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; Copyright/License: See the LICENSE file in this directory.
#| This is the .lisp file for the Abelian SemiGroup book.
John Cowles, University of Wyoming, Summer 1993
Last modified 29 July 1994.
Modified A. Flatau 2-Nov-1994
Added a :verify-guards t hint to PRED for Acl2 1.8.
Modified by Jared Davis, January 2014, to convert comments to XDOC
|#
(in-package "ACL2-ASG")
(include-book "xdoc/top" :dir :system)
(defsection abelian-semigroups
:parents (algebra)
:short "Axiomatization of an associative and commutative binary operation, developed by John Cowles."
:autodoc nil
:long "<h3>Theory of Abelian SemiGroups</h3>
<p>@('ACL2-ASG::op') is an associative and commutative binary operation on the
set (of equivalence classes formed by the equivalence relation,
@('ACL2-ASG::equiv'), on the set)</p>
@({
{ x | (ACL2-ASG::pred x) != nil }
})
<p>Exclusive-or on the set of Boolean values with the equivalence relation,
EQUAL, is an example.</p>
<p>Note, it is recommended that a second version of commutativity, called
Commutativity-2, be added as a :@(see rewrite) rule for any operation which has
both Associative and Commutative :@(see rewrite) rules. The macro
@(see ACL2-ASG::Add-Commutativity-2) may be used to add such a rule.</p>
<h3>Axioms of Abelian Semigroups</h3>
<p>Using @(see encapsulate), we introduce constrained functions:</p>
<ul>
<li>@(call equiv)</li>
<li>@(call pred)</li>
<li>@(call op)</li>
</ul>
<p>with the following, constraining axioms:</p>
@(def Equiv-is-an-equivalence)
@(def Equiv-implies-iff-pred-1)
@(def Equiv-implies-equiv-op-1)
@(def Equiv-implies-equiv-op-2)
@(def Closure-of-op-for-pred)
@(def Associativity-of-op)
@(def Commutativity-of-op)
<h3>Theorem of the theory of Abelian Groups</h3>
<p>Given the above constraints, we prove the following generic theorems.</p>
@(def Commutativity-2-of-op)"
(encapsulate
((equiv (x y) t)
(pred (x) t)
(op (x y) t))
(local (defun equiv (x y)
(equal x y)))
(local (defun pred (x)
(declare (xargs :verify-guards t))
(or (equal x t)
(equal x nil))))
(local (defun op (x y)
(declare (xargs :guard (and (pred x)
(pred y))))
(and (or x y)
(not (and x y)))))
(defequiv equiv)
(defcong equiv iff (pred x) 1)
(defcong equiv equiv (op x y) 1)
(defcong equiv equiv (op x y) 2)
(defthm Closure-of-op-for-pred
(implies (and (pred x)
(pred y))
(pred (op x y))))
(defthm Associativity-of-op
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op (op x y) z)
(op x (op y z)))))
(defthm Commutativity-of-op
(implies (and (pred x)
(pred y))
(equiv (op x y)
(op y x))))))
(defsection add-commutativity-2
:parents (abelian-semigroups)
:short "Macro for adding a second version of commutativity."
:autodoc nil
:long "<p>Examples:</p>
@({
(acl2-asg::add-commutativity-2 equal
rationalp
*
commutativity-of-*
commutativity-2-of-*)
(acl2-asg::add-commutativity-2 acl2-bag::bag-equal
true-listp
acl2-bag::bag-union
acl2-bag::commutativity-of-bag-union
commutativity-2-of-bag-union)
})
<p>General form:</p>
@({
(acl2-asg::add-commutativity-2 equiv-name
pred-name
op-name
commutativity-thm-name
commutativity-2-thm-name)
})
<p>where all the arguments are names.</p>
<ul>
<li><b>Equiv-name</b> is the name of an equivalence relation, @('equiv');</li>
<li><b>pred-name</b> is the name of a unary function, @('pred');</li>
<li><b>op-name</b> is the name of a binary function, @('op');</li>
<li><b>commutativity-thm-name</b> is the name of theorem which added a :@(see
rewrite) rule to the data base saying that the operation @('op') is commutative
on the set (of equivalence classes formed by the equivalence relation,
@('equiv'), on the set)
@({
SG = { x | (pred x) != nil }
})
There must be rules in the data base for the closure of SG under @('op') and
the associativity with respect to @('equiv') of @('op') on SG.</li>
</ul>
<p>The macro adds a rewrite rule for a second version of the commutativity with
respect to equiv of op on SG. This is done by proving a theorem named
<b>commutativity-2-thm-name</b>.</p>
<p>Here is the form of the rule added by the macro:</p>
@({
(defthm commutativity-2-thm-name
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op x (op y z))
(op y (op x z))))) .
})
<p>Here is what is meant by \"closure of SG under op\":</p>
@({
(implies (and (pred x)
(pred y))
(pred (op x y)))
})
<p>Here is what is meant by \"associativity with respect to equiv of
op on SG\":</p>
@({
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op (op x y) z)
(op x (op y z))))
})
<p>Here is the form of the commutativity rule:</p>
@({
(defthm commutativity-thm-name
(implies (and (pred x)
(pred y))
(equiv (op x y)
(op y x))))
})"
(local (defthm commutativity-2-of-op-lemma
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op (op x y) z)
(op (op y x) z)))
:rule-classes nil))
(defthm Commutativity-2-of-op
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op x (op y z))
(op y (op x z))))
:hints (("Goal"
:in-theory (acl2::disable commutativity-of-op)
:use commutativity-2-of-op-lemma)))
(defmacro add-commutativity-2 (equiv-name
pred-name
op-name
commutativity-thm-name
commutativity-2-thm-name)
(declare (xargs :guard (and (symbolp equiv-name)
(symbolp pred-name)
(symbolp op-name)
(symbolp commutativity-thm-name)
(symbolp commutativity-2-thm-name))))
`(encapsulate
nil
(local
(defthm Associativity-of-op-name
;; Temporarily ensure that the associativity rewrite rule comes after
;; the commutativity rewrite rule.
(IMPLIES (AND (,pred-name X)
(,pred-name Y)
(,pred-name Z))
(,equiv-name (,op-name (,op-name X Y) Z)
(,op-name X (,op-name Y Z))))
:hints (("Goal" :in-theory (acl2::disable ,commutativity-thm-name)))))
(defthm ,commutativity-2-thm-name
(implies (and (,pred-name x)
(,pred-name y)
(,pred-name z))
(,equiv-name (,op-name x (,op-name y z))
(,op-name y (,op-name x z))))
:hints (("Goal"
:use (:functional-instance
Commutativity-2-of-op
(equiv (lambda (x y) (,equiv-name x y)))
(pred (lambda (x) (,pred-name x)))
(op (lambda (x y) (,op-name x y))))))))))
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