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;;;***************************************************************
;;;An ACL2 Library of Floating Point Arithmetic
;;;David M. Russinoff
;;;Advanced Micro Devices, Inc.
;;;February, 1998
;;;***************************************************************
;this books contains rules which aren't used anywhere in lib/ or support/
(in-package "ACL2")
;(include-book "fp2")
;(local (include-book "even-odd"))
(local (include-book "basic")) ;yuck
(defthm exp+1-1
(implies (and (integerp m)
(integerp n)
(<= n m))
(<= (+ (expt 2 m) (expt 2 n))
(expt 2 (1+ m))))
:rule-classes ()
:hints (("Goal" :use ((:instance expt-weak-monotone)
(:instance expt-split (r 2) (i 1) (j m))))))
(defthm exp+1
(implies (and (integerp m)
(integerp n)
(<= n m))
(> (* (- 1 (expt 2 m)) (- 1 (expt 2 n)))
(- 1 (expt 2 (1+ m)))))
:rule-classes ()
:hints (("Goal" :in-theory (disable)
:use ((:instance exp+1-1)
))))
(defthm exp+2-1
(implies (and (integerp n)
(integerp m)
(<= n m)
(<= m 0))
(<= (* (expt 2 m) (expt 2 n))
(expt 2 m)))
:rule-classes ()
:hints (("Goal" :use ((:instance expt-weak-monotone (n (+ m n)))
(:instance expt-split (r 2))))))
(defthm exp+2-2
(implies (and (integerp n)
(integerp m)
(<= n m)
(<= m 0))
(<= (+ (expt 2 m) (expt 2 n) (* (expt 2 m) (expt 2 n)))
(* 3 (expt 2 m))))
:rule-classes ()
:hints (("Goal" :in-theory (disable)
:use ((:instance expt-weak-monotone)
(:instance exp+2-1)))))
(defthm exp+2-3
(implies (and (integerp n)
(integerp m)
(<= n m)
(<= m 0))
(< (+ (expt 2 m) (expt 2 n) (* (expt 2 m) (expt 2 n)))
(* 4 (expt 2 m))))
:rule-classes ()
:hints (("Goal" :in-theory (disable)
:use (
(:instance exp+2-2)
(:instance *-strongly-monotonic (x (expt 2 m)) (y 3) (y+ 4))))))
(defthm exp+2
(implies (and (integerp n)
(integerp m)
(<= n m)
(<= m 0))
(< (* (1+ (expt 2 m)) (1+ (expt 2 n)))
(1+ (expt 2 (+ m 2)))))
:rule-classes ()
:hints (("Goal" :use ((:instance exp+2-3)
(:instance expt-split (r 2) (i 2) (j m))))))
(defthm exp-invert-1
(implies (and (integerp n)
(<= n -1))
(<= (* (expt 2 n) (expt 2 (1+ n)))
(expt 2 n)))
:rule-classes ()
:hints (("Goal" :use ((:instance expt-weak-monotone (n (1+ n)) (m 0))
(:instance *-weakly-monotonic (x (expt 2 n)) (y (expt 2 (1+ n))) (y+ 1))))))
(defthm exp-invert-2
(implies (and (integerp n)
(<= n -1))
(>= (* (- 1 (expt 2 n))
(1+ (expt 2 (1+ n))))
1))
:rule-classes ()
:hints (("Goal" :use ((:instance expt-split (r 2) (i n) (j 1))
(:instance exp-invert-1)))))
(defthm cancel-x
(implies (and (rationalp x)
(rationalp y)
(> x 0)
(<= 1 (* x y)))
(<= (/ x) y))
:rule-classes ())
(defthm exp-invert
(implies (and (integerp n)
(<= n -1))
(<= (/ (- 1 (expt 2 n)))
(1+ (expt 2 (1+ n)))))
:rule-classes ()
:hints (("Goal" :use ((:instance cancel-x (x (- 1 (expt 2 n))) (y (1+ (expt 2 (1+ n)))))
(:instance exp-invert-1)
(:instance expt-weak-monotone (m -1))))))
(local
(defthm sq-sq-1
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(<= (* (* a b) (* a b))
(* (expt 2 (- (* 2 n) 2)) (* p p))))
:rule-classes ()
:hints (("Goal" :use ((:instance *-doubly-monotonic
(x (* a a)) (a (* b b))
(y p) (b (* (expt 2 (- (* 2 n) 2)) p))))))))
;not exported anywhere!
;rephrase?
(defthm sqrt<=
(implies (and (rationalp x)
(rationalp a)
(>= a 0)
(<= (* x x) (* a a)))
(<= x a))
:rule-classes ()
:hints (("Goal" :use ((:instance *-strongly-monotonic (y a) (y+ x))
(:instance *-strongly-monotonic (x a) (y a) (y+ x))))))
(local
(defthm sq-sq-2
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(<= (* (* (expt 2 (1- n)) p) (* (expt 2 (1- n)) p))
(* (expt 2 (- (* 2 n) 2)) (* p p))))
:rule-classes ()
:hints (("Goal" :use ((:instance expt-split (r 2) (i (1- n)) (j (1- n))))))))
(local
(defthm sq-sq-3
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(>= (* (expt 2 (1- n)) p) 0))
:rule-classes ()))
(local
(defthm sq-sq-4
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(<= (* a b)
(* (expt 2 (1- n)) p)))
:rule-classes ()
:hints (("Goal" :in-theory (enable a15)
:use ((:instance sq-sq-1)
(:instance sq-sq-3)
(:instance sqrt<= (x (* a b)) (a (* (expt 2 (1- n)) p)))
(:instance sq-sq-2))))))
(local
(defthm sq-sq-5
(implies (and (rationalp x)
(rationalp p)
(integerp n)
(<= x (* (expt 2 (1- n)) p)))
(<= (* 2 x) (* (expt 2 n) p)))
:rule-classes ()
:hints (("Goal" :in-theory (disable *-weakly-monotonic)
:use ((:instance expt-split (r 2) (i (1- n)) (j 1))
(:instance *-weakly-monotonic (x 2) (y x) (y+ (* (expt 2 (1- n)) p))))))))
(local
(defthm sq-sq-6
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(<= (* 2 a b)
(* (expt 2 n) p)))
:rule-classes ()
:hints (("Goal" :use ((:instance sq-sq-4)
(:instance sq-sq-5 (x (* a b))))))))
(local
(defthm sq-sq-7
(implies (and (rationalp a)
(rationalp b))
(>= (* (- a b) (- a b))
(- (* a a) (* 2 a b))))
:rule-classes ()))
(local
(defthm sq-sq-8
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(>= p 0)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(>= (* (- a b) (- a b))
(- (* (- 1 (expt 2 n)) p)
(* (expt 2 n) p))))
:rule-classes ()
:hints (("Goal" :use ((:instance sq-sq-6)
(:instance sq-sq-7))))))
(local
(defthm sq-sq-9
(implies (and (rationalp p)
(integerp n))
(= (- (* (- 1 (expt 2 n)) p)
(* (expt 2 n) p))
(* (- 1 (expt 2 (1+ n))) p)))
:rule-classes ()))
;where is this used?
;doesn't seem to be used anywhere or exported in lib?
(defthm sq-sq
(implies (and (rationalp a)
(rationalp b)
(rationalp p)
(integerp n)
(<= (* (- 1 (expt 2 n)) p) (* a a))
(<= (* a a) p)
(<= (* b b) (* (expt 2 (- (* 2 n) 2)) p)))
(>= (* (- a b) (- a b))
(* (- 1 (expt 2 (1+ n))) p)))
:rule-classes ()
:hints (("Goal" :use ((:instance sq-sq-8)
(:instance sq-sq-9)))))
;kill some of these 4 abs lemmas (they are from divsqrt and don't seem to be
;needed in support/ or exported in lib/) ?
(defthm abs-+
(implies (and (rationalp x)
(rationalp y)
(rationalp z))
(<= (abs (- x y))
(+ (abs (- x z))
(abs (- y z)))))
:rule-classes ())
(defthm abs->=
(implies (and (rationalp x)
(rationalp y))
(>= (abs (- y x)) (- (abs x) (abs y))))
:rule-classes ())
;kill?
(local
(defthm abs+
(implies (and (rationalp x)
(rationalp y))
(<= (abs (+ x y))
(+ (abs x) (abs y))))
:rule-classes ()
:hints (("goal" :in-theory (enable abs)))))
(defthm abs-
(implies (and (rationalp x)
(rationalp y))
(<= (abs (- x y))
(+ (abs x) (abs y))))
:rule-classes ()
:hints (("goal" :in-theory (enable abs))))
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