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|
(in-package "ACL2")
;see floor-proofs for all proofs (and todos?)
(local (include-book "floor-proofs"))
;;
;; Behavior of floor when its guards are violated
;;
;this looks like it might loop! add syntaxp hyp that j isn't 1?
(defthm floor-with-i-not-rational
(implies (not (rationalp i))
(equal (floor i j)
(if (and (complex-rationalp i) (complex-rationalp j) (rationalp (/ i j)))
(floor (/ i j) 1) ;yuck, defines floor in terms of itself
0))))
(defthm floor-with-j-not-rational
(implies (not (rationalp j))
(equal (floor i j)
(if (and (complex-rationalp i) (complex-rationalp j) (rationalp (/ i j)))
(floor (/ i j) 1) ;yuck, defines floor in terms of itself
0))))
;special case of floor-with-i-not-rational but contains no if
(defthm floor-with-i-not-rational-but-j-rational
(implies (and (not (rationalp i))
(rationalp j)
)
(equal (floor i j)
0)))
;special case of floor-with-j-not-rational but contains no if
(defthm floor-with-j-not-rational-but-i-rational
(implies (and (not (rationalp i))
(rationalp j)
)
(equal (floor i j)
0)))
;;
;; type prescriptions
;;
; (thm (integerp (floor i j)))) goes through
;to gen this, prove that the quotient of 2 positive complexes is either complex or positive. (that's a type!)
;I have a marvelous proof of that fact, but this buffer is too small to contain it.
;(Actually, it's in my green notebook.)
(defthm floor-non-negative-integerp-type-prescription
(implies (and (<= 0 i)
(<= 0 j)
(case-split (not (complex-rationalp j))) ;I think I can drop this hyp, but it will take some work.
)
(and (<= 0 (floor i j))
(integerp (floor i j))))
:rule-classes (:type-prescription))
;(floor #C(-4 -3) #C(4 3))= -1
(defthm floor-non-negative
(implies (and (<= 0 i)
(<= 0 j)
(case-split (not (complex-rationalp i)));(case-split (rationalp i));drop?
)
(<= 0 (floor i j))))
(defthm floor-compare-to-zero
(implies (and (case-split (rationalp i))
(case-split (rationalp j)))
(equal (< (floor i j) 0)
(or (and (< i 0) (< 0 j))
(and (< 0 i) (< j 0))
))))
(defthm floor-of-non-acl2-number
(implies (not (acl2-numberp i))
(and (equal (floor i j)
0)
(equal (floor j i)
0))))
;linear? how should it be phrased?
(defthm floor-upper-bound
(implies (and (case-split (rationalp i))
(case-split (rationalp j))
)
(<= (floor i j) (/ i j)))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j)))))
(defthm floor-equal-i-over-j-rewrite
(implies (and (case-split (not (equal j 0)))
(case-split (rationalp i))
(case-split (rationalp j))
)
(equal (equal (* j (floor i j)) i)
(integerp (* i (/ j))))))
;move
(defthm dumb
(equal (< x x)
nil))
(defthm floor-with-j-zero
(equal (floor i 0)
0))
(defthm floor-with-i-zero
(equal (floor 0 j)
0))
;(defthm floor-greater-than-zero-rewrite
; (equal (< 0 (floor i j))
; (
(defthm floor-upper-bound-2
(implies (and (<= 0 j)
(case-split (rationalp i))
(case-split (rationalp j))
(case-split (not (equal j 0)))
)
(<= (* j (floor i j)) i))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j)))))
(defthm floor-upper-bound-3
(implies (and (<= j 0) ;rarely true
(case-split (rationalp i))
(case-split (rationalp j))
(case-split (not (equal j 0)))
)
(<= i (* j (floor i j))))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j)))))
(defthm floor-lower-bound
(implies (and (case-split (rationalp i))
(case-split (rationalp j))
)
(< (+ -1 (* i (/ j))) (floor i j)))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j)))))
;a bit odd
(defthm floor-when-arg-quotient-isnt-rational
(implies (not (rationalp (* i (/ j))))
(equal (floor i j) 0)))
(defthm floor-of-non-rational-by-one
(implies (not (rationalp i))
(equal (floor i 1)
0)))
(defthm floor-of-rational-and-complex
(implies (and (rationalp i)
(not (rationalp j))
(case-split (acl2-numberp j)))
(and (equal (floor i j)
0)
(equal (floor j i)
0))))
#|
(defthm floor-of-two-complexes
(implies (and (complex-rationalp i)
(complex-rationalp j))
(equal (floor i j)
(if (rationalp (/ i j))
(floor (/ i j) 1)
0)))
:hints (("Goal" :in-theory (enable floor))))
|#
#|
(local
(defthm floor*2
(implies (integerp x)
(equal (floor (* 2 x) 2) x))
:hints (("Goal" :in-theory (enable floor)))
))
|#
(defthm floor-of-integer-by-1
(implies (integerp i)
(equal (floor i 1)
i))
:hints (("Goal" :in-theory (enable floor))))
|
