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|
; Alist-defuns.lisp - definitions of functions in the theory of alists.
; Copyright (C) 1997 Computational Logic, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by: Bill Bevier (bevier@cli.com)
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
(deflabel alist-defuns-section)
; * Structure of the Theory
;
; The functions which occur in the alist theory are selected from
; the ACL2 base theory (as defined in axioms.lisp) plus other functions
; which descend from earlier alist libraries.
;
; alist-defuns.lisp contains the definitions of functions which are not
; in the ACL2 base theory.
;
; alist-defthms.lisp contains theorems about the functions in the
; theory. Segregating the theory into two files allows one to load
; only the definitions when one is only interested in running
; simulations.
;
; * General Strategy for Theory Construction
;
; The goal of this theory is to provide useful alist-processing functions
; and useful theorems about those functions.
;
; * Enabled and Disabled functions
;
; We plan to leave all recursive functions enabled. The theorem prover
; is good at deciding when to open recursive functions. An expert user
; can choose to disable them explicitly.
;
; Non-recursive functions are globally disabled by the book alist-defthms.
;
; * Equality
;
; ACL2 (and Common Lisp) support three notions of equality: EQL, EQ and EQUAL.
; One uses EQL or EQ, rather than EQUAL, for efficiency. Many functions
; have three versions, each based on a different equality. ASSOC uses EQL,
; however ASSOC-EQ and ASSOC-EQUAL are also defined.
;
; We have followed this naming convention. When a function relies on equality.
; the default notion is EQL; -EQ and -EQUAL versions of the function are
; also provided.
;
; In list-defthms, the EQL and EQ versions of all functions are re-written to the
; EQUAL version. All interesting rewrite rules about the list functions are
; expressed in terms of the EQUAL versions of list functions.
;
; As a result, one can execute using the EQL or EQ versions, but one will reason
; using the EQUAL version.
; ------------------------------------------------------------
; Definitions
; ------------------------------------------------------------
; Acl2 In ACL2 type
; ---- ------- ----
;
; acons yes alist
; append yes alist
; alist-compose (eql) no alist
; alist-compose-eq no alist
; alist-compose-equal no alist
; bind (eql) no alist
; bind-eq no alist
; bind-equal no alist
; bind-all (eql) no alist
; bind-all-eq no alist
; bind-all-equal no alist
; bind-pairs (eql) no alist
; bind-pairs-eq no alist
; bind-pairs-equal no alist
; domain-restrict (eql) no alist
; domain-restrict-eq no alist
; domain-restrict-equal no alist
; pairlis$ yes alist
; rembind (eql) no alist
; rembind-eq no alist
; rembind-equal no alist
; rembind-all (eql) no alist
; rembind-all-eq no alist
; rembind-all-equal no alist
;
; assoc (eql) yes cons or NIL
; assoc-eq yes cons or NIL
; assoc-equal yes cons or NIL
; binding no value or NIL
; binding-eq no value or NIL
; binding-equal no value or NIL
;
; alistp yes boolean
; bound? no boolean
; bound?-eq no boolean
; bound?-equal no boolean
; all-bound? (eql) no boolean
; all-bound?-eq no boolean
; all-bound?-equal no boolean
;
; all-bindings (eql) no list
; all-bindings-eq no list
; all-bindings-equal no list
; collect-bound (eql) no list
; collect-bound-eq no list
; collect-bound-equal no list
; domain no list
; range yes list
(defun bind (x y a)
"The alist derived from A by binding X to Y."
(declare (xargs :guard (and (alistp a) (eqlablep x))))
(cond ((endp a) (list (cons x y)))
((eql x (car (car a))) (cons (cons x y) (cdr a)))
(t (cons (car a) (bind x y (cdr a))))))
(defun bind-eq (x y a)
"The alist derived from A by binding X to Y."
(declare (xargs :guard (and (alistp a) (symbolp x))))
(cond ((endp a) (list (cons x y)))
((eq x (car (car a))) (cons (cons x y) (cdr a)))
(t (cons (car a) (bind-eq x y (cdr a))))))
(defun bind-equal (x y a)
"The alist derived from A by binding X to Y."
(declare (xargs :guard (alistp a)))
(cond ((endp a) (list (cons x y)))
((equal x (car (car a))) (cons (cons x y) (cdr a)))
(t (cons (car a) (bind-equal x y (cdr a))))))
(defun bind-all (keys vals a)
"The alist whose domain is A's range, and whose range is A's domain."
(declare (xargs :guard (and (true-listp vals)
(eqlable-listp keys)
(eqlable-alistp a))))
(cond ((endp keys) a)
((endp vals) a)
(t (bind (car keys) (car vals)
(bind-all (cdr keys) (cdr vals) a)))))
(defun bind-all-eq (keys vals a)
"The alist whose domain is A's range, and whose range is A's domain."
(declare (xargs :guard (and (true-listp vals)
(symbol-listp keys)
(symbol-alistp a))))
(cond ((endp keys) a)
((endp vals) a)
(t (bind-eq (car keys) (car vals)
(bind-all-eq (cdr keys) (cdr vals) a)))))
(defun bind-all-equal (keys vals a)
"The alist whose domain is A's range, and whose range is A's domain."
(declare (xargs :guard (and (true-listp keys)
(true-listp vals)
(alistp a))))
(cond ((endp keys) a)
((endp vals) a)
(t (bind-equal (car keys) (car vals)
(bind-all-equal (cdr keys) (cdr vals) a)))))
(defun binding (x a)
"The value bound to X in alist A."
(declare (xargs :guard (and (alistp a)
(or (eqlablep x)
(eqlable-alistp a)))))
(cdr (assoc x a)))
(defun binding-eq (x a)
"The value bound to X in alist A."
(declare (xargs :guard (and (alistp a)
(or (symbolp x)
(symbol-alistp a)))))
(cdr (assoc-eq x a)))
(defun binding-equal (x a)
"The value bound to X in alist A."
(declare (xargs :guard (alistp a)))
(cdr (assoc-equal x a)))
(defun bound? (x a)
"The value bound to X in alist A."
(declare (xargs :guard (and (alistp a)
(or (eqlablep x) (eqlable-alistp a)))))
(consp (assoc x a)))
(defun bound?-eq (x a)
"The value bound to X in alist A."
(declare (xargs :guard (and (alistp a)
(or (symbolp x) (symbol-alistp a)))))
(consp (assoc-eq x a)))
(defun bound?-equal (x a)
"The value bound to X in alist A."
(declare (xargs :guard (alistp a)))
(consp (assoc-equal x a)))
(defun all-bound? (l a)
"Are all elements of list L bound in alist A?"
(declare (xargs :guard (and (true-listp l)
(alistp a)
(or (eqlable-listp l)
(eqlable-alistp a)))))
(cond ((endp l) t)
(t (and (bound? (car l) a)
(all-bound? (cdr l) a)))))
(defun all-bound?-eq (l a)
"Are all elements of list L bound in alist A?"
(declare (xargs :guard (and (true-listp l)
(alistp a)
(or (symbol-listp l)
(symbol-alistp a)))))
(cond ((endp l) t)
(t (and (bound?-eq (car l) a)
(all-bound?-eq (cdr l) a)))))
(defun all-bound?-equal (l a)
"Are all elements of list L bound in alist A?"
(declare (xargs :guard (and (true-listp l)
(alistp a))))
(cond ((endp l) t)
(t (and (bound?-equal (car l) a)
(all-bound?-equal (cdr l) a)))))
(defun all-bindings (l a)
"The list of bindings of elements of list L in alist A."
(declare (xargs :guard (and (true-listp l)
(alistp a)
(or (eqlable-listp l)
(eqlable-alistp a)))))
(cond ((endp l) nil)
((bound? (car l) a)
(cons (binding (car l) a)
(all-bindings (cdr l) a)))
(t (all-bindings (cdr l) a))))
(defun all-bindings-eq (l a)
"The list of bindings of elements of list L in alist A."
(declare (xargs :guard (and (true-listp l)
(alistp a)
(or (symbol-listp l)
(symbol-alistp a)))))
(cond ((endp l) nil)
((bound?-eq (car l) a)
(cons (binding-eq (car l) a)
(all-bindings-eq (cdr l) a)))
(t (all-bindings-eq (cdr l) a))))
(defun all-bindings-equal (l a)
"The list of bindings of elements of list L in alist A."
(declare (xargs :guard (and (true-listp l)
(alistp a))))
(cond ((endp l) nil)
((bound?-equal (car l) a)
(cons (binding-equal (car l) a)
(all-bindings-equal (cdr l) a)))
(t (all-bindings-equal (cdr l) a))))
(defun domain (a)
"The list of CARs of an alist."
(declare (xargs :guard (alistp a)))
(strip-cars a))
(defun domain-restrict (l a)
"An alist containing only those pairs in A whose CAR is in L."
(declare (xargs :guard (and (eqlable-listp l)
(eqlable-alistp a))))
(cond ((endp a) nil)
((member (car (car a)) l)
(cons (car a) (domain-restrict l (cdr a))))
(t (domain-restrict l (cdr a)))))
(defun domain-restrict-eq (l a)
"An alist containing only those pairs in A whose CAR is in L."
(declare (xargs :guard (and (symbol-listp l)
(symbol-alistp a))))
(cond ((endp a) nil)
((member-eq (car (car a)) l)
(cons (car a) (domain-restrict-eq l (cdr a))))
(t (domain-restrict-eq l (cdr a)))))
(defun domain-restrict-equal (l a)
"An alist containing only those pairs in A whose CAR is in L."
(declare (xargs :guard (and (true-listp l)
(alistp a))))
(cond ((endp a) nil)
((member-equal (car (car a)) l)
(cons (car a) (domain-restrict-equal l (cdr a))))
(t (domain-restrict-equal l (cdr a)))))
(defun range (a)
"The list of CDRs of an alist."
(declare (xargs :guard (alistp a)))
(strip-cdrs a))
(defun rembind (x a)
"The alist derived from A by removing any binding of X."
(declare (xargs :guard (and (alistp a)
(or (eqlablep x)
(eqlable-alistp a)))))
(cond ((endp a) nil)
((eql x (car (car a))) (rembind x (cdr a)))
(t (cons (car a) (rembind x (cdr a))))))
(defun rembind-eq (x a)
"The alist derived from A by removing any binding of X."
(declare (xargs :guard (and (alistp a)
(or (symbolp x)
(symbol-alistp a)))))
(cond ((endp a) nil)
((eq x (car (car a))) (rembind-eq x (cdr a)))
(t (cons (car a) (rembind-eq x (cdr a))))))
(defun rembind-equal (x a)
"The alist derived from A by removing any binding of X."
(declare (xargs :guard (alistp a)))
(cond ((endp a) nil)
((equal x (car (car a))) (rembind-equal x (cdr a)))
(t (cons (car a) (rembind-equal x (cdr a))))))
(defun rembind-all (l a)
(declare (xargs :Guard (and (eqlable-listp l)
(eqlable-alistp a))))
(cond ((endp l) a)
(t (rembind (car l) (rembind-all (cdr l) a)))))
(defun rembind-all-eq (l a)
(declare (xargs :Guard (and (symbol-listp l)
(symbol-alistp a))))
(cond ((endp l) a)
(t (rembind-eq (car l) (rembind-all-eq (cdr l) a)))))
(defun rembind-all-equal (l a)
(declare (xargs :Guard (and (true-listp l)
(alistp a))))
(cond ((endp l) a)
(t (rembind-equal (car l) (rembind-all-equal (cdr l) a)))))
(defun collect-bound (l a)
"Collect the sublist of L bound in A."
(declare (xargs :guard (and (eqlable-listp l)
(eqlable-alistp a))))
(cond ((endp l) nil)
((bound? (car l) a)
(cons (car l) (collect-bound (cdr l) a)))
(t (collect-bound (cdr l) a))))
(defun collect-bound-eq (l a)
"Collect the sublist of L bound in A."
(declare (xargs :guard (and (symbol-listp l)
(symbol-alistp a))))
(cond ((endp l) nil)
((bound?-eq (car l) a)
(cons (car l) (collect-bound-eq (cdr l) a)))
(t (collect-bound-eq (cdr l) a))))
(defun collect-bound-equal (l a)
"Collect the sublist of L bound in A."
(declare (xargs :guard (and (true-listp l)
(alistp a))))
(cond ((endp l) nil)
((bound?-equal (car l) a)
(cons (car l) (collect-bound-equal (cdr l) a)))
(t (collect-bound-equal (cdr l) a))))
(defun bind-pairs (a1 a2)
(declare (xargs :guard (and (eqlable-alistp a1)
(eqlable-alistp a2))))
(cond ((endp a1) a2)
(t (bind (caar a1) (cdar a1) (bind-pairs (cdr a1) a2)))))
(defun bind-pairs-eq (a1 a2)
(declare (xargs :guard (and (symbol-alistp a1)
(symbol-alistp a2))))
(cond ((endp a1) a2)
(t (bind-eq (caar a1) (cdar a1) (bind-pairs-eq (cdr a1) a2)))))
(defun bind-pairs-equal (a1 a2)
(declare (xargs :guard (and (alistp a1)
(alistp a2))))
(cond ((endp a1) a2)
(t (bind-equal (caar a1) (cdar a1) (bind-pairs-equal (cdr a1) a2)))))
(defun alist-compose-domain (dom a1 a2)
"X is bound to Z in (ALIST-COMPOSE-DOMAIN DOM A1 A2) if X occurs in
DOM, X is bound in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (eqlable-listp dom)
(eqlable-alistp a1)
(eqlable-alistp a2))
:verify-guards nil))
(cond ((endp dom) nil)
(t (let ((pair1 (assoc (car dom) a1)))
(if pair1
(let ((pair2 (assoc (cdr pair1) a2)))
(if pair2
(bind (car dom) (cdr pair2)
(alist-compose-domain (cdr dom) a1 a2))
(alist-compose-domain (cdr dom) a1 a2)))
(alist-compose-domain (cdr dom) a1 a2))))))
(local (defthm alistp-alist-compose-domain
(implies (alistp a1)
(alistp (alist-compose-domain dom a1 a2)))))
(local (defthm eqlable-alistp-alist-compose-domain
(implies (and (eqlable-listp dom)
(alistp a1))
(eqlable-alistp (alist-compose-domain dom a1 a2)))))
(local (defthm eqlable-alistp-bind
(implies (and (eqlablep x)
(eqlable-alistp a))
(eqlable-alistp (bind x y a)))))
(local
(defthm assoc-consp-or-nil
(implies (alistp a)
(or (consp (assoc x a))
(equal (assoc x a) nil)))
:rule-classes :type-prescription))
(verify-guards alist-compose-domain)
(defun alist-compose (a1 a2)
"X is bound to Z in (ALIST-COMPOSE A1 A2) if X is bound
in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (eqlable-alistp a1)
(eqlable-alistp a2))))
(alist-compose-domain (domain a1) a1 a2))
(defun alist-compose-domain-eq (dom a1 a2)
"X is bound to Z in (ALIST-COMPOSE-DOMAIN DOM A1 A2) if X occurs in
DOM, X is bound in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (symbol-listp dom)
(symbol-alistp a1)
(symbol-alistp a2))
:verify-guards nil))
(cond ((endp dom) nil)
(t (let ((pair1 (assoc-eq (car dom) a1)))
(if pair1
(let ((pair2 (assoc-eq (cdr pair1) a2)))
(if pair2
(bind-eq (car dom) (cdr pair2)
(alist-compose-domain-eq (cdr dom) a1 a2))
(alist-compose-domain-eq (cdr dom) a1 a2)))
(alist-compose-domain-eq (cdr dom) a1 a2))))))
(local (defthm alistp-alist-compose-domain-eq
(implies (alistp a1)
(alistp (alist-compose-domain-eq dom a1 a2)))))
(local (defthm symbol-alistp-alist-compose-domain-eq
(implies (and (symbol-listp dom)
(alistp a1))
(symbol-alistp (alist-compose-domain-eq dom a1 a2)))))
(local (defthm symbol-alistp-bind-equal
(implies (and (symbolp x)
(symbol-alistp a))
(symbol-alistp (bind-equal x y a)))))
(verify-guards alist-compose-domain-eq)
(defun alist-compose-eq (a1 a2)
"X is bound to Z in (ALIST-COMPOSE A1 A2) if X is bound
in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (symbol-alistp a1)
(symbol-alistp a2))))
(alist-compose-domain-eq (domain a1) a1 a2))
(defun alist-compose-domain-equal (dom a1 a2)
"X is bound to Z in (ALIST-COMPOSE-DOMAIN DOM A1 A2) if X occurs in
DOM, X is bound in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (true-listp dom)
(alistp a1)
(alistp a2))
:verify-guards nil))
(cond ((endp dom) nil)
(t (let ((pair1 (assoc-equal (car dom) a1)))
(if pair1
(let ((pair2 (assoc-equal (cdr pair1) a2)))
(if pair2
(bind-equal (car dom) (cdr pair2)
(alist-compose-domain-equal (cdr dom) a1 a2))
(alist-compose-domain-equal (cdr dom) a1 a2)))
(alist-compose-domain-equal (cdr dom) a1 a2))))))
(local (defthm alistp-alist-compose-domain-equal
(implies (alistp a1)
(alistp (alist-compose-domain-equal dom a1 a2)))))
(local (defthm alistp-bind-equal
(implies (alistp a)
(alistp (bind-equal x y a)))))
(local
(defthm assoc-equal-consp-or-nil
(implies (alistp a)
(or (consp (assoc-equal x a))
(equal (assoc-equal x a) nil)))
:rule-classes :type-prescription))
(verify-guards alist-compose-domain-equal)
(defun alist-compose-equal (a1 a2)
"X is bound to Z in (ALIST-COMPOSE A1 A2) if X is bound
in A1, and there exists a Y such that
(BINDING X A1) = Y, and (BINDING Y A2) = Z."
(declare (xargs :guard (and (alistp a1)
(alistp a2))))
(alist-compose-domain-equal (domain a1) a1 a2))
(deftheory alist-defuns
(set-difference-theories (current-theory :here)
(current-theory 'alist-defuns-section)))
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