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(in-package "ACL2")
;My approach (and I believe this is Russinoff's approach too) for reasoning about floor and related
;functions is to write everything in terms of fl. Unlike floor, fl takes only 1 argument. Furthermore, all
;calls to floor can be rewritten to calls to fl using floor-fl
(include-book "negative-syntaxp")
(local (include-book "fl-proofs"))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
;remove syntaxp hyp?
;weird rule...
(defthm integerp-<-non-integerp
(implies (and (and (syntaxp (quotep x)))
(not (integerp x))
(integerp n) ;backchain limit?
(case-split (rationalp x))
)
(equal (< n x)
(<= n (fl x)))))
;remove syntaxp hyp?
;weird rule...
(defthm non-integerp-<-integerp
(implies (and (and (syntaxp (quotep x)))
(not (integerp x))
(integerp n) ;backchain limit?
(case-split (rationalp x))
)
(equal (< x n)
(< (fl x) n))))
(defthm a10
(and (implies (integerp i) (equal (fl i) i))
(implies (and (integerp i)
(case-split (rationalp x1))) ;drop?
(and (equal (fl (+ i x1)) (+ i (fl x1)))
(equal (fl (+ x1 i)) (+ i (fl x1)))))
(implies (and (integerp i)
(case-split (rationalp x1))
(case-split (rationalp x2)))
(and (equal (fl (+ x1 (+ i x2)))
(+ i (fl (+ x1 x2))))
(equal (fl (+ x1 (+ x2 i)))
(+ i (fl (+ x1 x2))))))
(implies (and (integerp i)
(case-split (rationalp x1))
(case-split (rationalp x2))
(case-split (rationalp x3)))
(and (equal (fl (+ x1 (+ x2 (+ i x3))))
(+ i (fl (+ x1 x2 x3))))
(equal (fl (+ x1 (+ x2 (+ x3 i))))
(+ i (fl (+ x1 x2 x3))))))
(implies (and (integerp i)
(case-split (rationalp x1))
(case-split (rationalp x2))
(case-split (rationalp x3))
(case-split (rationalp x4)))
(and (equal (fl (+ x1 (+ x2 (+ x3 (+ i x4)))))
(+ i (fl (+ x1 x2 x3 x4))))
(equal (fl (+ x1 (+ x2 (+ x3 (+ x4 i)))))
(+ i (fl (+ x1 x2 x3 x4))))))))
(defthm a12
(implies (and (integerp i)
(integerp j)
(< 1 j)
(< j i))
(and (< (acl2-count (fl (/ i j))) (acl2-count i))
(< (acl2-count (fl (* (/ j) i))) (acl2-count i))))
:rule-classes :linear)
;why "fl-def" in the names? this isn't a definition...
;make a separate rewrite-version
(defthm fl-def-linear-part-1
(implies (case-split (not (complex-rationalp x)))
(<= (fl x) x))
:rule-classes (:rewrite (:linear :trigger-terms ((fl x)))))
(defthm fl-def-linear-part-2
(implies (case-split (not (complex-rationalp x)))
(< x (1+ (fl x))))
:rule-classes (:rewrite (:linear :trigger-terms ((fl x)))))
;later, drop the hyp completely
;disabled since we have the rules above
;drop this whole rule
(defthmd a13
(implies (case-split (rationalp x)) ;this hyp isn't needed for the first conslusion?
(and (< (1- x) (fl x))
(<= (fl x) x)))
:rule-classes :linear)
;disabled since we have the next rules above
(defthmd fl-def-linear
(implies (case-split (rationalp x)) ;gen?
(and (<= (fl x) x)
(< x (1+ (fl x)))))
:rule-classes :linear)
;note that FL is not strongly monotonic. That is, x<x+ does not always imply (fl x) < (fl x+)
;change param names to x, x+
;try match-free :all
(defthm fl-weakly-monotonic
(implies (and (<= y y+)
(case-split (rationalp y)) ;drop?
(case-split (rationalp y+)) ;drop?
)
(<= (fl y) (fl y+)))
:rule-classes ((:forward-chaining :trigger-terms ((fl y) (fl y+)))
(:linear)
(:forward-chaining
:trigger-terms ((fl y) (fl y+))
:corollary (implies (and (< y y+)
(case-split (rationalp y))
(case-split (rationalp y+)))
(<= (fl y) (fl y+))))
(:linear
:corollary (implies (and (< y y+)
(case-split (rationalp y))
(case-split (rationalp y+)))
(<= (fl y) (fl y+))))))
;add a separate "fl of complex is 0" ?
;because of this, I can put make (rationalp x) into (case-split (rationalp x)) for all arguments x of fl
; (since we always can simplify the other case, the split is nice)
(defthm fl-of-non-rational
(implies (not (rationalp x))
(equal (fl x)
0)))
(defthm fl-minus
(implies (rationalp x) ;could be (not (complex-rationalp x))
(equal (fl (* -1 x))
(if (integerp x)
(* -1 (fl x))
(+ -1 (- (fl x))))))
:rule-classes nil)
;bad? free var.
(defthm fl-monotone-linear
(implies (and (<= x y)
(rationalp x)
(rationalp y))
(<= (fl x) (fl y)))
:rule-classes :linear)
(defthm n<=fl-linear
(implies (and (<= n x)
(rationalp x)
(integerp n))
(<= n (fl x)))
:rule-classes :linear)
;may need to disable?
(defthm fl+int-rewrite
(implies (and (integerp n)
(rationalp x))
(equal (fl (+ x n)) (+ (fl x) n))))
(defthm fl/int-rewrite
(implies (and (integerp n)
(<= 0 n)
(rationalp x))
(equal (fl (* (fl x) (/ n)))
(fl (/ x n)))))
(defthm fl/int-rewrite-alt
(implies (and (integerp n)
(<= 0 n)
(rationalp x))
(equal (fl (* (/ n) (fl x)))
(fl (/ x n)))))
(defthm fl-integer-type
(integerp (fl x))
:rule-classes (:type-prescription))
;this rule is no better than fl-integer-type and might be worse
(in-theory (disable (:type-prescription fl)))
(defthm fl-int
(implies (integerp x)
(equal (fl x) x)))
;bad name?
(defthm fl-integerp
(equal (equal (fl x) x)
(integerp x)))
(defthm fl-unique
(implies (and (rationalp x)
(integerp n)
(<= n x)
(< x (1+ n)))
(equal (fl x) n))
:rule-classes ())
;ACL2 already knows these facts about FL, but we include them anyway
(defthm fl-rational
(rationalp (fl x)))
(defthm fl-integer
(integerp (fl x)))
;add "fl of negative is negative" type rule? (actually 2 posibilities?)
(defthm fl-non-negative-integer-type-prescription
(implies (<= 0 x)
(and (<= 0 (fl x))
(integerp (fl x))))
:rule-classes (:type-prescription))
(defthm fl-less-than-zero
(implies (case-split (rationalp x))
(equal (< (fl x) 0)
(< x 0))))
;use rifx?
;prove a version for fl negative? (also t-p rules for that?)
(defthm fl-non-negative-linear
(implies (<= 0 x)
(<= 0 (fl x)))
:rule-classes (:linear))
;rename to start with fl-
;needed? - any constant, not just integers
;replace the rule to pull out an integer?
;BOZO do we want to use rem here???
(defthm pull-constant-out-of-fl
(implies (and (syntaxp (and (quotep c1) (>= (abs (cadr c1)) 1)))
(rationalp c1)
(rationalp x))
(equal (fl (+ c1 x))
(+ (truncate c1 1) (fl (+ (rem c1 1) x))))))
;fl-minus?
;this can loop with fl-minus-gen (when the result of applying fl-minus-gen doesn't get simplfied before we
;build the linear pot list)??
;maybe fl-minus-gen is enough?
;why disabled?
(defthmd fl-minus-factor
(implies (and (syntaxp (quotep k))
(< k 0)
(rationalp k)
(rationalp x))
(equal (fl (* k x))
(if (integerp (* k x))
(* -1 (fl (* (- k) x)))
(+ -1 (- (fl (* (- k) x))))))))
(defthm fl-<-integer
(implies (and (integerp y)
(case-split (rationalp x)))
(equal (< (fl x) y)
(< x y))))
(defthm fl->-integer
(implies (and (integerp y)
(case-split (rationalp x)))
(equal (< y (fl x))
(<= (+ 1 y) x))))
;should this be disabled?
(defthm fl-equal-0
(implies (case-split (rationalp x))
(equal (equal (fl x) 0)
(and (<= 0 x)
(< x 1)))))
;kill this? or is this nice b/c it makes no change if its hyps fail to be satisfied?
(defthmd fl-equal-0-alt
(implies (and (< x 1)
(<= 0 x)
(case-split (rationalp x))
)
(equal (fl x) 0)))
;bad names?
;fl-def-linear isn't rewrite!
;remove this??
(defthm fl-strong-monotone
(implies (and (< x y)
(rationalp x)
(rationalp y)
)
(< (fl x) y)))
;remove this??
;make linear?
(defthm fl-weak-monotone
(implies (and (<= x y)
(rationalp x)
(rationalp y)
)
(<= (fl x) y)))
;kill one of these?
(defthm fl-def-linear-quotient
(implies (and (< 0 y)
(case-split (rationalp x))
(case-split (rationalp y))
)
(and (not (< X (* Y (FL (* X (/ Y))))))
(not (< X (* Y (FL (* (/ Y) X))))))))
;Our scheme for dealing with FLOOR is to always rewrite calls of it to FL
(defthm floor-fl
(equal (floor m n)
(fl (/ m n))))
(theory-invariant (incompatible (:rewrite floor-fl)
(:definition fl))
:key floor-fl--and--fl--conflict)
;perhaps always split on even/odd for fl(x/2)
;needed? was in proof.lisp for x87 recip proof
(defthm fl-of-odd/2
(implies (and (integerp x)
(not (integerp (* 1/2 x)))
)
(equal (fl (* 1/2 x))
(- (* 1/2 x) 1/2))))
(defthm fl-of-even/2
(implies (and (INTEGERP (* X (/ 2))))
(equal (fl (* 1/2 x))
(* 1/2 x))))
;new version
;may be expensive
(defthmd fl-force-to-0
(implies (case-split (rationalp x))
(equal (equal 0 (fl x))
(and (<= 0 x)
(< x 1)))))
;generalize to any amount of shifting each time and to any base (2,3, etc.)?
;is there a linear rule missing? why did we need to :use fl-def-linear?
;(both expressions shift x right n+1 spots and chop)
;attempted addition 1/7/02:
;make linear?
;bad name?
(defthm fl-factor-out-integer-bound
(implies (and (integerp n)
(> n 0)
(rationalp x)
)
(<= (* n (fl x))
(fl (* n x)))))
;make linear?
;bad name?
(defthm fl-factor-out-integer-bound-2
(implies (and (integerp n)
(> n 0)
(rationalp m)
)
(<= (* n (fl (* m (/ n))))
(fl m))))
;see sse-div.lisp for better versions of the above
#| for reference:
(DEFTHM FL-DEF-LINEAR
(IMPLIES (RATIONALP X)
(AND (<= (FL X) X) (< X (1+ (FL X)))))
:RULE-CLASSES :LINEAR)
|#
;these thms are about getting rid of one of two (roughly) "nested" calls to fl
;why??
(defthm fl-plus-integer-eric
(implies (and (integerp n)
(case-split (rationalp x)) ;not true if x is a complex-rationalp
)
(equal (fl (+ x n))
(+ n (fl x)))))
(local (in-theory (disable floor-fl)))
;BOZO move!
;strictly better than the version in the arithmetic books
(defthm quotient-numer-denom-eric
(implies (and (integerp x)
(<= 0 x) ; was (< 0 x)
(integerp y)
(<= 0 y)) ;was (< 0 y)
(equal (nonnegative-integer-quotient (numerator (/ x y))
(denominator (/ x y)))
(nonnegative-integer-quotient x y))))
;(in-theory (disable QUOTIENT-NUMER-DENOM))
; rewrites things like (EQUAL (* 32768 (FL (* 1/32768 x))) x)
(defthmd fl-claim-rewrite-to-integerp-claim-gen
(implies (and (equal k-inverse (/ k))
(case-split (acl2-numberp k))
(case-split (not (equal k 0)))
(case-split (acl2-numberp x))
)
(and (equal (EQUAL (* k (FL (* k-inverse X))) X)
(integerp (* k-inverse x)))
(equal (EQUAL (* k (FL (* X k-inverse))) X)
(integerp (* k-inverse x))))))
(defthm fl-claim-rewrite-to-integerp-claim-gen-2
(implies (and (equal k-inverse (/ k))
(case-split (acl2-numberp k))
(case-split (not (equal k 0)))
(case-split (acl2-numberp x))
(case-split (acl2-numberp y))
)
(and (equal (EQUAL (* k (FL (* k-inverse X y))) (* X y))
(integerp (* k-inverse x y)))
(equal (EQUAL (* k (FL (* X k-inverse y))) (* X y))
(integerp (* k-inverse x y)))
(equal (EQUAL (* k (FL (* X y k-inverse))) (* X y))
(integerp (* k-inverse x y))))))
(defthm fl-claim-rewrite-to-integerp-claim
(implies (and (syntaxp (and (quotep k-inverse)
(quotep k)))
(equal k-inverse (/ k))
(case-split (acl2-numberp k))
(case-split (not (equal k 0)))
(case-split (acl2-numberp x))
)
(and (equal (EQUAL (* k (FL (* k-inverse X))) X)
(integerp (* k-inverse x)))
(equal (EQUAL (* k (FL (* X k-inverse))) X)
(integerp (* k-inverse x))))))
;move! rename?
(defthm y-is-odd
(equal (EQUAL Y (+ 1 (* 2 (FL (* 1/2 Y)))))
(and (integerp y)
(not (integerp (* 1/2 y))))))
(defthm fl-minus-gen
(implies (and (syntaxp (negative-syntaxp x))
(case-split (rationalp x))
)
(EQUAL (FL x)
(IF (INTEGERP X)
(* -1 (FL (* -1 X)))
(+ -1 (- (FL (* -1 X))))))))
(defthmd fl-of-fraction-max-change-case-1
(implies (and (not (integerp (/ p q))) ;this case
(integerp p)
(integerp q)
(< 0 q)
)
(>= (+ 1 (fl (/ p q)))
(+ (/ p q) (/ q)))))
(defthmd fl-of-fraction-max-change-case-2
(implies (and (integerp (/ p q)) ;this case
(integerp p)
(integerp q)
(< 0 q)
)
(>= (+ 1 (fl (/ p q)))
(+ (/ p q) (/ q)))))
;fl(p/q) + 1 >= p/q + 1/q
;similar to fl-of-fraction-upper-bound
;rephrase the conclusion
; if fl changes it argument, it does so by at most 1-1/q
(defthm fl-of-fraction-max-change
(implies (and (< 0 q)
(integerp p)
(integerp q)
)
(>= (+ 1 (fl (/ p q)))
(+ (/ p q) (/ q)))))
; Two integers, both in the interval (x,y], whose width is at most 1, must be equal.
;make other versions?
;move?
(defthm int-unique
(implies (and (integerp i)
(integerp j)
(rationalp x)
(rationalp y)
(<= x y)
(< x i) (<= i y)
(< x j) (<= j y)
(<= (- y x) 1)
)
(equal i j))
:rule-classes nil
)
;replace fl-unique?
(defthm fl-unique-2
(implies (rationalp x)
(equal (and (integerp n)
(<= n x)
(< x (1+ n)))
(equal (fl x) n)))
:rule-classes nil)
(defthm fl-m+1
(implies (and (integerp m)
(integerp n)
(>= m 0)
(> n 0))
(= (fl (- (/ (1+ m) n)))
(1- (- (fl (/ m n))))))
:rule-classes ())
;this was the point of nniq-eric-5 to nniq-8 in basic. how does this get proved without nniq-eric-8?
;if fl changes its argument of p/q, it does so by at least a qth
;rephrase concl?
;make a linear rule?
(defthmd fl-of-fraction-min-change
(implies (and (not (integerp (/ p q)))
(integerp p)
(integerp q)
)
(<= (/ q)
(- (/ p q) (fl (/ p q))))) ;the amt of change made by fl
)
;bad name? improve this?
;BOZO call quot-bnd-2? or fl-bnd-2?
(defthm fl-bound
(implies (and (< 0 y)
(rationalp x)
(rationalp y)
)
(<= (* y (FL (* x (/ y)))) x)))
;see fl-bound
;BOZO rename?, params
(defthm quot-bnd
(implies (and (<= 0 n)
(<= 0 m)
(rationalp m)
(rationalp n)
)
(<= (* n (fl (/ m n))) m))
:rule-classes :linear)
;move!
;i just added this; is it expensive?
;this was causing problems, so I disabled it.
(defthmd x-times-something>=1
(implies (and (case-split (<= 1 y))
(case-split (rationalp y))
(case-split (rationalp x))
(case-split (<= 0 x)))
(<= x (* x y)))
:rule-classes :linear
)
(defthm fl-<=-y
(implies (and (<= x y)
(case-split (not (complex-rationalp x))))
(<= (fl x) y)))
;make a version where n is a constant?
(defthmd fl-equal-rewrite
(implies (and (rationalp x)
(integerp n)) ;move to conclusion?
(equal (equal (fl x) n)
(and (<= n x)
(< x (+ 1 n))))))
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