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|
(in-package "ACL2")
(include-book "rtl")
(include-book "bvecp-lemmas") ;bvecp and type lemmas for the RTL primitives
(local (include-book "../arithmetic/top"))
(local (include-book "bits"))
;would like to remove some of this stuff
;;;;;;;;;;;;;;;;;;; other helpful lemmas
(defthm nonneg-+
(implies (and (<= 0 x)
(<= 0 y))
(<= 0 (+ x y))))
(defthm integerp-+
(implies (and (integerp x)
(integerp y))
(integerp (+ x y))))
#|
;should be a forward-chaining rule?
(defthm bvecp-implies-natp
(implies (bvecp x k)
(and (integerp x)
(>= x 0)))
:hints (("Goal" :in-theory (enable bvecp))))
;free var
;should be a forward-chaining rule?
(defthm bvecp-implies-rationalp
(implies (bvecp x k)
(rationalp x))
:hints (("Goal" :in-theory (enable bvecp))))
|#
;why do we have this?
(defthm unknown-upper-bound
(< (unknown key size n) (expt 2 size))
:hints
(("goal" :use bvecp-unknown
:in-theory (set-difference-theories
(enable bvecp)
'(bvecp-unknown))))
:rule-classes
(:rewrite (:linear :trigger-terms ((unknown key size n)))))
(defthm bv-arrp-implies-nonnegative-integerp
(implies (bv-arrp obj size)
(and (INTEGERP (ag index obj))
(<= 0 (ag index obj))))
:rule-classes (:rewrite :type-prescription)
:hints (("Goal" :use (:instance
ag-maps-bv-arr-to-bvecp (a index) (r obj) (k size))
:in-theory (set-difference-theories
(enable bvecp)
'(ag-maps-bv-arr-to-bvecp)))))
;(local (in-theory (enable floor-fl)))
;These next two are for the bus unit bvecp lemmas:
;could use (local (in-theory (enable expt-compare-with-double)))
;remove?
(defthm bits-does-nothing-hack
(implies (and (< x (expt 2 i))
(integerp x)
(<= 0 x)
(integerp i)
(<= 0 i))
(equal (BITS (* 2 x) i 0)
(* 2 x)))
:hints (("Goal" :use (:instance bits-tail (x (* 2 x)) (i i))
:in-theory (set-difference-theories
(enable bvecp)
'(bits-tail)))))
;remove?
(defthm bits-does-nothing-hack-2
(implies (and (< x (expt 2 i))
(integerp x)
(<= 0 x)
(integerp i)
(<= 0 i))
(equal (bits (+ 1 (* 2 x)) i 0)
(+ 1 (* 2 x))))
:hints (("goal" :in-theory (set-difference-theories
(enable expt-split)
'(bits-tail
))
:use (:instance bits-tail (x (+ 1 (* 2 x))) (i i)))))
;is this one too expensive?
(defthm bvecp-def
(implies (and (< x (expt 2 k))
(integerp x)
(<= 0 x)
)
(bvecp x k))
:hints (("Goal" :in-theory (enable bvecp)))
:rule-classes ((:rewrite :backchain-limit-lst (1 nil nil))))
(defthm if1-0
(equal (if1 0 y z)
z))
(defthm if1-non-0
(implies (not (equal x 0))
(equal (if1 x y z)
y)))
(defthm if1-x-x
(equal (if1 tst x x)
x))
(defthm bvecp-if1
(equal (bvecp (if1 x y z) n)
(if1 x (bvecp y n) (bvecp z n))))
(defthm bv-arrp-if1
(equal (bv-arrp (if1 x y z) n)
(if1 x (bv-arrp y n) (bv-arrp z n))))
;remove these?
#|
(defthm bvecp-1-values
(implies (and (bvecp x 1)
(not (equal x 0)))
(equal (equal x 1) t))
:hints (("Goal" :in-theory (enable bvecp))))
(defthm bvecp-2-values
(implies (and (bvecp x 2)
(not (equal x 2))
(not (equal x 1))
(not (equal x 0)))
(equal (equal x 3) t))
:hints (("Goal" :in-theory (enable bvecp))))
(defthm bvecp-3-values
(implies (and (bvecp x 3)
(not (equal x 6))
(not (equal x 5))
(not (equal x 4))
(not (equal x 3))
(not (equal x 2))
(not (equal x 1))
(not (equal x 0)))
(equal (equal x 7) t))
:hints (("Goal" :in-theory (enable bvecp))))
(defthm bvecp-4-values
(implies (and (bvecp x 4)
(not (equal x 14))
(not (equal x 13))
(not (equal x 12))
(not (equal x 11))
(not (equal x 10))
(not (equal x 9))
(not (equal x 8))
(not (equal x 7))
(not (equal x 6))
(not (equal x 5))
(not (equal x 4))
(not (equal x 3))
(not (equal x 2))
(not (equal x 1))
(not (equal x 0)))
(equal (equal x 15) t))
:hints (("Goal" :in-theory (enable bvecp))))
(defthm bvecp-5-values
(implies (and (bvecp x 5)
(not (equal x 30))
(not (equal x 29))
(not (equal x 28))
(not (equal x 27))
(not (equal x 26))
(not (equal x 25))
(not (equal x 24))
(not (equal x 23))
(not (equal x 22))
(not (equal x 21))
(not (equal x 20))
(not (equal x 19))
(not (equal x 18))
(not (equal x 17))
(not (equal x 16))
(not (equal x 15))
(not (equal x 14))
(not (equal x 13))
(not (equal x 12))
(not (equal x 11))
(not (equal x 14))
(not (equal x 13))
(not (equal x 12))
(not (equal x 11))
(not (equal x 10))
(not (equal x 9))
(not (equal x 8))
(not (equal x 7))
(not (equal x 6))
(not (equal x 5))
(not (equal x 4))
(not (equal x 3))
(not (equal x 2))
(not (equal x 1))
(not (equal x 0)))
(equal (equal x 31) t))
:hints (("Goal" :in-theory (enable bvecp))))
;can remove these two?
(defthm natp-*
(implies (and (integerp x)
(>= x 0)
(integerp y)
(>= y 0))
(and (integerp (* x y))
(>= (* x y) 0))))
(defthm natp-+
(implies (and (integerp x)
(>= x 0)
(integerp y)
(>= y 0))
(and (integerp (+ x y))
(>= (+ x y) 0))))
;drop?
(DEFTHM BITS-bvecp-FW
(IMPLIES (EQUAL N (- (1+ I) J))
(BVECP (BITS X I J) N))
:RULE-CLASSES
((:FORWARD-CHAINING :TRIGGER-TERMS ((BITS X I J)))))
|#
;BOZO where all should this go? make an if1 book!
(defthmd if1-lnot
(implies (bvecp tst 1)
(equal (if1 (lnot tst 1) x y)
(if1 tst y x)))
:hints (("Goal" :in-theory (enable if1 bvecp))))
|
