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;;
;; Copyright (C) 2021, Collins Aerospace
;; All rights reserved.
;;
;; This software may be modified and distributed under the terms
;; of the 3-clause BSD license. See the LICENSE file for details.
;;
(in-package "ACL2")
(include-book "coi/util/defun" :dir :system)
(include-book "tools/pattern-match" :dir :system)
(include-book "coi/util/in-conclusion" :dir :system)
(include-book "std/lists/equiv" :dir :system)
(include-book "coi/util/deffix" :dir :system)
(include-book "std/util/defines" :dir :system)
(defun memberp (a x)
(declare (type (satisfies true-listp) x))
(if (member-equal a x) t nil))
(defthmd equal-boolean
(implies
(booleanp x)
(equal (equal x y)
(and (booleanp y)
(iff x y)))))
(defthm equal-cons-reduction
(equal (equal (cons a x) y)
(and (consp y)
(equal a (car y))
(equal x (cdr y)))))
(defthm equal-plus-len
(implies
(and
(syntaxp (and (quotep c) (quotep d)))
(integerp c)
(integerp d)
(equal e (- d c)))
(equal (equal (+ c (len x)) d)
(equal (len x) e))))
(defthm equal-len-into-consp
(implies
(and (syntaxp (quotep c))
(integerp c))
(equal (equal (len x) c)
(if (equal c 0) (not (consp x))
(and (consp x)
(equal (len (cdr x)) (1- c)))))))
(defun list-equiv-induction (x y)
(if (and (consp x) (consp y))
(list-equiv-induction (cdr x) (cdr y))
(list x y)))
(def::un tcons (list x)
(declare (xargs :signature ((t t) true-listp)))
(if (not (consp list)) (list x)
(cons (car list) (tcons (cdr list) x))))
(in-theory (disable (:type-prescription tcons)))
(defthmd open-list-equiv-on-consp
(implies
(consp x)
(iff (list-equiv x y)
(and (consp y)
(equal (car x) (car y))
(list-equiv (cdr x) (cdr y)))))
:hints (("Goal" :in-theory (enable true-list-fix list-equiv))))
(defthm list-equiv-induction-rule t
:rule-classes ((:induction :pattern (list-equiv x y)
:scheme (list-equiv-induction x y))))
(defcong list-equiv equal (tcons list x) 1
:hints (("Goal" :in-theory (enable open-list-equiv-on-consp))))
(defthm len-tcons
(equal (len (tcons list x))
(+ 1 (len list))))
(defun tcar (list)
(declare (type t list))
(if (not (consp list)) nil
(if (not (consp (cdr list))) (car list)
(tcar (cdr list)))))
(def::un tcdr (list)
(declare (xargs :signature ((t) true-listp)))
(if (not (consp list)) nil
(if (not (consp (cdr list))) nil
(cons (car list) (tcdr (cdr list))))))
(defthm len-tcdr
(equal (len (tcdr list))
(nfix (1- (len list)))))
(defthm tcar-tcons
(equal (tcar (tcons list x))
x))
(defthm tcdr-tcons
(equal (tcdr (tcons list x))
(list-fix list)))
(defthm tcons-tcdr-tcar
(implies
(consp x)
(equal (tcons (tcdr x) (tcar x))
(list-fix x))))
(def::un tnth (i list)
(declare (type t i list))
(if (not (consp list)) nil
(let ((i (nfix i)))
(if (zp i) (tcar list)
(tnth (1- i) (tcdr list))))))
(encapsulate
()
(local
(defthm nth-tcdr
(equal (nth i (tcdr list))
(if (< (nfix i) (len (tcdr list)))
(nth i list)
nil))))
(local
(defthm nth-to-tcar
(equal (nth (+ -1 (len list)) list)
(tcar list))))
(defthmd tnth-as-nth
(equal (tnth i list)
(if (< (nfix i) (len list))
(nth (+ (len list) -1 (- (nfix i))) list)
nil)))
)
;; ==================================================================
;; nz-list abstraction
;; ==================================================================
;; ----------------------------------------
;; nfix-equiv
;; ----------------------------------------
(defun nfix-equiv (x y)
(declare (type t x y))
(equal (nfix x)
(nfix y)))
(defequiv nfix-equiv)
(defthm nfix-equiv-nfix
(nfix-equiv (nfix x) x))
(defthm equal-nfix-0
(equal (equal (nfix i) 0)
(zp i)))
(defthm zp-implies-not-positive
(implies
(zp x)
(not (< 0 (nfix x))))
:rule-classes (:rewrite :linear))
(defthm zp-implies-equal-zero
(implies
(zp x)
(equal (nfix x) 0))
:rule-classes (:forward-chaining))
(defthm not-zp-implication
(implies
(not (zp x))
(and (natp x) (< 0 (nfix x)) (< 0 x)))
:rule-classes (:forward-chaining))
;; ----------------------------------------
;; non-zero (+/-)1
;; ----------------------------------------
(defun nzp (x)
(declare (type t x))
(or (equal x 1)
(equal x -1)))
(defthm nzp-implies
(implies
(nzp x)
(and (integerp x)
(or (equal x 1)
(equal x -1))))
:rule-classes (:forward-chaining))
(defthmd nzp-lt-0
(implies
(nzp x)
(iff (< x 0)
(equal x -1))))
(def::un nz-fix (x)
(declare (xargs :signature ((t) nzp)))
(if (not (nzp x)) 1
x))
(defthm nzp-negation
(implies
(nzp x)
(equal (nz-fix (- x))
(- x))))
(defthm nzp-nz-fix-identity
(implies
(nzp x)
(equal (nz-fix x) x)))
(defun nz-equiv (x y)
(declare (type t x y))
(equal (nz-fix x)
(nz-fix y)))
(defequiv nz-equiv)
(defcong nz-equiv equal (nz-fix x) 1)
(defthm nz-equiv-nz-fix
(nz-equiv (nz-fix x) x))
(defthmd equal-nz-fix
(equal (equal (nz-fix x) y)
(and (nzp y)
(nz-equiv x y))))
(defthm nzp--
(implies
(nzp x)
(nzp (- x))))
(defthm nz-equiv-negation
(implies
(and
(syntaxp (quotep c))
(nzp x)
(nzp c))
(equal (nz-equiv (- x) c)
(nz-equiv x (- c)))))
(defthm integer--
(implies
(integerp x)
(equal (- (- x)) x)))
(defthm nzp-*
(implies
(and (nzp x) (nzp y))
(nzp (* x y))))
(defthmd rewrite-negated-nz-equality
(implies
(and
(syntaxp (quotep a))
(in-conclusion-check (nz-equiv x a) :top :negated)
(nzp a))
(equal (nz-equiv x a)
(not (nz-equiv x (- a))))))
(in-theory (disable nzp nz-fix nz-equiv))
;; ----------------------------------------
(defun nz-listp (list)
(if (not (consp list)) (null list)
(and (nzp (car list))
(nz-listp (cdr list)))))
(def::signature tcons (nz-listp nzp) nz-listp)
(defthm nz-listp-cons
(equal (nz-listp (cons a x))
(and (nzp a)
(nz-listp x))))
(def::un fix-nz-list (list)
(declare (xargs :signature ((t) nz-listp)))
(if (not (consp list)) nil
(cons (nz-fix (car list))
(fix-nz-list (cdr list)))))
(defthm fix-nz-list-nz-listp
(implies
(nz-listp list)
(equal (fix-nz-list list) list)))
(defun nz-list-equiv (x y)
(equal (fix-nz-list x)
(fix-nz-list y)))
(defequiv nz-list-equiv)
(defcong nz-list-equiv equal (fix-nz-list x) 1)
(defthm nz-list-equiv-fix-nz-list
(nz-list-equiv (fix-nz-list x) x))
(defthmd open-nz-list-equiv
(and
(implies
(or (consp x) (consp y))
(equal (nz-list-equiv x y)
(and (iff (consp x) (consp y))
(nz-equiv (car x) (car y))
(nz-list-equiv (cdr x) (cdr y)))))
(implies
(or (not (consp x)) (not (consp y)))
(equal (nz-list-equiv x y)
(and (not (consp x)) (not (consp y))))))
:hints (("Goal" :expand (fix-nz-list y)
:in-theory (enable equal-nz-fix))))
(in-theory (disable nz-list-equiv))
(defthm weak-open-nz-list-equiv
(and
(equal (nz-list-equiv (cons a z) y)
(and (consp y)
(nz-equiv a (car y))
(nz-list-equiv z (cdr y))))
(equal (nz-list-equiv y (cons a z))
(and (consp y)
(nz-equiv a (car y))
(nz-list-equiv z (cdr y))))
(implies
(or (not (consp x)) (not (consp y)))
(equal (nz-list-equiv x y)
(and (not (consp x)) (not (consp y))))))
:hints (("Goal" :in-theory (enable open-nz-list-equiv
equal-boolean))))
(defthm nz-list-equiv-induction t
:rule-classes ((:induction :pattern (nz-list-equiv x y)
:scheme (list-equiv-induction x y))))
(defcong nz-list-equiv equal (consp x) 1
:hints (("Goal" :in-theory (enable equal-boolean open-nz-list-equiv))))
(defcong nz-list-equiv nz-equiv (car x) 1)
(defcong nz-list-equiv nz-list-equiv (cdr x) 1)
(defcong nz-list-equiv nz-list-equiv (cons a x) 2)
(defcong nz-equiv nz-list-equiv (cons a x) 1)
(defcong nz-list-equiv nz-list-equiv (append x y) 1)
(defcong nz-list-equiv nz-list-equiv (append x y) 2)
(defcong nz-list-equiv equal (len x) 1)
(defrefinement list-equiv nz-list-equiv
:hints (("Goal" :in-theory (enable open-list-equiv-on-consp))))
(def::und nz-bit ()
(declare (xargs :signature (() nzp)))
1)
(def::un nth-nz (x list)
(declare (xargs :signature ((t t) nzp)
:congruence ((nfix-equiv nz-list-equiv) equal)))
(if (not (consp list)) (nz-bit)
(if (zp (nfix x)) (nz-fix (car list))
(nth-nz (1- (nfix x)) (cdr list)))))
(defthm open-nth-nz
(implies
(not (zp i))
(equal (nth-nz i x)
(if (not (consp x)) (nz-bit)
(nth-nz (1- (nfix i)) (cdr x))))))
(defthm nth-nz-out-of-range
(implies
(<= (len x) (nfix i))
(equal (nth-nz i x) (nz-bit))))
(defthm open-nth-nz-cons
(equal (nth-nz i (cons a x))
(if (zp i) (nz-fix a)
(nth-nz (1- i) x))))
(defthm nth-nz-tcons
(equal (nth-nz i (tcons x a))
(let ((i (nfix i)))
(if (< i (len x))
(nth-nz i x)
(if (equal i (len x))
(nz-fix a)
(nz-bit)))))
:hints (("GOal" :induct (list (tcons x a)
(nth-nz i x)))))
;; ----------------------------------------
;;
;; nz-list-equiv reduction
;;
;; ----------------------------------------
(defun nth-nz-bad-guy (x y)
(declare (xargs :measure (max (len x) (len y))))
(if (or (not (consp x)) (not (consp y))) 0
(if (not (nz-equiv (car x) (car y))) 0
(1+ (nth-nz-bad-guy (cdr x) (cdr y))))))
(defthm natp-nth-nz-bad-guy
(natp (nth-nz-bad-guy x y)))
(defthm nth-nz-bad-guy-equiv
(implies
(nz-list-equiv x y)
(equal (nth-nz-bad-guy x y)
(min (len x) (len y)))))
(defthm nth-nz-len-x
(equal (nth-nz (len x) x)
(nz-bit)))
(defthm nz-equiv-nth-nz-bad-guy
(implies
(and
(not (nz-list-equiv x y))
(equal (len x) (len y)))
(not (equal (nth-nz (nth-nz-bad-guy x y) x)
(nth-nz (nth-nz-bad-guy x y) y))))
:hints (("Goal" :in-theory (enable equal-nz-fix)
:induct (nth-nz-bad-guy x y))))
(defthmd nz-list-equiv-reduction
(iff (nz-list-equiv x y)
(and (equal (len x) (len y))
(equal (nth-nz (nth-nz-bad-guy x y) x)
(nth-nz (nth-nz-bad-guy x y) y))))
:hints (("Goal" :do-not-induct t)))
;; ----------------------------------------
;; step nz-list
;; ----------------------------------------
(def::und step-nz (L v R)
(declare (xargs :signature ((t t t) nzp)))
(let ((L (nz-fix L))
(v (nz-fix v))
(R (nz-fix R)))
(if (< v 0)
(if (not (< L 0)) (- v) v)
(if (< R 0) (- v) v))))
(def::un step-nz-list-rec (L v x)
(declare (xargs :signature ((t t t) nzp nzp nz-listp)
;;:congruence ((nz-equiv nz-equiv nz-list-equiv) equal equal equal)
))
(let ((L (nz-fix L))
(v (nz-fix v)))
(if (not (consp x)) (mv L v nil)
(let ((R (nz-fix (car x))))
(let ((z (step-nz L v R)))
(mv-let (L v res) (step-nz-list-rec v R (cdr x))
(mv L v (cons z res))))))))
(defthm nz-equiv-1-equal-step-nz-list-rec
(implies
(nz-equiv L1 L2)
(equal (step-nz-list-rec L1 v x)
(step-nz-list-rec L2 v x)))
:rule-classes :congruence)
(defthm nz-equiv-2-equal-step-nz-list-rec
(implies
(nz-equiv v1 v2)
(equal (step-nz-list-rec L v1 x)
(step-nz-list-rec L v2 x)))
:rule-classes :congruence)
(defthm nz-equiv-3-equal-step-nz-list-rec
(implies
(nz-list-equiv x1 x2)
(equal (step-nz-list-rec L v x1)
(step-nz-list-rec L v x2)))
:rule-classes :congruence)
(defthm len-step-nz-list-rec
(equal (len (val 2 (step-nz-list-rec L v x)))
(len x)))
(defthm nth-nz-step-nz-list-rec
(equal (nth-nz i (val 2 (step-nz-list-rec L v x)))
(let ((i (nfix i)))
(if (and (<= 0 i) (< i (len x)))
(let ((L (nth-nz i (list* L v x)))
(M (nth-nz (+ i 1) (list* L v x)))
(R (nth-nz (+ i 2) (list* L v x))))
(step-nz L M R))
(nz-bit))))
:hints (("Goal" :induct (list (nth-nz i x)
(step-nz-list-rec L v x)))))
(def::un step-nz-list-suffix (L v list)
(declare (xargs :signature ((t t t) nz-listp)
:congruence ((nz-equiv nz-equiv nz-list-equiv) equal)))
(let ((L (nz-fix L))
(v (nz-fix v)))
(mv-let (L v res) (step-nz-list-rec L v list)
(tcons res (step-nz L v (- v))))))
(defthm len-step-nz-list-suffix
(equal (len (step-nz-list-suffix L v list))
(1+ (len list))))
(defthm val0-STEP-NZ-LIST-REC
(equal (VAL 0 (STEP-NZ-LIST-REC L V X))
(nth-nz (len x) (list* L V X))))
(defthm val1-STEP-NZ-LIST-REC
(equal (VAL 1 (STEP-NZ-LIST-REC L V X))
(nth-nz (len x) (list* V X))))
(defthm nth-nz-step-nz-list-suffix
(equal (nth-nz i (step-nz-list-suffix L v x))
(let ((i (nfix i)))
(let ((L (nth-nz i (list* L v x)))
(M (nth-nz (+ i 1) (list* L v x)))
(R (nth-nz (+ i 2) (list* L v x))))
(if (and (<= 0 i) (< i (len x)))
(step-nz L M R)
(if (equal i (len x))
(step-nz L M (- M))
(nz-bit))))))
:hints (("Goal" :do-not-induct t
:in-theory (disable open-nth-nz
nth-nz))
(and stable-under-simplificationp
'(:in-theory (current-theory :here)))))
(def::un step-nz-list (list)
(declare (xargs :signature ((t) nz-listp)
:congruence ((nz-list-equiv) equal)))
(if (not (consp list)) nil
(let ((v (nz-fix (car list))))
(if (not (consp (cdr list))) (list (step-nz (- v) v (- v)))
(step-nz-list-suffix (- v) v (cdr list))))))
(defthm len-step-nz-list
(equal (len (step-nz-list x))
(len x)))
(defthm nth-nz-step-nz-list
(equal (nth-nz i (step-nz-list list))
(let ((i (nfix i)))
(let ((L (nth-nz (- i 1) list))
(M (nth-nz (+ i 0) list))
(R (nth-nz (+ i 1) list)))
(if (equal (len list) 0)
(nz-bit)
(if (equal (len list) 1)
(if (equal i 0)
(step-nz (- M) M (- M))
(nz-bit))
(if (and (<= 0 i) (< i (1- (len list))))
(if (equal i 0)
(step-nz (- M) M R)
(step-nz L M R))
(if (equal i (1- (len list)))
(step-nz L M (- M))
(nz-bit))))))))
:hints (("Goal" :do-not-induct t)))
(in-theory (disable val0-STEP-NZ-LIST-REC
val1-STEP-NZ-LIST-REC))
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