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|
; S-Expressions for 4-Valued Logic
; Copyright (C) 2010-2012 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Sol Swords <sswords@centtech.com>
; sexpr-fixpoint-spec.lisp
;
(in-package "ACL2")
(include-book "sexpr-fixpoint")
(include-book "sexpr-advanced")
(include-book "misc/bash" :dir :system)
(local (in-theory (disable set::double-containment
set::nonempty-means-set)))
(set-default-hints '('(:do-not-induct t)))
(local (in-theory (disable append-of-nil)))
;; BOZO probably move all of this stuff into libraries ---------------------
(defthm keys-equiv-cdr-when-not-consp-car
(implies (not (consp (car a)))
(keys-equiv (cdr a) a))
:hints (("Goal" :in-theory (enable hons-assoc-equal))
(witness))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthmd keys-equiv-iff-set-equiv-alist-keys
;; BOZO move to fast-alists book?
(iff (keys-equiv env1 env2)
(set-equiv (alist-keys env1) (alist-keys env2)))
:hints(("goal" :in-theory (e/d (hons-assoc-equal-iff-member-alist-keys)
(set-equiv
alist-keys-member-hons-assoc-equal)))
(witness)))
;; (defthm member-equal-remove
;; (iff (member-equal j (remove k x))
;; (and (not (equal j k))
;; (member-equal j x)))
;; :hints (("goal" :induct (len x))))
(encapsulate
()
(local (defthmd lemma
(iff (set-equiv a (cons b c))
(and (member-equal b a)
(set-equiv (remove b a) (remove b c))))
:hints ((set-reasoning))))
(defthmd set-equiv-breakdown-cons
(equal (set-equiv a (cons b c))
(and (member-equal b a)
(set-equiv (remove b a) (remove b c))))
:hints(("Goal" :use ((:instance lemma))))))
(defthmd set-equiv-breakdown-cons2
(equal (set-equiv (cons b c) a)
(and (member-equal b a)
(set-equiv (remove b a) (remove b c))))
:hints(("Goal" :use ((:instance set-equiv-breakdown-cons)))))
(local
(defthm remove-when-not-member
(implies (not (member-equal k x))
(equal (remove k x)
(append x nil)))
:hints(("Goal" :induct t
:in-theory (enable remove)))))
;;; Matt K. mod, 12/2018: I'm renaming remove-assoc, formerly just below, as
;;; remove-assoc-hons (not to conflict with hons-remove-assoc in
;;; std/alists/hons-remove-assoc.lisp), to avoid conflict with remove-assoc now
;;; built into ACl2. I'm also replacing "remove-assoc" everywhere below by
;;; "remove-assoc-hons".
(defun remove-assoc-hons (key al)
(declare (xargs :guard t))
(if (atom al)
nil
(if (or (atom (car al))
(hons-equal (caar al) key))
(remove-assoc-hons key (cdr al))
(cons (car al) (remove-assoc-hons key (cdr al))))))
(encapsulate nil
(local (set-default-hints nil))
(defthm lookup-in-remove-assoc-hons
(equal (hons-assoc-equal k (remove-assoc-hons key al))
(and (not (equal key k))
(hons-assoc-equal k al)))))
(defthm alist-keys-remove-assoc-hons
(equal (alist-keys (remove-assoc-hons k a))
(remove k (alist-keys a)))
:hints (("goal" :induct t)))
(defthm keys-equiv-cdr-remove-assoc-hons-car
(implies (and (keys-equiv a b)
(no-duplicatesp-equal (alist-keys a))
(consp (car a)))
(equal (keys-equiv (cdr a)
(remove-assoc-hons (caar a) b))
t))
:hints (("goal" :in-theory (enable keys-equiv-iff-set-equiv-alist-keys
alist-keys)
:do-not-induct t)
(set-reasoning))
:otf-flg t)
(defthm alist-equiv-cons-append-remove
(implies (hons-assoc-equal k a)
(alist-equiv (cons (cons k (cdr (hons-assoc-equal k a)))
(append (remove-assoc-hons k a)
b))
(append a b)))
:hints(("Goal" :in-theory (enable alist-equiv-iff-agree-on-bad-guy))))
;; --------------- end of library stuff ----------------------------
(encapsulate ()
(local
(defthm hons-assoc-equal-caar
(implies (and (keys-equiv x y)
(consp (car x)))
(hons-assoc-equal (caar x) y))))
(defthm 4v-env-equiv-remove-assoc-hons
(implies (and (not (4v-env-equiv a (remove-assoc-hons k b)))
(not (hons-assoc-equal k a)))
(not (4v-env-equiv (cons (cons k v) a)
b)))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-fix))
(witness :ruleset 4v-env-equiv-witnessing)
(witness :ruleset 4v-env-equiv-hons-assoc-equal-ex))
:otf-flg t))
(defthm 4v-alist-<=-append
(implies (and (keys-equiv a b)
(4v-alist-<= a b)
(4v-alist-<= c d))
(4v-alist-<= (append a c)
(append b d)))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-fix))
(witness)
(witness :ruleset (4v-alist-<=-4v-hons-assoc-equal-example))))
(defthm 4v-sexpr-alist-<=-acons
(implies (not (hons-assoc-equal k al1))
(iff (4v-sexpr-alist-<= (cons (cons k v) al1) al2)
(and (4v-sexpr-<= v (cdr (hons-assoc-equal k al2)))
(4v-sexpr-alist-<= al1 al2))))
:hints (("Goal" :do-not-induct t)
(witness :ruleset (4v-sexpr-alist-<=-hons-assoc-equal-example
4v-sexpr-alist-<=-witnessing))))
(defthm 4v-sexpr-<=-restrict
(implies (and (4v-sexpr-alist-<= a b)
(keys-equiv a b))
(4v-sexpr-<= (4v-sexpr-restrict x a)
(4v-sexpr-restrict x b)))
:hints(("Goal" :in-theory (e/d (keys-equiv-iff-set-equiv-alist-keys)
(4v-alist-<= 4v-sexpr-restrict 4v-sexpr-eval
4v-<=)))
(witness)))
(defthm 4v-sexpr-<=-restrict-x
(implies (4v-sexpr-<= a b)
(4v-sexpr-<= (4v-sexpr-restrict a (list (cons k *4vx-sexpr*)))
b))
:hints (("goal" :in-theory (e/d (4v-<=-trans2) (4v-<=)))
(Witness) (witness)))
(defthm 4v-sexpr-restrict-4v-sexpr-restrict
(4v-sexpr-equiv (4v-sexpr-restrict (4v-sexpr-restrict x al1) al2)
(4v-sexpr-restrict x (append (4v-sexpr-restrict-alist al1 al2)
al2)))
:hints (("goal" :do-not-induct t)
(witness))
:otf-flg t)
(defquant 4v-sexpr-fixpointp (ups vals)
(forall k (implies (hons-assoc-equal k ups)
(4v-sexpr-equiv (4v-sexpr-restrict
(cdr (hons-assoc-equal k ups))
vals)
(cdr (hons-assoc-equal k vals))))))
(verify-guards 4v-sexpr-fixpointp)
(defexample 4v-sexpr-fixpointp-lookup-example
:pattern (4v-sexpr-restrict (cdr (hons-assoc-equal k updatefns)) env)
:templates (k)
:instance-rulename 4v-sexpr-fixpointp-instancing)
(encapsulate nil
(local (witness-disable set-consp-witnessing))
(defthm 4v-sexpr-fixpointp-cdr-updatefns
(implies (and (4v-sexpr-fixpointp updatefns vals)
(no-duplicatesp-equal (alist-keys updatefns)))
(4v-sexpr-fixpointp (cdr updatefns) vals))
:hints (("goal" :in-theory (enable hons-assoc-equal alist-keys no-duplicatesp-equal))
(witness))))
(defun 4v-sexpr-fixpoint-spec (ups)
(if (atom ups)
nil
(b* ((rest (4v-sexpr-fixpoint-spec (cdr ups)))
((when (atom (car ups))) rest)
(name (caar ups))
(upfn (cdar ups))
(composed-with-rest
(4v-sexpr-restrict upfn rest))
(fixpoint (4v-sexpr-restrict composed-with-rest
`((,name x))))
(rest-restrict (4v-sexpr-restrict-alist
rest `((,name . ,fixpoint)))))
(acons name fixpoint rest-restrict))))
(defthm sexpr-fixpoint-spec-hons-assoc-equal-iff
(iff (hons-assoc-equal x (4v-sexpr-fixpoint-spec ups))
(hons-assoc-equal x ups))
:hints (("goal" :induct t)))
(defthmd 4v-sexpr-equiv-fixpoint-eval
(let ((ev-bottom (4v-sexpr-eval x (cons (cons name 'x) env))))
(equal (4v-sexpr-eval x (cons (cons name ev-bottom) env))
ev-bottom))
:hints (("goal" :use ((:instance 4v-sexpr-eval-monotonicp
(env (cons (cons name 'x) env))
(env1 (cons (cons name (4v-sexpr-eval x (cons
(cons name 'x)
env)))
env)))))))
(defexample 4v-sexpr-fixpointp-hons-assoc-equal-ex
:pattern (hons-assoc-equal k ups)
:templates (k)
:instance-rulename 4v-sexpr-fixpointp-instancing)
(defcong 4v-sexpr-alist-equiv iff (4v-sexpr-fixpointp ups fixpoint) 2
:hints ((witness :ruleset (4v-sexpr-fixpointp-witnessing
4v-sexpr-fixpointp-hons-assoc-equal-ex))))
(defthm atom-fixpoint
(implies (not (consp ups))
(4v-sexpr-fixpointp ups x))
:hints ((witness :ruleset 4v-sexpr-fixpointp-witnessing)))
(defthm atom-car-fixpoint
(implies (not (consp (car ups)))
(iff (4v-sexpr-fixpointp (cdr ups) x)
(4v-sexpr-fixpointp ups x)))
:hints (("Goal" :in-theory (enable hons-assoc-equal))
(witness :ruleset (4v-sexpr-fixpointp-witnessing
4v-sexpr-fixpointp-hons-assoc-equal-ex))))
(local (defthm switch-append
(implies (not (hons-assoc-equal x al))
(alist-equiv (append al (list (cons x v)))
(cons (cons x v) al)))
:hints (("goal" :in-theory (enable
alist-equiv-iff-agree-on-bad-guy)))))
(local (defthm switch-append-cons
(implies (not (hons-assoc-equal x al))
(alist-equiv (append al (cons (cons x v) al2))
(cons (cons x v) (append al al2))))
:hints (("goal" :in-theory (enable alist-equiv-iff-agree-on-bad-guy)))))
(encapsulate
nil
(local (set-default-hints nil))
(make-event
(b* (((er res)
(simplify-with-prover
'(implies (not (hons-assoc-equal name env1))
(let* ((x (4v-sexpr-restrict x env1))
(ev-bottom (4v-sexpr-eval x (cons (cons name 'x) env))))
(equal (4v-sexpr-eval x (cons (cons name ev-bottom)
env))
ev-bottom)))
nil '4v-sexpr-equiv-fixpoint-eval-special state))
(res (prettyify-clause-lst res t (w state))))
(value
`(defthm 4v-sexpr-equiv-fixpoint-eval-special
,(car res)
:hints (("goal" :use ((:instance 4v-sexpr-equiv-fixpoint-eval
(x (4v-sexpr-restrict x env1)))))))))))
(local (defthm fixpoint-eval-lookup
(implies (and (4v-sexpr-fixpointp ups fixpoint)
(hons-assoc-equal k ups))
(4v-sexpr-equiv
(cdr (hons-assoc-equal k fixpoint))
(4v-sexpr-restrict (cdr (hons-assoc-equal k ups))
fixpoint)))
:hints ((witness :ruleset 4v-sexpr-fixpointp-hons-assoc-equal-ex))))
(defthm 4v-sexpr-fixpointp-4v-sexpr-fixpoint-spec
(implies (no-duplicatesp-equal (alist-keys ups))
(4v-sexpr-fixpointp ups (4v-sexpr-fixpoint-spec ups)))
:hints (("goal" :induct (4v-sexpr-fixpoint-spec ups)
:in-theory (e/d (no-duplicatesp-equal alist-keys
hons-assoc-equal)
(;sexpr-eval-4v-sexpr-restrict
;4v-sexpr-restrict-4v-sexpr-restrict
4v-sexpr-restrict-alist
4v-sexpr-eval 4v-sexpr-restrict
default-car default-cdr))
:do-not-induct t)
(witness :ruleset 4v-sexpr-fixpointp-witnessing)
(witness :ruleset 4v-sexpr-equiv-witnessing))
:otf-flg t)
(defquant 4v-sexpr-fixpoint-lower-boundp (ups lb)
(forall (fp env)
(implies (and (keys-equiv ups fp)
(4v-alist-<= ;; 4v-env-equiv
(4v-sexpr-eval-alist ups (make-fal fp env))
fp))
(4v-alist-<= (4v-sexpr-eval-alist lb env) fp))))
(verify-guards 4v-sexpr-fixpoint-lower-boundp)
(defun 4v-sexpr-fixpointp-strong (ups fixpoint)
(declare (xargs :guard t))
(and (4v-sexpr-fixpointp ups fixpoint)
(keys-equiv ups fixpoint)
(not (intersectp-equal (alist-keys ups)
(4v-sexpr-vars-list (alist-vals fixpoint))))))
;; This was my previous definition for a fixpoint lower bound. However, it's
;; not quite strong enough. The previous definition implies this one (as we'll
;; prove.)
(defquant 4v-sexpr-fixpoint-lower-boundp2 (ups vals)
(forall fixpoint (implies (4v-sexpr-fixpointp-strong ups fixpoint)
(4v-sexpr-alist-<= vals fixpoint))))
(defun 4v-sexpr-fixpointp-alt1 (ups fp)
(4v-sexpr-alist-equiv (4v-sexpr-restrict-alist ups fp)
fp))
(defthmd 4v-sexpr-alist-equiv-fixpoint
(implies (and (4v-sexpr-fixpointp ups fixpoint)
(keys-equiv ups fixpoint))
(4v-sexpr-alist-equiv (4v-sexpr-restrict-alist ups fixpoint)
fixpoint))
:hints ((witness :ruleset
(4v-sexpr-alist-equiv-witnessing
4v-sexpr-fixpointp-hons-assoc-equal-ex))))
(defthm 4v-sexpr-fixpointp-is-alt1
(implies (keys-equiv ups fixpoint)
(iff (4v-sexpr-fixpointp ups fixpoint)
(4v-sexpr-fixpointp-alt1 ups fixpoint)))
:hints(("Goal" :in-theory (enable 4v-sexpr-alist-equiv-fixpoint)
:do-not-induct t)
(witness :ruleset (4v-sexpr-fixpointp-witnessing
4v-sexpr-alist-equiv-example))))
(defexample 4v-sexpr-fixpoint-lower-boundp-eval-alist-ex
:pattern (4v-sexpr-eval-alist x (append fp env))
:templates (fp env)
:instance-rulename 4v-sexpr-fixpoint-lower-boundp-instancing)
(defthm 4v-sexpr-fixpoint-lower-boundp2-if-lower-boundp
(implies (4v-sexpr-fixpoint-lower-boundp ups lb)
(4v-sexpr-fixpoint-lower-boundp2 ups lb))
:hints (("goal" :do-not-induct t
:in-theory (e/d (4v-sexpr-alist-equiv-is-alt
4v-sexpr-alist-<=-is-alt)
(4v-sexpr-eval)))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-witnessing
4v-sexpr-fixpoint-lower-boundp2-witnessing))
(witness :ruleset (4v-sexpr-alist-<=-alt-witnessing
4v-sexpr-alist-<=-alt-eval-alist-ex))
(witness :ruleset 4v-sexpr-alist-equiv-alt-eval-alist-ex)
;; (witness :ruleset 4v-sexpr-<=-witnessing)
(and stable-under-simplificationp
'(:use ((:instance 4v-sexpr-fixpoint-lower-boundp-necc
(fp (4v-sexpr-eval-alist fixpoint0 env0))
(env env0)))))
)
:otf-flg t)
(defthm lower-boundp-nil-for-no-update-fns
(4v-sexpr-fixpoint-lower-boundp ups nil)
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-sexpr-eval))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-witnessing))))
(defcong 4v-sexpr-alist-equiv iff (4v-sexpr-fixpoint-lower-boundp ups lb) 1
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-sexpr-eval))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-witnessing))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-eval-alist-ex))
))
(defthm 4v-sexpr-alist-equiv-nil-any-atom
(implies (atom x)
(equal (4v-sexpr-alist-equiv x nil) t))
:hints (("goal" :do-not-induct t)
(witness :ruleset 4v-sexpr-alist-equiv-witnessing))
:rule-classes nil)
(defthm 4v-sexpr-fixpoint-lower-boundp-atom-ups
(implies (and (syntaxp (not (equal ups ''nil)))
(atom ups))
(iff (4v-sexpr-fixpoint-lower-boundp ups lb)
(4v-sexpr-fixpoint-lower-boundp nil lb)))
:hints (("goal" :use ((:instance 4v-sexpr-alist-equiv-nil-any-atom
(x ups))))))
(defthm lower-boundp-when-atom-car
(implies (and (not (4v-sexpr-fixpoint-lower-boundp ups lb))
(not (consp (car ups))))
(not (4v-sexpr-fixpoint-lower-boundp (cdr ups) lb)))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-sexpr-eval)
:cases ((consp ups)))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-witnessing))
(witness :ruleset (4v-sexpr-fixpoint-lower-boundp-eval-alist-ex))
)
:otf-flg t)
(defthm alist-keys-4v-sexpr-fixpoint-spec
(equal (alist-keys (4v-sexpr-fixpoint-spec ups))
(alist-keys ups))
:hints (("goal" :induct t)))
;; (defthm keys-equiv-4v-sexpr-eval-alist
;; (keys-equiv (4v-sexpr-eval-alist al env)
;; (double-rewrite al))
;; :hints(("Goal" :in-theory (enable keys-equiv-iff-set-equiv-alist-keys))))
;; (defthm keys-equiv-4v-sexpr-fixpoint-spec
;; (keys-equiv (4v-sexpr-fixpoint-spec ups)
;; (double-rewrite ups))
;; :hints(("Goal" :in-theory (enable keys-equiv-iff-set-equiv-alist-keys))))
;; (defcong keys-equiv keys-equiv (cons a b) 2
;; :hints (("goal" :do-not-induct t)
;; (witness)))
;; (local
;; (progn
;; (defthmd replace-fixpoint-in-hons-assoc-equal
;; (implies (key-and-env-equiv (cons (cons k v) a) b)
;; (4v-equiv (cdr (hons-assoc-equal k b))
;; v)))
;; (defthm 4v-<=-cons-append-env
;; (implies (and (4v-alist-<= (cons (cons k v) a) fp0)
;; (keys-equiv (double-rewrite (cons (cons k v) a)) fp0))
;; (4v-<=
;; (4v-sexpr-eval
;; x (cons (cons k v) (append a env)))
;; (4v-sexpr-eval
;; x (append fp0 env))))
;; :hints (("goal" :use ((:instance 4v-alist-<=-append
;; (a (cons (cons k v) a))
;; (b fp0)
;; (c env) (d env)))
;; :do-not-induct t
;; :in-theory (disable 4v-alist-<=-append 4v-fix 4v-<=)))
;; :otf-flg t)
;; (defthm keys-equiv-cons-caar
;; (implies (consp (car x))
;; (keys-equiv (cons (cons (caar x) v) (cdr x))
;; (double-rewrite x))))
;; (defthm key-and-env-equiv-remove-assoc-hons
;; (implies (and (not (4v-env-equiv a (remove-assoc-hons k b)))
;; (not (hons-assoc-equal k a)))
;; (not (key-and-env-equiv (cons (cons k v) a) b)))
;; :hints (("goal" :do-not-induct t
;; :in-theory (e/d (key-and-env-equiv)
;; (4v-fix
;; 4v-env-equiv-to-key-and-env-equiv)))
;; (witness :ruleset 4v-env-equiv-witnessing)
;; (witness :ruleset 4v-env-equiv-hons-assoc-equal-ex))
;; :otf-flg t)
;; (defthm keys-equiv-cons-eval-alist
;; (implies (and (keys-equiv ups fp0)
;; (consp (car ups)))
;; (equal (keys-equiv (cons (cons (caar ups) v)
;; (4v-sexpr-eval-alist (cdr ups) a))
;; fp0)
;; t))
;; :hints (("goal" :in-theory (enable keys-equiv-iff-set-equiv-alist-keys))))
;; (defthmd 4v-alist-<=-acons->remove
;; (implies (not (hons-assoc-equal k a))
;; (iff (4v-alist-<= (cons (cons k v) a) b)
;; (and (4v-<= v (4v-lookup k b))
;; (4v-alist-<= a (remove-assoc-hons k b)))))
;; :hints (("goal" :do-not-induct t
;; :in-theory (disable 4v-fix))
;; (witness :ruleset 4v-alist-<=-witnessing)
;; (witness :ruleset 4v-alist-<=-hons-assoc-equal-example))
;; :otf-flg t)
;; (defthm hack
;; (implies
;; (and
;; (4v-alist-<=
;; (4v-sexpr-eval-alist (4v-sexpr-fixpoint-spec (cdr update-fns))
;; (cons (cons (caar update-fns)
;; (cdr (hons-assoc-equal (caar update-fns)
;; fp0)))
;; env0))
;; (remove-assoc-hons (caar update-fns) fp0))
;; (consp update-fns)
;; (consp (car update-fns))
;; (4v-sexpr-fixpoint-lower-boundp (cdr update-fns)
;; (4v-sexpr-fixpoint-spec (cdr update-fns)))
;; (not (hons-assoc-equal (caar update-fns)
;; (cdr update-fns)))
;; (no-duplicatesp-equal (alist-keys (cdr update-fns)))
;; (keys-equiv update-fns fp0)
;; (key-and-env-equiv (cons (cons (caar update-fns)
;; (4v-sexpr-eval (cdar update-fns)
;; (append fp0 env0)))
;; (4v-sexpr-eval-alist (cdr update-fns)
;; (append fp0 env0)))
;; fp0))
;; (4v-<=
;; (4v-sexpr-eval
;; (cdar update-fns)
;; (cons
;; (cons (caar update-fns) 'x)
;; (append (4v-sexpr-eval-alist (4v-sexpr-fixpoint-spec (cdr update-fns))
;; (cons (cons (caar update-fns) 'x) env0))
;; env0)))
;; (CDR (HONS-ASSOC-EQUAL (CAAR UPDATE-FNS)
;; FP0))))
;; :hints (("goal" :do-not-induct t
;; :in-theory (e/d (alist-keys
;; 4v-alist-<=-acons->remove
;; 4v-alist-<=-trans2
;; replace-fixpoint-in-hons-assoc-equal)
;; (4v-sexpr-restrict-alist
;; 4v-sexpr-eval
;; 4v-alist-<=-acons-1
;; 4v-fix 4v-<=)))))))
(defthmd 4v-alist-<=-acons->remove
(implies (not (hons-assoc-equal k a))
(iff (4v-alist-<= (cons (cons k v) a) b)
(and (4v-<= v (4v-lookup k b))
(4v-alist-<= a (remove-assoc-hons k b)))))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-fix))
(witness :ruleset 4v-alist-<=-witnessing)
(witness :ruleset 4v-alist-<=-hons-assoc-equal-example))
:otf-flg t)
;; Notation: We'll use
;; - * to apply a substitution (aka 4v-sexpr-restrict(-alist))
;; - : to construct a substitution pair
;; - :: to append substitutions.
;; - \ to remove a binding from a substitution
;; Sketch of proof that we compute a least fixpoint:
;; Induct on length of update-fns. Base case is trivial.
;; In the inductive case, let update-fns = ((s1 : up1) :: upsr), i.e. s1
;; is the first signal bound and up1 is its update function. By IH we
;; have computed lbr, lower bound for upsr. Meaning, for any fpr, envr
;; where fpr has the same keys as upsr and satisfying
;; upsr (fpr :: envr) <= fpr,
;; we have lbr(envr) <= fpr.
;; Our result is:
;; lbf = (s1 : ((up1 * lbr) * (s1 : X)))
;; :: (lbr * (s1 : ((up1 * lbr) * (s1 : X)))).
;; We need to show that for any fp, env where the bound signals of fp are
;; the same as those of ups and ups (fp :: env) <= fp,
;; we have lbf(env) <= fp.
;; Suppose we have such an fp, env. Decomposing, we have
;; upsr (fp :: env) <= fp\s1, and
;; up1 (fp :: env) <= s1(fp).
;; and we need to show
;; ((up1 * lbr) * (s1 : X))(env) <= s1(fp) and
;; (lbr * (s1 : ((up1 * lbr) * (s1 : X)))) (env) <= fp\s1.
;; Evaluation rule for substitution (4v-sexpr-eval-4v-sexpr-restrict(-alist)):
;; (a * b)(env) = a(b(env) :: env).
;; So we reduce the above obligations to:
;; (up1 * lbr)((s1 : X) :: env)
;; = up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp)
;; and
;; lbr ((s1 : ((up1 * lbr) * (s1 : X)) (env)) :: env) =
;; lbr ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= s1\fp1.
;; Looking at the second obligation, we need to show
;; lbr ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= s1\fp1.
;;
;; Using the IH, let fpr, envr as above be fp\s1 and
;; ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env).
;; If we can show
;; upsr (fpr :: envr) <= fpr, i.e.
;; upsr (fp\s1 :: (s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1,
;; then we have lbr(envr) <= fpr, i.e.
;; lbr ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1,
;; and we'll be done.
;; But we already know
;; upsr (fp :: env) <= fp\s1. By monotonicity of upsr and transitivity
;; of <=, we can just show
;; fp\s1 :: (s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) <= fp.
;; This decomposes to:
;; fp\s1 <= fp\s1 (trivial)
;; and up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp),
;; which is just the first obligation above.
;; Looking at the first obligation, we need to show
;; up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp)m
;; but we have
;; up1 (fp :: env) <= s1(fp). It suffices to show that
;; up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) has the
;; same keys as fp (trivial) and ... <= fp.
;; That is,
;; lbr ((s1 : X) :: env) :: (s1 : X) <= fp,
;; which decomposes to
;; X <= s1(fp), (trivial),
;; and
;; lbr ((s1 : X) :: env) <= fp\s1.
;; Here we apply the IH again: let fpr, envr as above be fp\s1 and
;; ((s1 : X) :: env).
;; If we can show
;; upsr (fpr :: envr) <= fpr, i.e.
;; upsr (fp\s1 :: (s1 : X) :: env) <= fp\s1,
;; then we have lbr(envr) <= fpr, i.e.
;; lbr ((s1 : X) :: env) <= fp\s1,
;; and we'll be done.
;; But we already know
;; upsr (fp :: env) <= fp\s1. By monotonicity of upsr and transitivity
;; of <=, we can just show
;; fp\s1 :: (s1 : X) <= fp.
;; This decomposes to:
;; fp\s1 <= fp\s1 (trivial)
;; and X <= s1(fp), also trivial.
;; Working from bottom up:
(defthm |fp\s1 :: (s1 : X) <= fp|
(4v-alist-<= (append (remove-assoc-hons s1 fp)
(list (cons s1 *4vx*)))
fp)
:hints ((witness :ruleset 4v-alist-<=-witnessing)))
(defthmd 4v-alist-<=-append-cons-append
(implies (and (set-equiv (double-rewrite (cons bk (alist-keys a)))
(double-rewrite (alist-keys d)))
(4v-alist-<= (append a (list (cons bk bv)))
d)
(4v-alist-<= c e))
(4v-alist-<= (append a (cons (cons bk bv) c))
(append d e)))
:hints (("goal" :do-not-induct t
:in-theory (e/d* (hons-assoc-equal-iff-member-alist-keys
;; hons-assoc-equal-when-not-member-alist-keys
)
(alist-keys-member-hons-assoc-equal
4v-fix 4v-<= default-car default-cdr
append member-when-atom member-equal
alist-keys-when-atom
;; set-equiv-asym set-equiv-trans
(:rules-of-class :type-prescription :here))))
(witness :ruleset (4v-alist-<=-witnessing
4v-alist-<=-4v-lookup-example))
(witness :ruleset set-equiv-member-template)))
;(why 4v-alist-<=-trans2)
;(why 4V-ALIST-<=-SEXPR-EVAL-ALIST-MONOTONIC-ENV)
;(why 4v-alist-<=-append-cons-append)
;; Assuming
;; upsr (fp :: env) <= fp\s1,
;; prove
;; upsr (fp\s1 :: (s1 : X) :: env) <= fp\s1.
(defthm |upsr (fp\s1 :: (s1 : X) :: env) <= fp\s1|
(implies (and (4v-alist-<= (4v-sexpr-eval-alist upsr (append fp env))
(remove-assoc-hons s1 fp))
(set-equiv (alist-keys fp)
(cons s1 (alist-keys upsr))))
(4v-alist-<= (4v-sexpr-eval-alist upsr
(append (remove-assoc-hons s1 fp)
(cons (cons s1 *4vx*)
env)))
(remove-assoc-hons s1 fp)))
:hints(("Goal" :in-theory
(e/d (4v-alist-<=-trans2
hons-assoc-equal-iff-member-alist-keys
set-equiv-breakdown-cons
4v-alist-<=-append-cons-append)
(switch-append-cons
switch-append
alist-keys-member-hons-assoc-equal))
:do-not-induct t)))
;; Given
;; upsr (fp :: env) <= fp\s1
;; and lbr a FLB of upsr,
;; show
;; lbr ((s1 : X) :: env) <= fp\s1.
(defthm |lbr ((s1 : X) :: env) <= fp\s1|
(implies (and (4v-sexpr-fixpoint-lower-boundp upsr lbr)
(4v-alist-<=
(4v-sexpr-eval-alist upsr (append fp env))
(remove-assoc-hons s1 fp))
(set-equiv (alist-keys fp)
(cons s1 (alist-keys upsr)))
(not (member-equal s1 (alist-keys upsr))))
(4v-alist-<=
(4v-sexpr-eval-alist
lbr
(cons (cons s1 *4vx*) env))
(remove-assoc-hons s1 fp)))
:hints (("goal" :use
((:instance 4v-sexpr-fixpoint-lower-boundp-necc
(ups upsr)
(lb lbr)
(fp (remove-assoc-hons s1 fp))
(env (cons (cons s1 *4vx*) env))))
:in-theory (e/d (set-equiv-breakdown-cons)
(alist-keys-member-hons-assoc-equal
switch-append
switch-append-cons))
:do-not-induct t))
:otf-flg t)
;; Looking at the first obligation, we need to show
;; up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp)
;; but we have
;; up1 (fp :: env) <= s1(fp). It suffices to show that "..." has the
;; same keys as fp (trivial) and ... <= fp.
;; That is,
;; lbr ((s1 : X) :: env) :: (s1 : X) <= fp,
;; which decomposes to
;; lbr ((s1 : X) :: env) <= fp\s1 (same as the second obligation above)
;; and
;; X <= s1(fp), trivial.
(defthm |lbr ((s1 : X) :: env) :: (s1 : X) <= fp|
(implies (4v-alist-<=
(4v-sexpr-eval-alist
lbr (cons (cons s1 *4vx*) env))
(remove-assoc-hons s1 fp))
(4v-alist-<=
(append (4v-sexpr-eval-alist
lbr (cons (cons s1 *4vx*) env))
(list (cons s1 *4vx*)))
fp))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-fix 4v-sexpr-eval))
(witness :ruleset (4v-alist-<=-witnessing
4v-alist-<=-4v-lookup-example))))
;; given lbr ((s1 : X) :: env) <= fp\s1
;; and up1 (fp :: env) <= s1(fp),
;; show up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp).
(defthm |up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp)|
(implies (and (4v-alist-<=
(4v-sexpr-eval-alist
lbr (cons (cons s1 *4vx*) env))
(remove-assoc-hons s1 fp))
(4v-<= (4v-sexpr-eval up1 (append fp env))
(4v-lookup s1 fp))
(set-equiv (alist-keys fp)
(cons s1 (alist-keys lbr))))
(4v-<=
(4v-sexpr-eval
up1
(append (4v-sexpr-eval-alist
lbr (cons (cons s1 *4vx*) env))
(cons (cons s1 *4vx*) env)))
(4v-lookup s1 fp)))
:hints (("goal" :in-theory
(e/d (4v-<=-trans2
4v-alist-<=-append-cons-append)
(4v-<= switch-append
switch-append-cons 4v-lookup))
:do-not-induct t)))
(defthmd 4v-alist-<=-append->remove
(implies (not (hons-assoc-equal k a))
(iff (4v-alist-<= (append a (list (cons k v))) b)
(and (4v-<= v (4v-lookup k b))
(4v-alist-<= a (remove-assoc-hons k b)))))
:hints (("goal" :do-not-induct t
:in-theory (disable 4v-fix))
(witness :ruleset 4v-alist-<=-witnessing)
(witness :ruleset 4v-alist-<=-hons-assoc-equal-example))
:otf-flg t)
;; But we already know
;; upsr (fp :: env) <= fp\s1. By monotonicity of upsr and transitivity
;; of <=, we can just show
;; fp\s1 :: (s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) <= fp.
;; This decomposes to:
;; fp\s1 <= fp\s1 (trivial)
;; and up1(lbr((s1 : X) :: env) :: (s1 : X) :: env) <= s1(fp),
;; which is just the first obligation above.
(defthm
|upsr (fp\s1 :: (s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1|
(implies (and (4v-alist-<=
(4v-sexpr-eval-alist upsr (append fp env))
(remove-assoc-hons s1 fp))
(4v-<= (4v-sexpr-eval up1 (append fp env))
(4v-lookup s1 fp))
(set-equiv (alist-keys fp)
(cons s1 (alist-keys upsr)))
(set-equiv (alist-keys lbr)
(alist-keys upsr))
(not (member-equal s1 (alist-keys upsr)))
(4v-sexpr-fixpoint-lower-boundp upsr lbr))
(4v-alist-<=
(4v-sexpr-eval-alist
upsr
(append (remove-assoc-hons s1 fp)
(cons (cons s1 (4v-sexpr-eval
up1
(append (4v-sexpr-eval-alist
lbr
(cons (cons s1 *4vx*) env))
(cons (cons s1 *4vx*)
env))))
env)))
(remove-assoc-hons s1 fp)))
:hints(("Goal" :in-theory
(e/d (4v-alist-<=-trans2
hons-assoc-equal-iff-member-alist-keys
set-equiv-breakdown-cons
set-equiv-breakdown-cons2
4v-alist-<=-append->remove
4v-alist-<=-append-cons-append)
(switch-append-cons
switch-append 4v-<= 4v-fix 4v-lookup
alist-keys-member-hons-assoc-equal))
:do-not-induct t)))
;; Using the IH, let fpr, envr as above be fp\s1 and
;; ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env).
;; If we can show
;; upsr (fpr :: envr) <= fpr, i.e.
;; upsr (fp\s1 :: (s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1,
;; then we have lbr(envr) <= fpr, i.e.
;; lbr ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1,
;; and we'll be done.
(defthm |lbr ((s1 : up1(lbr((s1 : X) :: env) :: (s1 : X) :: env)) :: env) <= fp\s1|
(implies (and (4v-alist-<=
(4v-sexpr-eval-alist upsr (append fp env))
(remove-assoc-hons s1 fp))
(4v-<= (4v-sexpr-eval up1 (append fp env))
(4v-lookup s1 fp))
(set-equiv (alist-keys fp)
(cons s1 (alist-keys upsr)))
(set-equiv (alist-keys lbr)
(alist-keys upsr))
(not (member-equal s1 (alist-keys upsr)))
(4v-sexpr-fixpoint-lower-boundp upsr lbr))
(4v-alist-<=
(4v-sexpr-eval-alist
lbr
(cons (cons s1 (4v-sexpr-eval
up1
(append (4v-sexpr-eval-alist
lbr
(cons (cons s1 *4vx*) env))
(cons (cons s1 *4vx*)
env))))
env))
(remove-assoc-hons s1 fp)))
:hints (("goal" :use ((:instance 4v-sexpr-fixpoint-lower-boundp-necc
(ups upsr)
(lb lbr)
(fp (remove-assoc-hons s1 fp))
(env (cons (cons s1 (4v-sexpr-eval
up1
(append (4v-sexpr-eval-alist
lbr
(cons (cons s1 *4vx*) env))
(cons (cons s1 *4vx*)
env))))
env))))
:in-theory
(e/d (4v-alist-<=-trans2
hons-assoc-equal-iff-member-alist-keys
set-equiv-breakdown-cons
set-equiv-breakdown-cons2
4v-alist-<=-append->remove
4v-alist-<=-append-cons-append)
(switch-append-cons
switch-append 4v-<= 4v-fix 4v-lookup
alist-keys-member-hons-assoc-equal)))))
;; Our result is:
;; lbf = (s1 : ((up1 * lbr) * (s1 : X)))
;; :: (lbr * (s1 : ((up1 * lbr) * (s1 : X)))).
(defthm inductive-step
(implies (and (4v-sexpr-fixpoint-lower-boundp upsr lbr)
(set-equiv (alist-keys upsr)
(alist-keys lbr))
(not (member-equal s1 (alist-keys upsr))))
(4v-sexpr-fixpoint-lower-boundp
(cons (cons s1 up1) upsr)
(cons (cons s1 (4v-sexpr-restrict
(4v-sexpr-restrict up1 lbr)
(list (cons s1
*4vx-sexpr*))))
(4v-sexpr-restrict-alist
lbr (list (cons s1 (4v-sexpr-restrict
(4v-sexpr-restrict up1 lbr)
(list (cons s1
*4vx-sexpr*)))))))))
:hints (("goal" :do-not-induct t
:in-theory (e/d (4v-alist-<=-acons->remove
keys-equiv-iff-set-equiv-alist-keys
hons-assoc-equal-iff-member-alist-keys)
(4v-<= 4v-fix 4v-lookup
switch-append switch-append-cons
alist-keys-member-hons-assoc-equal
4v-sexpr-eval)))
(witness :ruleset 4v-sexpr-fixpoint-lower-boundp-witnessing))
:otf-flg t)
(defthm 4v-sexpr-fixpoint-lower-boundp-4v-sexpr-fixpoint-spec
(implies (no-duplicatesp-equal (alist-keys update-fns))
(4v-sexpr-fixpoint-lower-boundp
update-fns
(4v-sexpr-fixpoint-spec update-fns)))
:hints (("goal" :induct t
:in-theory (disable 4v-sexpr-restrict-4v-sexpr-restrict))))
(defthm 4v-sexpr-fixpoint-lower-boundp2-4v-sexpr-fixpoint-spec
(implies (no-duplicatesp-equal (alist-keys update-fns))
(4v-sexpr-fixpoint-lower-boundp2
update-fns
(4v-sexpr-fixpoint-spec update-fns))))
;; (defthm 4v-sexpr-fixpoint-lower-boundp-4v-sexpr-fixpoint-spec
;; (implies (no-duplicatesp-equal (alist-keys update-fns))
;; (4v-sexpr-fixpoint-lower-boundp
;; update-fns
;; (4v-sexpr-fixpoint-spec update-fns)))
;; :hints (("goal" :induct (4v-sexpr-fixpoint-spec update-fns)
;; :in-theory (e/d (alist-keys
;; 4v-alist-<=-acons->remove
;; 4v-alist-<=-trans2)
;; (4v-sexpr-restrict-alist
;; 4v-sexpr-eval
;; 4v-sexpr-restrict
;; 4v-alist-<=-acons-1
;; default-car default-cdr
;; keys-equiv-when-alist-keys
;; subsetp-car-member
;; member-equal
;; 4v-fix 4v-<=))
;; :do-not-induct t)
;; (witness :ruleset 4v-sexpr-fixpoint-lower-boundp-witnessing)
;; (and stable-under-simplificationp
;; '(:use ((:instance 4v-sexpr-fixpoint-lower-boundp-necc
;; (ups (cdr update-fns))
;; (fp (remove-assoc-hons (caar update-fns) fp0))
;; (env (cons (cons (caar update-fns)
;; (cdr (hons-assoc-equal
;; (caar update-fns)
;; fp0)))
;; env0))
;; (lb (4v-sexpr-fixpoint-spec
;; (cdr update-fns))))))))
;; :otf-flg t)
;; (defthmd 4v-sexpr-fixpointp-iff-4v-sexpr-alist-equiv
;; (implies (keys-equiv ups fixpoint)
;; (iff (4v-sexpr-fixpointp ups fixpoint)
;; (4v-sexpr-alist-equiv (4v-sexpr-restrict-alist ups fixpoint)
;; fixpoint)))
;; :hints(("Goal" :in-theory (enable 4v-sexpr-alist-equiv-fixpoint)
;; :do-not-induct t)
;; (witness :ruleset (4v-sexpr-fixpointp-witnessing
;; 4v-sexpr-alist-equiv-example))))
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