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(in-package "ACL2")
;This book introduces the function cg (for "ceiling"), which is in many ways analogous to fl and which is used
;in the definition of the "away" rounding mode.
;todo: prove more thms about cg analogous to those about fl (maybe not worth doing since only fl is used to
;define, for example, bits).
(local (include-book "fl"))
(local (include-book "fp2"))
(local (include-book "integerp"))
(local (include-book "integerp"))
(local (include-book "arith2"))
(local (include-book "common-factor"))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund cg (x)
(declare (xargs :guard (real/rationalp x)))
(- (fl (- x))))
(defthm cg-def-linear
(implies (case-split (rationalp x))
(and (>= (cg x) x)
(> (1+ x) (cg x))))
:hints (("Goal" :in-theory (enable cg)))
:rule-classes :linear)
(defthm cg-monotone-linear
(implies (and (rationalp x)
(rationalp y)
(<= x y))
(<= (cg x) (cg y)))
:rule-classes :linear)
(defthm n>=cg-linear
(implies (and (>= n x)
(rationalp x)
(integerp n))
(>= n (cg x)))
:rule-classes :linear)
(defthm cg+int-rewrite
(implies (and (integerp n)
(rationalp x))
(equal (cg (+ x n)) (+ (cg x) n))))
(local
(defthm cg/int-1
(implies (and (rationalp x)
(integerp n)
(> n 0))
(>= (cg (/ (cg x) n))
(cg (/ x n))))
:rule-classes ()
:hints (("Goal" :use ((:instance cg-def-linear)
(:instance cg-monotone-linear (x (/ x n)) (y (/ (cg x) n))))))))
(local
(defthm cg/int-2
(implies (and (rationalp x)
(integerp n)
(> n 0))
(<= (cg (/ (cg x) n))
(cg (/ x n))))
:rule-classes ()
:hints (("Goal" :use ((:instance n>=cg-linear (n (* n (cg (/ x n)))))
(:instance n>=cg-linear (n (cg (/ x n))) (x (/ (cg x) n)))
(:instance cg-def-linear (x (/ x n))))))))
(defthm cg/int-rewrite
(implies (and (integerp n)
(> n 0)
(rationalp x))
(equal (cg (* (cg x) (/ n)))
(cg (/ x n))))
:hints (("Goal" :use ((:instance cg/int-1)
(:instance cg/int-2)))))
(defthm cg/int-rewrite-alt
(implies (and (integerp n)
(> n 0)
(rationalp x))
(equal (cg (* (/ n) (cg x)))
(cg (/ x n))))
:hints (("Goal" :use ((:instance cg/int-1)
(:instance cg/int-2)))))
(defthm int-cg-rules
(implies (rationalp x)
(integerp (cg x)))
:rule-classes (:rewrite :type-prescription))
(defthm cg-int
(implies (integerp x)
(equal (cg x) x)))
(defthm cg-integerp
(implies (rationalp x)
(equal (equal (cg x) x)
(integerp x))))
(defthm cg-unique
(implies (and (rationalp x)
(integerp n)
(>= n x)
(> (1+ x) n))
(equal (cg x) n))
:rule-classes ())
(defthm fl-cg
(implies (and (rationalp x)
(not (integerp x)))
(equal (cg x) (1+ (fl x))))
:rule-classes ())
(defthm cg-integer-type
(integerp (cg x))
:rule-classes ( :type-prescription))
(defthm cg-positive
(implies (case-split (not (complex-rationalp x)))
(equal (< 0 (cg x))
(< 0 x)))
:hints (("Goal" :in-theory (enable cg)))
)
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