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|
;; Copyright (C) 2017, Regents of the University of Texas
;; Written by Cuong Chau (derived from the FM9001 work of Brock and Hunt)
;; License: A 3-clause BSD license. See the LICENSE file distributed with
;; ACL2.
;; The ACL2 source code for the FM9001 work is available at
;; https://github.com/acl2/acl2/tree/master/books/projects/fm9001.
;; Cuong Chau <ckcuong@cs.utexas.edu>
;; May 2019
(in-package "ADE")
(include-book "index")
(include-book "std/lists/repeat" :dir :system)
;; ======================================================================
(defun assoc-eq-value (x alist)
(declare (xargs :guard (and (symbolp x)
(alistp alist))))
(cdr (assoc-eq x alist)))
(defthm assoc-eq-value-cons-1
(implies (equal x (car pair))
(equal (assoc-eq-value x (cons pair alist))
(cdr pair))))
(defthm assoc-eq-value-cons-2
(implies (not (equal x (car pair)))
(equal (assoc-eq-value x (cons pair alist))
(assoc-eq-value x alist))))
;; Necessary for some proofs with lots of signals, otherwise stack overflow
;; occurs from deeply nested rewriting! This lemma also speeds up proofs
;; significantly if the modules have lots of single output gates.
(defthm rewrite-assoc-eq-value-4x
(equal (assoc-eq-value a
(cons (cons b c)
(cons (cons d e)
(cons (cons g h)
(cons (cons i j)
k)))))
(if (equal a b)
c
(if (equal a d)
e
(if (equal a g)
h
(if (equal a i)
j
(assoc-eq-value a k)))))))
(defthm assoc-eq-value-append-pairlis$
(equal (assoc-eq-value a (append (pairlis$ b c) alist))
(if (member a b)
(assoc-eq-value a (pairlis$ b c))
(assoc-eq-value a alist))))
;; Not sure we always want this one firing
(defthmd assoc-eq-value-pairlis$-member
(implies (member name names)
(equal (assoc-eq-value name (pairlis$ names values))
(car (nthcdr (position1 name names) values))))
:hints (("Goal" :in-theory (enable position1))))
(defthm assoc-eq-value-pairlis$-not-member
(implies (not (member name names))
(not (assoc-eq-value name (pairlis$ names values)))))
(defthm assoc-eq-value-pairlis$-append-member
(implies (member a x)
(equal (assoc-eq-value a (pairlis$ (append x y) l))
(assoc-eq-value a (pairlis$ x l)))))
(local
(defun assoc-eq-value-pairlis$-append-not-member-induct (x l)
(if (atom x)
l
(assoc-eq-value-pairlis$-append-not-member-induct (cdr x) (cdr l)))))
(defthm assoc-eq-value-pairlis$-append-not-member
(implies (not (member a x))
(equal (assoc-eq-value a (pairlis$ (append x y) l))
(assoc-eq-value a (pairlis$ y (nthcdr (len x) l)))))
:hints (("Goal"
:induct (assoc-eq-value-pairlis$-append-not-member-induct x l))))
(defthm assoc-eq-value-update-alist-1
(implies (and (equal occ-name1 occ-name2)
(consp (assoc-eq occ-name2 st-alist)))
(equal (assoc-eq-value occ-name1
(update-alist occ-name2 st st-alist))
st))
:hints (("Goal" :in-theory (enable update-alist))))
(defthm assoc-eq-value-update-alist-2
(implies (not (equal occ-name1 occ-name2))
(equal (assoc-eq-value occ-name1
(update-alist occ-name2 st st-alist))
(assoc-eq-value occ-name1
st-alist)))
:hints (("Goal" :in-theory (enable update-alist))))
(defthmd assoc-eq-value-nth-pairlis$
(implies (and (<= 0 n)
(< n (len keys))
(no-duplicatesp keys)
(<= (len keys) (len vals)))
(equal (assoc-eq-value (nth n keys)
(pairlis$ keys vals))
(nth n vals))))
(defthm assoc-eq-value-of-si-pairlis$-sis
(implies (and (natp i)
(natp m)
(posp n)
(<= m i)
(< i (+ m n))
(<= n (len vals)))
(equal (assoc-eq-value (si s i)
(pairlis$ (sis s m n) vals))
(nth (- i m) vals)))
:hints (("Goal"
:use (si-is-nth-of-sis
(:instance assoc-eq-value-nth-pairlis$
(n (- i m))
(keys (sis s m n)))))))
(in-theory (disable assoc-eq-value))
(defun assoc-eq-values (syms alist)
(declare (xargs :guard (and (symbol-listp syms)
(alistp alist))))
(if (atom syms)
nil
(cons (assoc-eq-value (car syms) alist)
(assoc-eq-values (cdr syms) alist))))
(defthm len-assoc-eq-values
(equal (len (assoc-eq-values syms alist))
(len syms)))
(defthm pairlis$-assoc-eq-values
(implies (and (no-duplicatesp syms)
(alistp alist)
(equal syms (strip-cars alist)))
(equal (pairlis$ syms (assoc-eq-values syms alist))
alist)))
(defthm assoc-eq-values-cons-not-member
(implies (not (member (car pair) syms))
(equal (assoc-eq-values syms (cons pair alist))
(assoc-eq-values syms alist))))
(defthm assoc-eq-values-append-pairlis$
(implies (and (no-duplicatesp x)
(true-listp a)
(equal (len x) (len a)))
(equal (assoc-eq-values x (append (pairlis$ x a) alist))
a)))
(defthm assoc-eq-values-append-when-disjoint
(implies (or (disjoint syms (strip-cars x))
(disjoint (strip-cars x) syms))
(equal (assoc-eq-values syms (append x y))
(assoc-eq-values syms y))))
(defthm assoc-eq-values-atom
(implies (atom args)
(not (assoc-eq-values args alist))))
(defthm consp-assoc-eq-values
(equal (consp (assoc-eq-values args alist))
(consp args)))
(defthm assoc-eq-values-append
(equal (assoc-eq-values (append a b) alist)
(append (assoc-eq-values a alist)
(assoc-eq-values b alist))))
(defthm assoc-eq-values-rev
(equal (assoc-eq-values (rev syms) alist)
(rev (assoc-eq-values syms alist))))
(defthm assoc-eq-values-append-pairlis$-when-subset
(implies (subsetp args1 args2)
(equal (assoc-eq-values args1 (append (pairlis$ args2 answer)
x))
(assoc-eq-values args1 (pairlis$ args2 answer)))))
(defthm assoc-eq-values-append-pairlis$-when-disjoint
(implies (or (disjoint args1 args2)
(disjoint args2 args1))
(equal (assoc-eq-values args1 (append (pairlis$ args2 a)
b))
(assoc-eq-values args1 b))))
(defthm assoc-eq-values-pairlis$-append-when-subset
(implies (subsetp x y)
(equal (assoc-eq-values x (pairlis$ (append y z) l))
(assoc-eq-values x (pairlis$ y (take (len y) l)))))
:hints (("Goal" :in-theory (enable pairlis$-of-take))))
(defthm assoc-eq-values-pairlis$-append-when-disjoint
(implies
(or (disjoint x y)
(disjoint y x))
(equal (assoc-eq-values x (pairlis$ (append y z) l))
(assoc-eq-values x (pairlis$ z (take (len z)
(nthcdr (len y) l))))))
:hints (("Goal" :in-theory (enable disjoint pairlis$-of-take))))
(defthm assoc-eq-values-take-1
(implies (and (no-duplicatesp l)
(<= n (len l))
(true-listp x)
(equal (len l) (len x)))
(equal (assoc-eq-values (take n l)
(pairlis$ l x))
(take n x))))
(defthm assoc-eq-values-take-2
(implies (and (no-duplicatesp args)
(<= n (len args)))
(equal (assoc-eq-values (take n args)
(pairlis$ args answer))
(assoc-eq-values (take n args)
(pairlis$ (take n args)
(take n answer))))))
(defthm assoc-eq-values-nthcdr-1
(implies (and (no-duplicatesp l)
(true-listp x)
(equal (len l) (len x)))
(equal (assoc-eq-values (nthcdr n l)
(pairlis$ l x))
(nthcdr n x))))
(defthm assoc-eq-values-nthcdr-2
(implies (no-duplicatesp args)
(equal (assoc-eq-values (nthcdr n args)
(pairlis$ args answer))
(assoc-eq-values (nthcdr n args)
(pairlis$ (nthcdr n args)
(nthcdr n answer))))))
(defthm not-member-take-nthcdr-lemma
(implies (and (not (member x l))
(<= 0 n2)
(<= (+ n1 n2) (len l)))
(not (member x (take n1 (nthcdr n2 l))))))
(defthm assoc-eq-values-cons
(equal (assoc-eq-values (cons a b) alist)
(cons (assoc-eq-value a alist)
(assoc-eq-values b alist))))
(defthm singleton-assoc-eq-values
(implies (equal (len a) 1)
(equal (assoc-eq-values a alist)
(list (assoc-eq-value (car a) alist)))))
(defthm assoc-eq-values-args-pairlis$-args
(implies (and (no-duplicatesp args)
(equal (len args) (len list))
(true-listp list))
(equal (assoc-eq-values args (pairlis$ args list))
list)))
;; ASSOC-EQ-VALUES-SPLITTING-CROCK is used to force a rewrite which goes
;; "against the grain" of the normal rewriting direction of
;; ASSOC-EQ-VALUES-APPEND.
(defthmd assoc-eq-values-splitting-crock
(implies (<= n (len l))
(equal (assoc-eq-values l alist)
(append (assoc-eq-values (take n l) alist)
(assoc-eq-values (nthcdr n l) alist)))))
(defthm assoc-eq-values-make-list
(equal (assoc-eq-values (make-list n :initial-element name) alist)
(make-list n :initial-element (assoc-eq-value name alist)))
:hints (("Goal" :in-theory (enable repeat))))
(defthm assoc-eq-values-take-nthcdr
(implies (and (no-duplicatesp l)
(<= 0 n2)
(<= (+ n1 n2) (len l))
(true-listp x)
(<= (len l) (len x)))
(equal (assoc-eq-values (take n1 (nthcdr n2 l))
(pairlis$ l x))
(take n1 (nthcdr n2 x))))
:hints (("Subgoal *1/1"
:in-theory (disable assoc-eq-values-take-1)
:use (:instance assoc-eq-values-take-1
(n n1)))))
(defthm assoc-eq-values-subseq-args-pairlis$-args
(implies (and (<= 0 m)
(no-duplicatesp args)
(equal (len args) (len list))
(true-listp list)
(<= n (len args)))
(equal (assoc-eq-values (subseq args m n) (pairlis$ args list))
(subseq list m n))))
(defthm assoc-eq-values-update-alist-not-member
(implies (not (member name names))
(equal (assoc-eq-values names
(update-alist name value alist))
(assoc-eq-values names alist))))
(defthm assoc-eq-values-cdr
(equal (assoc-eq-values (cdr keys) alist)
(cdr (assoc-eq-values keys alist))))
(defthm assoc-eq-values-of-repeat-cons-1
(implies (equal sym (car x))
(equal (assoc-eq-values (repeat n sym)
(cons x l))
(repeat n (cdr x))))
:hints (("Goal" :in-theory (enable repeat))))
(defthm assoc-eq-values-of-repeat-cons-2
(implies (not (equal sym (car x)))
(equal (assoc-eq-values (repeat n sym)
(cons x l))
(assoc-eq-values (repeat n sym) l)))
:hints (("Goal" :in-theory (enable repeat))))
(defthm assoc-eq-values-of-repeat-append
(implies (not (member sym (strip-cars x)))
(equal (assoc-eq-values (repeat n sym)
(append x y))
(assoc-eq-values (repeat n sym) y)))
:hints (("Goal" :in-theory (enable repeat))))
(in-theory (disable assoc-eq-values))
(defthm assoc-eq-values-of-sis-pairlis$-sis
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(<= (+ i j) (+ m n))
(true-listp vals)
(<= n (len vals)))
(equal (assoc-eq-values (sis s i j)
(pairlis$ (sis s m n) vals))
(take j (nthcdr (- i m) vals))))
:hints (("Goal" :use sis-of-subset)))
|
