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; See the top-level arithmetic-3 LICENSE file for authorship,
; copyright, and license information.
;;
;; basic-arithmetic.lisp
;;
(in-package "ACL2")
(local (include-book "basic-arithmetic-helper"))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Extra
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; We include this definition here. It will be used in
; inequalities and prefer-times.
(defun nonlinearp-default-hint-pass1 (stable-under-simplificationp hist pspv)
(cond (stable-under-simplificationp
(if (access rewrite-constant
(access prove-spec-var pspv :rewrite-constant)
:nonlinearp)
nil
'(:computed-hint-replacement t
:nonlinearp t)))
((access rewrite-constant
(access prove-spec-var pspv :rewrite-constant)
:nonlinearp)
(if (equal (caar hist) 'SETTLED-DOWN-CLAUSE)
nil
'(:computed-hint-replacement t
:nonlinearp nil)))
(t
nil)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Facts about + (and -)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm commutativity-2-of-+
(equal (+ x (+ y z))
(+ y (+ x z))))
(defthm functional-self-inversion-of-minus
(equal (- (- x))
(fix x)))
(defthm distributivity-of-minus-over-+
(equal (- (+ x y))
(+ (- x) (- y))))
(defthm minus-cancellation-on-right
(equal (+ x y (- x))
(fix y)))
(defthm minus-cancellation-on-left
(equal (+ x (- x) y)
(fix y)))
; Note that the cancellation rules below (and similarly for *) aren't
; complete, in the sense that the element to cancel could be on the
; left side of one expression and the right side of the other. But
; perhaps those situations rarely arise in practice. (?)
(defthm right-cancellation-for-+
(equal (equal (+ x z)
(+ y z))
(equal (fix x) (fix y))))
(defthm left-cancellation-for-+
(equal (equal (+ x y)
(+ x z))
(equal (fix y) (fix z))))
(defthm equal-minus-0
(equal (equal (- x) 0)
(equal (fix x) 0)))
;; This rule causes trouble occasionally. So I restrict its usage
;; to the simple case of two things being added by the syntaxp
;; hypothesis.
(defthm equal-+-x-y-0
(implies (syntaxp (not (and (consp y)
(equal (car y) 'binary-+))))
(equal (equal (+ x y) 0)
(and (equal (fix x) (- y))
(equal (- x) (fix y))))))
(defthm equal-+-x-y-x
(equal (equal (+ x y) x)
(and (acl2-numberp x)
(equal (fix y) 0))))
(defthm equal-+-x-y-y
(equal (equal (+ x y) y)
(and (acl2-numberp y)
(equal (fix x) 0))))
(defthm equal-minus-minus
(equal (equal (- a) (- b))
(equal (fix a) (fix b))))
(defthm fold-consts-in-+
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (+ x (+ y z))
(+ (+ x y) z))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Facts about * (and /)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Some of the following are actually proved in basics.lisp.
(defthm commutativity-2-of-*
(equal (* x (* y z))
(* y (* x z))))
(defthm functional-self-inversion-of-/
(equal (/ (/ x)) (fix x)))
(defthm distributivity-of-/-over-*
(equal (/ (* x y))
(* (/ x) (/ y))))
(defthm /-cancellation-on-right
(equal (* y x (/ y))
(if (not (equal (fix y) 0))
(fix x)
0)))
(defthm /-cancellation-on-left
(equal (* y (/ y) x)
(if (not (equal (fix y) 0))
(fix x)
0)))
(defthm right-cancellation-for-*
(equal (equal (* x z) (* y z))
(or (equal 0 (fix z))
(equal (fix x) (fix y)))))
(defthm left-cancellation-for-*
(equal (equal (* z x) (* z y))
(or (equal 0 (fix z))
(equal (fix x) (fix y)))))
(defthm equal-/-0
(equal (equal (/ x) 0)
(equal (fix x) 0)))
(defthm equal-*-x-y-0
(equal (equal (* x y) 0)
(or (equal (fix x) 0)
(equal (fix y) 0))))
(defthm equal-*-x-y-x
(equal (equal (* x y) x)
(and (acl2-numberp x)
(or (equal x 0)
(equal y 1))))
:hints (("Goal" :use
((:instance right-cancellation-for-*
(z x)
(x y)
(y 1))))))
(defthm equal-*-y-x-x
(equal (equal (* y x) x)
(and (acl2-numberp x)
(or (equal x 0)
(equal y 1)))))
(defthm equal-/-/
(equal (equal (/ a) (/ b))
(equal (fix a) (fix b))))
(defthm fold-consts-in-*
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (* x (* y z))
(* (* x y) z))))
;; Note that the inverse rule can also be usefull.
;; We cannot include both of them though since this
;; may cause looping. The inverse rule:
;;
;; (defthm Uniqueness-of-*-inverses
;; (equal (equal (* x y)
;; 1)
;; (and (not (equal (fix x) 0))
;; (equal y (/ x))))
;;
;; is included in mini-theories.
(defthm equal-/
(equal (equal (/ x) y)
(if (not (equal (fix x) 0))
(equal 1 (* x y))
(equal y 0))))
; The following hack helps in the application of equal-/ when
; forcing is turned off.
(defthm numerator-nonzero-forward
(implies (not (equal (numerator r) 0))
(and (not (equal r 0))
(acl2-numberp r)))
:rule-classes
((:forward-chaining :trigger-terms
((numerator r)))))
;; We could prove an analogous rule about non-numeric coefficients, but
;; this one has efficiency advantages: it doesn't match too often, it has
;; no hypothesis, and also we know that the 0 is the first argument so we
;; don't need two versions. Besides, we won't need this too often; it's
;; a type-reasoning fact. But it did seem useful in the proof of a meta
;; lemma about times cancellation, so we include it here.
(defthm times-zero
(equal (* 0 x) 0))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Facts about + (or -) and * (or /) together
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm functional-commutativity-of-minus-*-right
(implies (syntaxp (not (quotep y)))
(equal (* x (- y))
(- (* x y)))))
(defthm functional-commutativity-of-minus-*-left
(implies (syntaxp (not (quotep x)))
(equal (* (- x) y)
(- (* x y)))))
(defthm reciprocal-minus-a
(equal (/ (- x))
(- (/ x))))
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