-1-1
(implies (and (natp x)
(natp k)
(= (bitn x k) 0))
(< (bits x k 0)
(expt 2 k)))
:hints (("Goal" :use ((:instance bitn-plus-bits (n k) (m 0))
(:theorem (or (zp k) (< 0 k))))))
:rule-classes :linear)
(local-defthm bits-sum-swallow-1-1-1-2
(implies (and (natp y)
(natp k)
(<= y (expt 2 k)))
(equal (bits y k 0) y))
:hints (("Goal" :in-theory (enable bvecp)
:expand ((expt 2 (1+ k))))))
(local-defthm bits-sum-swallow-1-1-1
(implies (and (natp x)
(natp y)
(natp k)
(= (bitn x k) 0)
(<= y (expt 2 k)))
(< (+ (bits x k 0) (bits y k 0))
(expt 2 (1+ k))))
:hints (("Goal" :expand ((expt 2 (1+ k)))))
:rule-classes nil)
(local-defthm bits-sum-swallow-1-1
(implies (and (natp x)
(natp y)
(natp k)
(= (bitn x k) 0)
(<= y (expt 2 k)))
(equal (bitn (+ (bits x k 0) (bits y k 0))
(1+ k))
0))
:hints (("Goal" :in-theory (enable bvecp) ; for bvecp-bitn-0
:use bits-sum-swallow-1-1-1))
:rule-classes nil)
(local-defthm bits-sum-swallow-1-2
(implies (and (natp y)
(natp k)
(>= i k)
(<= y (expt 2 k)))
(equal (bits y i (1+ k))
0))
:hints (("Goal" :in-theory (enable bvecp-bits-0 bvecp)
:expand ((expt 2 (1+ k))))))
(local-defthm bits-sum-swallow-1
(implies (and (natp x)
(natp y)
(natp k)
(>= i k)
(= (bitn x k) 0)
(<= y (expt 2 k)))
(equal (bits (+ x y) i (1+ k))
(bits x i (1+ k))))
:hints (("Goal" :use ((:instance bits-sum (j (1+ k)))
bits-sum-swallow-1-1))))
(defthmd bits-sum-swallow
(implies (and (equal (bitn x k) 0)
(natp x)
(natp y)
(integerp i)
(integerp j)
(integerp k)
(>= i j)
(> j k)
(>= k 0)
(<= y (expt 2 k)))
(equal (bits (+ x y) i j)
(bits x i j)))
:hints (("Goal"
;; We deliberately leave bits-bits enabled because we need another
;; instance of it too.
:use ((:instance bits-bits
(x (+ x y))
(i i)
(j (1+ k))
(k (1- (- i k)))
(l (1- (- j k))))))))
(defthmd bits-sum-of-bits
(implies (and (integerp x)
(integerp y)
(natp i))
(equal (bits (+ x (bits y i 0)) i 0)
(bits (+ x y) i 0)))
:hints (("Goal" :use ((:instance bits-sum
(x x)
(y (bits y i 0))
(i i)
(j 0))
(:instance bits-sum
(x x)
(y y)
(i i)
(j 0))))))
(defthm bits-sum-3
(implies (and (integerp x)
(integerp y)
(integerp z)
(integerp i)
(integerp j))
(equal (bits (+ x y z) i j)
(bits (+ (bits x i j)
(bits y i j)
(bits z i j)
(bitn (+ (bits x (1- j) 0) (bits y (1- j) 0))
j)
(bitn (+ (bits (+ x y) (1- j) 0) (bits z (1- j) 0))
j))
(- i j) 0)))
:hints (("Goal" :use ((:instance bits-sum
(x x)
(y y)
(i (1- j))
(j 0))
(:instance bits-sum
(x x)
(y y)
(i i)
(j j))
(:instance bits-sum
(x (+ x y))
(y z)
(i i)
(j j))
(:instance bits-sum-of-bits
(x (+ (bits z i j)
(bitn (+ (bits z (1- j) 0)
(bits (+ x y) (1- j) 0))
j)))
(y (+ (bits x i j)
(bits y i j)
(bitn (+ (bits x (1- j) 0)
(bits y (1- j) 0))
j)))
(i (+ i (* -1 j)))))))
:rule-classes ())
(local-defthm bits-sum-3-with-gen-normal-case
(implies (and (integerp x)
(integerp y)
(integerp z)
(integerp i)
(integerp j)
(< 0 j))
(equal (bits (+ x y z) i j)
(bits (+ (bits x i j)
(bits y i j)
(bits z i j)
(gen x y (1- j) 0)
(gen (+ x y) z (1- j) 0))
(- i j) 0)))
:hints (("Goal" :use bits-sum-3
:in-theory (enable gen-as-bitn)))
:rule-classes ())
(defthm bits-sum-3-with-gen
(implies (and (integerp x)
(integerp y)
(integerp z)
(integerp i)
(integerp j))
(equal (bits (+ x y z) i j)
(bits (+ (bits x i j)
(bits y i j)
(bits z i j)
(gen x y (1- j) 0)
(gen (+ x y) z (1- j) 0))
(- i j) 0)))
:hints (("Goal" :use (bits-sum-3 bits-sum-3-with-gen-normal-case)))
:rule-classes ())
(defthmd lior-bits-1
(equal (lior (bits x (1- n) 0)
y
n)
(lior x y n))
:hints (("Goal" :in-theory (e/d (lior) (logior lior-commutative)))))
(defthmd lior-bits-2
(equal (lior x
(bits y (1- n) 0)
n)
(lior x y n))
:hints (("Goal" :in-theory (e/d (lior) (logior lior-commutative)))))
(defthmd land-bits-1
(equal (land (bits x (1- n) 0)
y
n)
(land x y n))
:hints (("Goal" :in-theory (e/d (land) (logior land-commutative)))))
(defthmd land-bits-2
(equal (land x
(bits y (1- n) 0)
n)
(land x y n))
:hints (("Goal" :in-theory (e/d (land) (logior land-commutative)))))
(defthmd lxor-bits-1
(equal (lxor (bits x (1- n) 0)
y
n)
(lxor x y n))
:hints (("Goal" :in-theory (e/d (lxor) (logior lxor-commutative)))))
(defthmd lxor-bits-2
(equal (lxor x
(bits y (1- n) 0)
n)
(lxor x y n))
:hints (("Goal" :in-theory (e/d (lxor) (logior lxor-commutative)))))
; Proof of lior-slice:
; Note that (2^i - 2^j) = 2^j(2^(i-j)-1) = {2^(i-j)-1,j'b0}
; x[n-1:0] | (2^i - 2^j) =
; {x[n-1:i], x[i-1:j], x[j-1:0]} | {(n-i)'b0, 2^(i-j), j'b0} =
; [by LIOR-CAT, LIOR-0, and LIOR-ONES]
; {{x[n-1:i], 2^(i-j)-1, x[j-1:0]}.
(local-defthm lior-slice-1
(implies (and (<= j i)
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j))
(equal (cat (bits x (1- n) i) (- n i)
(bits x (1- i) j) (- i j)
(bits x (1- j) 0) j)
(bits x (1- n) 0)))
:hints (("Goal" :use ((:instance cat-bits-bits
(x (bits x (1- i) 0))
(i (1- i))
(j j)
(k (1- j))
(l 0)
(m (- i j))
(n j))
(:instance cat-bits-bits
(x x)
(i (1- n))
(j i)
(k (1- i))
(l 0)
(m (- n i))
(n i)))))
:rule-classes nil)
(local-defthm bvecp-power-of-2-minus-1
(implies (and (integerp k)
(equal k k2)
(< 0 k))
(bvecp (1- (expt 2 k))
k2))
:hints (("Goal" :in-theory (enable bvecp))))
(local-defthm bvecp-power-of-2-minus-1-alt
(implies (and (natp k)
(equal k2 (1+ k)))
(bvecp (1- (* 2 (expt 2 k)))
k2))
:hints (("Goal" :in-theory (enable bvecp expt))))
(local-defthm lior-slice-2
(implies (and (<= j i)
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j))
(equal (cat 0 (- n i)
(1- (expt 2 (- i j))) (- i j)
0 j)
(- (expt 2 i) (expt 2 j))))
;; The following hint is very weird!! It can't be on Goal or Goal'. The
;; idea came from a proof-checker proof. I should investigate....
:hints (("Goal''" :in-theory (enable cat expt)))
:rule-classes nil)
(local
(defthmd lior-slice-3-1
(implies (and (<= j i)
(integerp i)
(integerp j)
(<= 0 j)
(equal k (1- (expt 2 (- i j))))
(equal diff (- i j)))
(equal (lior k
(bits x (1- i) j)
diff)
(1- (expt 2 (- i j)))))
:hints (("Goal" :use ((:instance lior-ones
(x (bits x (1- i) j))
(n (- i j))))))))
(local-defthm lior-slice-3
(implies (and (<= j i)
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j))
(equal (lior (cat (bits x (1- n) i) (- n i)
(bits x (1- i) j) (- i j)
(bits x (1- j) 0) j)
(cat 0 (- n i)
(1- (expt 2 (- i j))) (- i j)
0 j)
n)
(cat (bits x (1- n) i) (- n i)
(1- (expt 2 (- i j))) (- i j)
(bits x (1- j) 0) j)))
:hints (("Goal" :in-theory (e/d (lior-cat lior-slice-3-1)
(cat-bits-bits cat-0))
:cases ((and (equal j i) (equal j 0) (equal i n))
(and (equal j i) (not (equal j 0)) (equal i n))
(and (not (equal j i)) (equal j 0) (equal i n))
(and (equal j i) (equal j 0) (not (equal i n)))
(and (equal j i) (not (equal j 0)) (not (equal i n)))
(and (not (equal j i)) (equal j 0) (not (equal i n)))
(and (not (equal j i)) (not (equal j 0)) (not (equal i n))))))
:rule-classes nil)
(local-defthm lior-slice-almost
(implies (and (<= j i)
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j))
(equal (lior (bits x (1- n) 0)
(- (expt 2 i) (expt 2 j))
n)
(cat (bits x (1- n) i) (- n i)
(1- (expt 2 (- i j))) (- i j)
(bits x (1- j) 0) j)))
:hints (("Goal" :use (lior-slice-1 lior-slice-2 lior-slice-3)
:in-theory (theory 'ground-zero)))
:rule-classes nil)
(defthmd lior-slice
(implies (and (<= j i)
(<= i n)
(integerp n)
(integerp i)
(integerp j)
(<= 0 j))
(equal (lior x
(- (expt 2 i) (expt 2 j))
n)
(cat (bits x (1- n) i) (- n i)
(1- (expt 2 (- i j))) (- i j)
(bits x (1- j) 0) j)))
:hints (("Goal" :use (lior-slice-almost)
:in-theory (enable lior-bits-1))))
(local-defthm land-base-lemma
(implies (and (bvecp x 1) (bvecp y 1))
(equal (land x y 1)
(if (and (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0)))
:rule-classes nil)
(defthm land-base
(equal (land x y 1)
(if (and (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0))
:hints (("Goal" :use ((:instance land-base-lemma
(x (bits x 0 0))
(y (bits y 0 0)))
(:instance land-bits-1
(x x)
(y (bits y 0 0))
(n 1))
(:instance land-bits-2 (n 1)))))
:rule-classes nil)
(local-defthm lior-base-lemma
(implies (and (bvecp x 1) (bvecp y 1))
(equal (lior x y 1)
(if (or (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0)))
:rule-classes nil)
(defthm lior-base
(equal (lior x y 1)
(if (or (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
1
0))
:hints (("Goal" :use ((:instance lior-base-lemma
(x (bits x 0 0))
(y (bits y 0 0)))
(:instance lior-bits-1
(x x)
(y (bits y 0 0))
(n 1))
(:instance lior-bits-2 (n 1)))))
:rule-classes nil)
(local-defthm lxor-base-lemma
(implies (and (bvecp x 1) (bvecp y 1))
(equal (lxor x y 1)
(if (iff (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
0
1)))
:rule-classes nil)
(defthm lxor-base
(equal (lxor x y 1)
(if (iff (equal (bitn x 0) 1)
(equal (bitn y 0) 1))
0
1))
:hints (("Goal" :use ((:instance lxor-base-lemma
(x (bits x 0 0))
(y (bits y 0 0)))
(:instance lxor-bits-1
(x x)
(y (bits y 0 0))
(n 1))
(:instance lxor-bits-2 (n 1)))))
:rule-classes nil)
(local (in-theory (theory 'lib-top1)))
(local-defthm lxor-1-not-2
(equal (equal 2 (lxor i j 1))
nil)
:hints (("Goal" :expand ((lxor i j 1))
:use ((:instance acl2::bvecp-1-rewrite
(acl2::x (bitn i 0)))
(:instance acl2::bvecp-1-rewrite
(acl2::x (bitn j 0)))))))
(local
(encapsulate
()
(local-defthm bitn-of-1-less-than-power-of-2-lemma-1
(implies (natp j)
(equal (bitn (1- (expt 2 (1+ j))) j)
1))
:hints (("Goal" :in-theory (enable acl2::bvecp-bitn-1 bvecp)
:expand ((expt 2 (+ 1 j))))))
(defthm bitn-of-1-less-than-power-of-2
(implies (and (integerp i)
(natp j)
(< j i))
(equal (bitn (1- (expt 2 i)) j)
1))
:hints (("Goal" :use ((:instance acl2::bitn-plus-mult
(acl2::x (1- (expt 2 (1+ j))))
(acl2::k (1- (expt 2 (- i (1+ j)))))
(acl2::m (1+ j))
(acl2::n j))))))))
(local-defun prop-as-lxor-thm (i j x y)
(implies (and (natp i)
(natp j)
(<= j i)
(natp x)
(natp y))
(equal (prop x y i j)
(log= (lxor (bits x i j) (bits y i j) (1+ (- i j)))
(1- (expt 2 (1+ (- i j))))))))
(local-defthm prop-as-lxor-thm-proved-1
(implies (not (and (natp i) (natp j) (<= j i)))
(prop-as-lxor-thm i j x y)))
(local-defthm prop-as-lxor-thm-proved-3-1
(implies (and (integerp i)
(<= 0 i)
(integerp j)
(<= 0 j)
(<= j i)
(equal (bitn x i) (bitn y i))
(integerp x)
(<= 0 x)
(integerp y)
(<= 0 y))
(not (equal (bitn (+ -1 (expt 2 (+ 1 i (* -1 j)))) (- i j))
(bitn (lxor (bits x i j)
(bits y i j)
(+ 1 i (* -1 j)))
(- i j)))))
:rule-classes nil)
(local-defthm prop-as-lxor-thm-proved-3
(implies (and (and (natp i) (natp j) (<= j i))
(= (bitn x i) (bitn y i)))
(prop-as-lxor-thm i j x y))
:hints (("Goal" :use prop-as-lxor-thm-proved-3-1
:in-theory (e/d (log=)
(bitn-of-1-less-than-power-of-2)))))
(local (in-theory (enable bvecp-0)))
(local-defthm prop-as-lxor-thm-proved-2
(implies (and (and (natp i) (natp j) (<= j i))
(not (= (bitn x i) (bitn y i)))
(prop-as-lxor-thm (+ -1 i) j x y))
(prop-as-lxor-thm i j x y))
:hints (("Goal" :in-theory (enable log=)
:expand ((expt 2 (+ 1 i (* -1 j))))
:use ((:instance acl2::bitn-plus-bits (acl2::m 0)
(n (- i j))
(acl2::x (lxor (bits x i j)
(bits y i j)
(+ 1 i (* -1 j)))))
(:instance acl2::bvecp-1-rewrite
(acl2::x (bitn x i)))
(:instance acl2::bvecp-1-rewrite
(acl2::x (bitn y i)))
(:instance acl2::bvecp-1-rewrite
(acl2::x (bitn x 0)))
(:instance acl2::bvecp-1-rewrite
(acl2::x (bitn y 0)))))))
(local-defthm prop-as-lxor-thm-proved
(prop-as-lxor-thm i j x y)
:hints (("Goal" :induct (prop x y i j)
:in-theory (disable prop-as-lxor-thm)))
:rule-classes nil)
(defthmd prop-as-lxor
(implies (and (natp i)
(natp j)
(<= j i)
(natp x)
(natp y))
(equal (prop x y i j)
(log= (lxor (bits x i j) (bits y i j) (1+ (- i j)))
(1- (expt 2 (1+ (- i j)))))))
:hints (("Goal" :use prop-as-lxor-thm-proved)))
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