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|
(in-package "ACL2")
(local (include-book "ground-zero"))
(local (include-book "fp2"))
(local (include-book "denominator"))
(local (include-book "numerator"))
(local (include-book "predicate"))
(local (include-book "nniq"))
(local (include-book "product"))
(local (include-book "unary-divide"))
(local (include-book "rationalp"))
(local (include-book "inverted-factor"))
(local (include-book "meta/meta-plus-lessp" :dir :system))
; (thm (rationalp (floor i j)))) goes through
(defthm floor-non-negative-integerp-type-prescription
(implies (and (<= 0 i)
(<= 0 j)
(case-split (not (complex-rationalp j))) ;gen?
)
(and (<= 0 (floor i j))
(integerp (floor i j))))
:rule-classes (:type-prescription)
:hints (("Goal" :in-theory (set-difference-theories
(enable floor)
'()))
))
;nope. (floor #C(0 -1) #C(0 -1)) = 1
;(defthm floor-with-j-non-rational
; (implies (not (rationalp j))
; (equal (floor i j)
; 0))
; :hints (("Goal" :in-theory (set-difference-theories
; (enable floor)
; '(a13 FL-WEAKLY-MONOTONIC)))
; ))
(defthm floor-non-negative
(implies (and (<= 0 i)
(<= 0 j)
(case-split (not (complex-rationalp i)));drop?
;(case-split (rationalp j))
)
(<= 0 (floor i j)))
:hints (("Goal" :in-theory (set-difference-theories
(enable floor)
'()))
))
(defthm floor-with-i-not-rational-but-j-rational
(implies (and (not (rationalp i))
(rationalp j)
)
(equal (floor i j)
0))
:hints (("Goal" :in-theory (enable floor)))
)
(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))
)))
:hints (("Goal" :in-theory (enable floor)))
)
(defthm floor-of-non-acl2-number
(implies (not (acl2-numberp i))
(and (equal (floor i j)
0)
(equal (floor j i)
0)))
:hints (("Goal" :in-theory (enable floor)))
)
;linear? how should it be phrased?
;too many hints. without the frac-coeff rule, things worked out here
(defthm floor-upper-bound
(implies (and (case-split (rationalp i))
(case-split (rationalp j))
)
(<= (floor i j) (/ i j)))
:hints (("Goal" :use ( (:instance nonnegative-integer-quotient-lower-bound-rewrite
(i (* -1 (NUMERATOR (* I (/ J)))))
(j (DENOMINATOR (* I (/ J)))))
(:instance nonnegative-integer-quotient-upper-bound-rewrite
(i (* -1 (NUMERATOR (* I (/ J)))))
(j (DENOMINATOR (* I (/ J)))))
(:instance nonnegative-integer-quotient-lower-bound-rewrite
(i (NUMERATOR (* I (/ J))))
(j (DENOMINATOR (* I (/ J)))))
(:instance nonnegative-integer-quotient-upper-bound-rewrite
(i (NUMERATOR (* I (/ J))))
(j (DENOMINATOR (* I (/ J))))))
:in-theory (set-difference-theories
(enable floor)
'( nonnegative-integer-quotient-lower-bound-rewrite
nonnegative-integer-quotient-upper-bound-rewrite
))))
: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)))))
:otf-flg t
:hints (("Goal" :in-theory (set-difference-theories
(enable floor)
'( nonnegative-integer-quotient-lower-bound-rewrite
nonnegative-integer-quotient-max-value-rewrite))
:use(
(:instance nonnegative-integer-quotient-max-value-rewrite
(i (* -1 (NUMERATOR (* I (/ J)))))
(j (DENOMINATOR (* I (/ J)))))
(:instance nonnegative-integer-quotient-lower-bound-rewrite
(i (* -1 (NUMERATOR (* I (/ J)))))
(j (DENOMINATOR (* I (/ J))))))
) )
)
(defthm dumb
(equal (< x x)
nil))
(defthm floor-with-j-zero
(equal (floor i 0)
0)
:hints (("Goal" :in-theory (enable floor)))
)
;(defthm floor-greater-than-zero-rewrite
; (equal (< 0 (fl 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))
:hints (("Goal" :in-theory (disable floor-upper-bound)
:use floor-upper-bound))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j))))
)
(defthm floor-upper-bound-3
(implies (and (<= j 0)
(case-split (rationalp i))
(case-split (rationalp j))
(case-split (not (equal j 0)))
)
(<= i (* j (floor i j))))
:hints (("Goal" :in-theory (disable floor-upper-bound)
:use floor-upper-bound))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j))))
)
;BOZO remove the disables (and prove better nniq rules, and disable nniq!)
(defthm floor-lower-bound
(implies (and (case-split (rationalp i))
(case-split (rationalp j))
)
(< (+ -1 (* i (/ j))) (floor i j)))
:otf-flg t
:hints (("Goal"
:in-theory (set-difference-theories
(enable floor)
'( ;why do these disables help so much?
less-than-multiply-through-by-inverted-factor-from-left-hand-side
less-than-multiply-through-by-inverted-factor-from-right-hand-side
EQUAL-MULTIPLY-THROUGH-BY-INVERTED-FACTOR-FROM-RIGHT-HAND-SIDE
)
)))
:rule-classes (:rewrite (:linear :trigger-terms ((floor i j)))))
(defthm floor-when-arg-quotient-isnt-rational
(IMPLIES (NOT (RATIONALP (* i (/ j))))
(EQUAL (FLOOR i j) 0))
:hints (("Goal" :in-theory (enable floor))))
(defthm floor-of-non-rational-by-one
(implies (not (rationalp i))
(equal (floor i 1)
0))
:hints (("Goal" :in-theory (enable floor))))
(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)))
:hints (("Goal" :in-theory (enable floor))))
#|
(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))))
|#
(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)
0)))
:hints (("Goal" :in-theory (enable floor))))
(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)
0)))
:hints (("Goal" :in-theory (enable floor))))
(defthm floor-with-j-not-rational-but-i-rational
(implies (and (not (rationalp i))
(rationalp j)
)
(equal (floor i j)
0)))
#|
(defthm floor-by-one-equal-zero
(implies (and (rationalp i)
(rationalp j))
(equal (EQUAL 0 (FLOOR (* i (/ j)) 1))
(integerp (* i (/ j)))))
:hints (("Goal" :in-theory (enable floor)))
)
|#
(defthm floor-of-zero
(equal (floor 0 j)
0))
(defthm floor-of-integer-by-1
(implies (integerp i)
(equal (floor i 1)
i))
:hints (("Goal" :in-theory (enable floor))))
|
