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; See the top-level arithmetic-3 LICENSE file for authorship,
; copyright, and license information.
;;
;; expt.lisp
;;
(in-package "ACL2")
(local (include-book "basic-arithmetic"))
(local (include-book "inequalities"))
(local (include-book "prefer-times"))
(local (include-book "expt-helper"))
(defmacro fc (x)
x)
; Much of this is adapted from John Cowles's acl2-exp.lisp book.
; There are various modifications, however.
(defthm expt-type-prescription-rationalp
(implies (rationalp r)
(rationalp (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-positive-1
(implies (and (< 0 r)
(rationalp r))
(< 0 (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-positive-2
(implies (and (<= 0 r)
(rationalp r))
(<= 0 (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-nonzero
(implies (and (fc (acl2-numberp r))
(not (equal r 0)))
(not (equal 0 (expt r i))))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-integerp
(implies (and (<= 0 i)
(integerp r))
(integerp (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm equal-expt-0
(equal (equal (expt x i) 0)
(and (equal (fix x) 0)
(not (equal (ifix i) 0)))))
(defthm expt-0
(and (equal (expt x 0)
1)
(equal (expt 0 i)
(if (zip i)
1
0))))
(defthm expt-1
(and (equal (expt x 1)
(fix x))
(equal (expt 1 i)
1)))
(local
(defthm expt-minus
(equal (expt r (- i))
(/ (expt r i)))))
(local
(defthm expt-minus-redux
(equal (expt r (* -1 i))
(/ (expt r i)))
:hints (("Goal" :use ((:theorem (equal (* -1 i)
(- i))))))))
(defthm exponents-add-1
(implies (and (fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(if (equal (+ i j) 0)
1
(* (expt r i)
(expt r j))))))
(defthm exponents-add-for-nonpos-exponents
(implies (and (<= i 0)
(<= j 0)
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm exponents-add-for-nonneg-exponents
(implies (and (<= 0 i)
(<= 0 j)
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm exponents-add-2
(implies (and (not (equal 0 r))
(fc (acl2-numberp r))
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm distributivity-of-expt-over-*
(equal (expt (* a b) i)
(* (expt a i)
(expt b i))))
(defthm functional-commutativity-of-expt-/-base
(equal (expt (/ r) i)
(/ (expt r i))))
(defthm Exponents-multiply
(implies (and (fc (integerp i))
(fc (integerp j)))
(equal (expt (expt r i) j)
(expt r (* i j))))
:hints (("Subgoal *1/4'" :in-theory (enable prefer-*-to-/))))
(defthm expt-is-increasing-for-base>1
(implies (and (< 1 r)
(< i j)
(fc (rationalp r))
(fc (integerp i))
(fc (integerp j)))
(< (expt r i)
(expt r j)))
:rule-classes (:rewrite :linear))
(defthm Expt-is-decreasing-for-pos-base<1
(implies (and (< 0 r)
(< r 1)
(< i j)
(fc (rationalp r))
(fc (integerp i))
(fc (integerp j)))
(< (expt r j)
(expt r i)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :use
((:instance
expt-is-increasing-for-base>1
(r (/ r))))
:in-theory (enable prefer-*-to-/))))
;; Should the two following rules be linear rules?
(defthm Expt-is-weakly-increasing-for-base>1
(implies (and (<= 1 r)
(<= i j)
(fc (rationalp r))
(fc (integerp i))
(fc (integerp j)))
(<= (expt r i)
(expt r j)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :use expt-is-increasing-for-base>1
:in-theory (disable expt-is-increasing-for-base>1))))
(defthm Expt-is-weakly-decreasing-for-pos-base<1
(implies (and (< 0 r)
(<= r 1)
(<= i j)
(fc (rationalp r))
(fc (integerp i))
(fc (integerp j)))
(<= (expt r j)
(expt r i)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :use expt-is-decreasing-for-pos-base<1
:in-theory (disable expt-is-decreasing-for-pos-base<1))))
;; Should these be rewrite rules also? Probably not.
(local
(defthm stupid-hack
(implies (and (rationalp x)
(not (equal x 0)))
(equal (* x (/ x) (expt a b))
(expt a b)))))
(local
(in-arithmetic-theory '((:REWRITE COMMUTATIVITY-OF-*)
(:REWRITE COMMUTATIVITY-2-OF-*)
(:REWRITE STUPID-HACK)
(:REWRITE EXPT-1)
(:REWRITE EXPT-0))))
(defthm expt->-1-1
(implies (and (< 1 r)
(< 0 i)
(fc (rationalp r))
(fc (integerp i)))
(< 1 (expt r i)))
:rule-classes :linear)
(defthm expt->-1-2
(implies (and (< 0 r)
(< r 1)
(< i 0)
(fc (rationalp r))
(fc (integerp i)))
(< 1 (expt r i)))
:rule-classes :linear)
(defthm expt-<-1-1
(implies (and (< 0 r)
(< r 1)
(< 0 i)
(fc (rationalp r))
(fc (integerp i)))
(< (expt r i) 1))
:rule-classes :linear)
(defthm expt-<-1-2
(implies (and (< 1 r)
(< i 0)
(fc (rationalp r))
(fc (integerp i)))
(< (expt r i) 1))
:hints (("Goal" :in-theory (enable prefer-*-to-/)))
:rule-classes :linear)
|
