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; See the top-level arithmetic-3 LICENSE file for authorship,
; copyright, and license information.
;;
;; expt-helper.lisp
;;
(in-package "ACL2")
(local (include-book "basic-arithmetic"))
(local (include-book "inequalities"))
(local (include-book "prefer-times"))
(defmacro fc (x)
x)
(defthm expt-crock
(equal (expt x 1)
(fix x))
:hints (("Goal" :expand (expt x 1))))
(encapsulate
()
(local
(defun
Math-induction-start-at-k (k n)
(declare (xargs :measure (if (and (integerp k)
(integerp n)
(< k n))
(- n k)
0)))
(if (and (integerp k)
(integerp n)
(< k n))
(Math-induction-start-at-k k (+ -1 n))
t)))
(local
(defthm justify-induction
(implies (and (integerp i)
(< 1 r)
(rationalp r))
(< (expt r i) (expt r (+ 1 i))))
:hints (("Goal" :in-theory (enable prefer-*-to-/)))))
(defthm expt-is-increasing-for-base>1
(implies (and (< 1 r)
(< i j)
(rationalp r)
(integerp i)
(integerp j))
(< (expt r i)
(expt r j)))
:rule-classes (:rewrite :linear)
:hints (("Goal"
:do-not '(generalize)
:induct (Math-induction-start-at-k i j)
:in-theory (enable prefer-*-to-/))
("Subgoal *1/1" :use (:instance justify-induction
(i (+ -1 J))))))
)
;; The following four theorems need to be proved in the order
;; given. However, I want them to be in a different order so
;; I prove them here, and then include them in expt.lisp in the
;; right order.
(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))))
:hints (("Goal" :expand (expt r (+ i j)))))
(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)))))
:hints (("Goal" :do-not '(generalize))))
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