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|
; Std/basic - Basic definitions
; Copyright (C) 2008-2013 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: Jared Davis <jared@centtech.com>
(in-package "ACL2")
(include-book "xdoc/top" :dir :system)
(local (xdoc::set-default-parents std/basic))
(defsection bitp
:parents (std/basic logops-definitions)
:short "Bit recognizer. @('(bitp b)') recognizes 0 and 1."
:long "<p>This is a predicate form of the @(see type-spec) declaration
@('(TYPE BIT b)').</p>"
(defun-inline bitp (b)
(declare (xargs :guard t))
(or (eql b 0)
(eql b 1)))
)
(defsection bfix
:parents (std/basic logops-definitions bitp)
:short "Bit fix. @('(bfix b)') is a fixing function for @(see bitp)s. It
coerces any object to a bit (0 or 1) by coercing non-1 objects to 0."
:long "<p>See also @(see lbfix).</p>"
(defun-inline bfix (b)
(declare (xargs :guard t))
(if (eql b 1)
1
0)))
(defsection lbfix
:parents (std/basic logops-definitions bitp)
:short "Logical bit fix. @('(lbfix b)') is logically identical to @('(bfix
b)') but executes as the identity. It requires @('(bitp b)') as a guard, and
expands to just @('b')."
:long "@(def lbfix)"
(defmacro lbfix (x)
`(mbe :logic (bfix ,x) :exec ,x)))
(defsection bitp-basics
:extension bitp
(defthm bitp-bfix
(bitp (bfix b)))
(defthm bfix-bitp
(implies (bitp b)
(equal (bfix b) b))))
(defsection maybe-bitp
:parents (std/basic bitp)
:short "Recognizer for bits and @('nil')."
:long "<p>This is like an <a
href='https://en.wikipedia.org/wiki/Option_type'>option type</a>; when @('x')
satisfies @('maybe-bitp'), then either it is a @(see bitp) or nothing.</p>"
(defund-inline maybe-bitp (x)
(declare (xargs :guard t))
(or (not x)
(bitp x)))
(local (in-theory (enable maybe-bitp)))
(defthm maybe-bitp-compound-recognizer
(implies (maybe-bitp x)
(or (not x)
(natp x)))
:rule-classes :compound-recognizer))
(defsection lnfix
:short "@(call lnfix) is logically identical to @('(nfix x)'), but its guard
requires that @('x') is a natural number and, in the execution, it is just a
no-op that returns @('x')."
:long "<p>@(call lnfix) is an inlined, enabled function that just expands into</p>
@({
(mbe :logic (nfix x) :exec x)
})
<p>Why would you want this? When you defining a function whose @(see guard)
assumes @('(natp n)'), it is often a good idea to write the function so that it
logically treats non-naturals as 0. You might generally accomplish this by
rewriting, e.g.,:</p>
@({
(defun my-function (n ...)
(declare (xargs :guard (natp n)))
<body>)
--->
(defun my-function (n ...)
(declare (xargs :guard (natp n)))
(let ((n (nfix n)))
<body>))
})
<p>This leads to a nice @(see nat-equiv) @(see congruence) rule. But since
@(see nfix) has to check whether its argument is a natural number, and that
has some performance cost.</p>
<p>By using @(see lnfix) in place of @('nfix') here, you can get the same
logical definition without this overhead.</p>"
(defun-inline lnfix (x)
(declare (xargs :guard (natp x)))
(mbe :logic (nfix x) :exec x)))
(defsection lifix
:short "@(call lifix) is logically identical to @('(ifix x)'), but its guard
requires that @('x') is an integer and, in the execution, it is just a no-op
that returns @('x')."
:long "<p>@(call lifix) is an inlined, enabled function that just expands into</p>
@({
(mbe :logic (ifix x) :exec x)
})
<p>To understand why you might want this, see the analogous discussion about
@(see lnfix).</p>"
(defun-inline lifix (x)
(declare (xargs :guard (integerp x)))
(mbe :logic (ifix x) :exec x)))
(defsection true
:short "A function that always just returns the constant @('t')."
:long "<p>This is occasionally useful for @(see defattach), etc.</p>"
(defun true ()
(declare (xargs :guard t))
t))
(defsection false
:short "A function that just returns the constant @('nil')."
:long "<p>This is occasionally useful for @(see defattach), etc.</p>"
(defun false ()
(declare (xargs :guard t))
nil))
(defsection maybe-natp
:short "Recognizer for naturals and @('nil')."
:long "<p>This is like an <a
href='https://en.wikipedia.org/wiki/Option_type'>option type</a>; when @('x')
satisfies @('maybe-natp'), then either it is a natural number or nothing.</p>"
(defund-inline maybe-natp (x)
(declare (xargs :guard t))
(or (not x)
(natp x)))
(local (in-theory (enable maybe-natp)))
(defthm maybe-natp-compound-recognizer
(equal (maybe-natp x)
(or (not x)
(and (integerp x)
(<= 0 x))))
:rule-classes :compound-recognizer))
(defsection maybe-integerp
:short "Recognizer for integers and @('nil')."
:long "<p>This is like an <a
href='https://en.wikipedia.org/wiki/Option_type'>option type</a>; when @('x')
satisfies @('maybe-integerp'), then either it is a integer or nothing.</p>"
(defund-inline maybe-integerp (x)
(declare (xargs :guard t))
(or (not x)
(integerp x)))
(local (in-theory (enable maybe-integerp)))
(defthm maybe-integerp-compound-recognizer
(equal (maybe-integerp x)
(or (integerp x)
(not x)))
:rule-classes :compound-recognizer))
(defsection maybe-posp
:short "Recognizer for positive naturals and @('nil')."
:long "<p>This is like an <a
href='https://en.wikipedia.org/wiki/Option_type'>option type</a>; when @('x')
satisfies @('maybe-posp'), then either it is a positive, natural number or
nothing.</p>"
(defund-inline maybe-posp (x)
(declare (xargs :guard t))
(or (not x)
(posp x)))
(local (in-theory (enable maybe-posp)))
(defthm maybe-posp-compound-recognizer
(equal (maybe-posp x)
(or (and (integerp x)
(< 0 x))
(not x)))
:rule-classes :compound-recognizer))
(defsection maybe-stringp
:short "Recognizer for strings and @('nil')."
:long "<p>This is like an <a
href='https://en.wikipedia.org/wiki/Option_type'>option type</a>; when @('x')
satisfies @('maybe-stringp'), then either it is a string or nothing.</p>"
(defund-inline maybe-stringp (x)
(declare (xargs :guard t))
(or (not x)
(stringp x)))
(local (in-theory (enable maybe-stringp)))
(defthm maybe-stringp-compound-recognizer
(equal (maybe-stringp x)
(or (not x)
(stringp x)))
:rule-classes :compound-recognizer))
(defsection char-fix
:parents (std/basic str::equivalences)
:short "Coerce to a character."
:long "<p>@(call char-fix) is the identity on @(see acl2::characters), and
returns the NUL character (i.e., the character whose code is 0) for any
non-character.</p>
<p>This is similar to other fixing functions like @(see fix) and @(see nfix).
See also @(see chareqv).</p>"
(defund-inline char-fix (x)
(declare (xargs :guard t))
(if (characterp x)
x
(code-char 0)))
(local (in-theory (enable char-fix)))
(defthm characterp-of-char-fix
(characterp (char-fix x))
:rule-classes :type-prescription)
(defthm char-fix-default
(implies (not (characterp x))
(equal (char-fix x)
(code-char 0))))
(defthm char-fix-when-characterp
(implies (characterp x)
(equal (char-fix x)
x))))
(defsection chareqv
:parents (std/basic str::equivalences)
:short "Case-sensitive character equivalence test."
:long "<p>@(call chareqv) determines if @('x') and @('y') are equivalent when
interpreted as characters. That is, non-characters are first coerced to be the
NUL character (via @(see char-fix)), then we see if these coerced arguments are
equal.</p>
<p>See also @(see str::ichareqv) for a case-insensitive alternative.</p>"
(defund-inline chareqv (x y)
(declare (xargs :guard t))
(eql (char-fix x) (char-fix y)))
(local (in-theory (enable char-fix char< chareqv)))
(defequiv chareqv)
(defthm chareqv-of-char-fix
(chareqv (char-fix x) x))
(defcong chareqv equal (char-fix x) 1)
(defcong chareqv equal (char-code x) 1)
(defcong chareqv equal (char< x y) 1)
(defcong chareqv equal (char< x y) 2))
(defsection str-fix
:parents (std/basic str::equivalences)
:short "Coerce to a string."
:long "<p>@(call str-fix) is the identity on @(see acl2::stringp)s, or
returns the empty string, @('\"\"'), for any non-string.</p>
<p>This is similar to other fixing functions like @(see fix) and @(see nfix).
See also @(see streqv).</p>"
(defund-inline str-fix (x)
(declare (xargs :guard t))
(if (stringp x)
x
""))
(local (in-theory (enable str-fix)))
(defthm stringp-of-str-fix
(stringp (str-fix x))
:rule-classes :type-prescription)
(defthm str-fix-default
(implies (not (stringp x))
(equal (str-fix x)
"")))
(defthm str-fix-when-stringp
(implies (stringp x)
(equal (str-fix x)
x))))
(defsection streqv
:parents (std/basic str::equivalences)
:short "Case-sensitive string equivalence test."
:long "<p>@(call streqv) determines if @('x') and @('y') are equivalent when
interpreted as strings. That is, non-strings are first coerced to be the empty
string (via @(see str-fix)), then we see if these coerced arguments are
equal.</p>
<p>See also @(see str::istreqv) for a case-insensitive alternative.</p>"
(defund-inline streqv (x y)
(declare (xargs :guard t))
(equal (str-fix x) (str-fix y)))
(local (in-theory (enable str-fix streqv)))
(defequiv streqv)
(defthm streqv-of-str-fix
(streqv (str-fix x) x))
(defcong streqv equal (str-fix x) 1)
(defcong streqv equal (char x n) 1)
(defcong streqv equal (string-append x y) 1)
(defcong streqv equal (string-append x y) 2))
(defsection tuplep
:parents (std/basic)
:short "Recognizers for true-lists of some particular length."
:long "<p>@(call tuplep) recognizes @('n')-tuples. For instance:</p>
@({
(tuplep 3 '(1 2 3)) --> t
(tuplep 3 '(1 2)) --> nil (not long enough)
(tuplep 3 '(1 2 3 . 4)) --> nil (not a true-listp)
})
<p>We generally just leave this enabled.</p>"
(defun tuplep (n x)
(declare (xargs :guard (natp n)))
(mbe :logic (and (true-listp x)
(equal (len x) n))
:exec (and (true-listp x)
(eql (length x) n)))))
(defsection impossible
:parents (std/basic)
:short "Prove that some case is unreachable using @(see guard)s."
:long "<p>Logically, @('(impossible)') just returns @('nil').</p>
<p>But @('impossible') has a guard of @('nil'), so whenever you use it in a
function, you will be obliged to prove that it cannot be executed when the
guard holds.</p>
<p>What good is this? One use is to make sure that every possible case in a
sum type has been covered. For instance, you can write:</p>
@({
(define f ((x whatever-p))
(case (type-of x)
(:foo (handle-foo ...))
(:bar (handle-bar ...))
(otherwise (impossible))))
})
<p>This is a nice style in that, if we later extend @('x') so that its type can
also be @(':baz'), then the guard verification will fail and remind us that we
need to extend @('f') to handle @(':baz') types as well.</p>
<p>If somehow @('(impossible)') is ever executed (e.g., due to program mode
code that is violating guards), it just causes a hard error.</p>"
(defun impossible ()
(declare (xargs :guard nil))
(er hard 'impossible "Provably impossible")))
(defsection impliez
;; Added by Alessandro Coglio (coglio@kestrel.edu), Kestrel Institute.
:parents (std/basic)
:short "Logical implication defined via @(tsee if)."
:long
"<p>Since @(tsee implies) is a function,
guards in the consequent must be verified
without assuming the antecedent.
@('impliez') is a macro that expands to an @(tsee if),
so guards in the consequent can be verified
assuming the antecedent.</p>"
"<p>@(def impliez)</p>"
(defmacro impliez (p q)
`(if ,p ,q t)))
|
