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; Copyright/License: See the LICENSE file in this directory.
#| This is the .lisp file for the Abelian Group 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-AGP")
(include-book "acl2-asg")
(defsection abelian-groups
:parents (algebra)
:short "Axiomatization of an associative and commutative binary operation
with an identity and an unary inverse operation, developed by John Cowles."
:long "<h3>Theory of Abelian Groups</h3>
<p>@('ACL2-AGP::op') is an associative and commutative binary operation on the
set (of equivalence classes formed by the equivalence relation,
@('ACL2-AGP::equiv'), on the set)</p>
@({
GP = { x | (ACL2-AGP::pred x) != nil }
})
<p>@('ACL2-AGP::id') is a constant in the set GP which acts as an unit for
@('ACL2-AGP::op') in GP.</p>
<p>@('ACL2-AGP::inv') is an unary operation on the set (of equivalence classes
formed by the equivalence relation, @('ACL2-AGP::equiv'), on the set) GP which
acts as an @('ACL2-AGP::op-inverse') for @('ACL2-AGP:: id').</p>
<p>For example, let</p>
<ul>
<li>@('ACL2-AGP::pred') = Booleanp, </li>
<li>@('ACL2-AGP::op') = exclusive-or, </li>
<li>@('ACL2-AGP::id') = nil, and </li>
<li>@('ACL2-AGP::inv') = identity function. </li>
</ul>
<h3>Axioms of the theory of Abelian Groups</h3>
<p>Using @(see encapsulate), we introduce constrained functions:</p>
<ul>
<li>@(call equiv)</li>
<li>@(call pred)</li>
<li>@(call op)</li>
<li>@(call id)</li>
<li>@(call inv)</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 Equiv-implies-equiv-inv-1)
@(def Closure-of-op-for-pred)
@(def Closure-of-id-for-pred)
@(def Closure-of-inv-for-pred)
@(def Commutativity-of-op)
@(def Associativity-of-op)
@(def Left-unicity-of-id-for-op)
@(def Right-inverse-of-inv-for-op)
<h3>Theorems of the theory of Abelian Groups</h3>"
;; It looks like this doc ends abruptly, but see below; we extend it.
:autodoc nil
(encapsulate
((equiv (x y) t)
(pred (x) t)
(op (x y) t)
(id () t)
(inv (x) 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)))))
(local (defun id ()
nil))
(local (defun inv (x)
(declare (xargs :guard (pred x)))
x))
(defequiv equiv
:rule-classes (:equivalence
(:type-prescription
:corollary
(or (equal (equiv x y) t)
(equal (equiv x y) nil)))))
(defcong equiv iff (pred x) 1)
(defcong equiv equiv (op x y) 1)
(defcong equiv equiv (op x y) 2)
(defcong equiv equiv (inv x) 1)
(defthm Closure-of-op-for-pred
(implies (and (pred x)
(pred y))
(pred (op x y))))
(defthm Closure-of-id-for-pred
(pred (id)))
(defthm Closure-of-inv-for-pred
(implies (pred x)
(pred (inv x))))
(defthm Commutativity-of-op
(implies (and (pred x)
(pred y))
(equiv (op x y)
(op y x))))
(defthm Associativity-of-op
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op (op x y) z)
(op x (op y z)))))
(defthm Left-unicity-of-id-for-op
(implies (pred x)
(equiv (op (id) x)
x)))
(defthm Right-inverse-of-inv-for-op
(implies (pred x)
(equiv (op x (inv x))
(id))))))
(defsection abelian-groups-thms
:extension abelian-groups
(acl2-asg::add-commutativity-2 equiv
pred
op
commutativity-of-op
commutativity-2-of-op)
(defthm Right-unicity-of-id-for-op
(implies (pred x)
(equiv (op x (id))
x)))
(defthm Left-inverse-of-inv-for-op
(implies (pred x)
(equiv (op (inv x) x)
(id))))
(local (defthm Right-cancellation-for-op-iff
(implies (and (pred x)
(pred y)
(pred z))
(iff (equiv (op x z) (op y z))
(equiv x y)))
:rule-classes nil
:hints (("Subgoal 1"
:in-theory (disable Equiv-implies-equiv-op-1)
:use (:instance Equiv-implies-equiv-op-1
(x (op x z))
(x-equiv (op y z))
(y (inv z)))))))
(defthm Right-cancellation-for-op
(implies (and (pred x)
(pred y)
(pred z))
(equal (equiv (op x z) (op y z))
(equiv x y)))
:rule-classes nil
:hints (("Goal" :use Right-cancellation-for-op-iff)))
(local (defthm Left-cancellation-for-op-iff
(implies (and (pred x)
(pred y)
(pred z))
(iff (equiv (op x y) (op x z))
(equiv z y)))
:rule-classes nil
:hints (("Goal" :use ((:instance Right-cancellation-for-op
(x z)
(z x)))))))
(defthm Left-cancellation-for-op
(implies (and (pred x)
(pred y)
(pred z))
(equal (equiv (op x y) (op x z))
(equiv y z)))
:hints (("Goal" :use Left-cancellation-for-op-iff)))
(defthm Uniqueness-of-id-as-op-idempotent
(implies (and (pred x)
(equiv (op x x) x))
(equiv x (id)))
:rule-classes nil
:hints (("Goal"
:use (:instance Right-cancellation-for-op
(y (id))
(z x)))))
(defthm Uniqueness-of-op-inverses
(implies (and (pred x)
(pred y)
(equiv (op x y) (id)))
(equiv y (inv x)))
:rule-classes nil
:hints (("Goal"
:use (:instance Right-cancellation-for-op
(x y)
(y (inv x))
(z x)))))
(defthm Involution-of-inv
(implies (pred x)
(equiv (inv (inv x))
x))
:hints (("Goal" :use (:instance Uniqueness-of-op-inverses
(x (inv x))
(y x)))))
(defthm Uniqueness-of-op-inverses-1
(implies (and (pred x)
(pred y)
(equiv (op x (inv y)) (id)))
(equiv x y))
:rule-classes nil
:hints (("Goal" :use (:instance Uniqueness-of-op-inverses
(y x)
(x (inv y))))))
(defthm Distributivity-of-inv-over-op
(implies (and (pred x)
(pred y))
(equiv (inv (op x y))
(op (inv x)
(inv y))))
:hints (("Goal" :use (:instance Uniqueness-of-op-inverses
(x (op x y))
(y (op (inv x)(inv y)))))))
(defthm id-is-its-own-invese
(equiv (inv (id))
(id))
:hints (("Goal" :use (:instance Uniqueness-of-op-inverses
(x (id))
(y (id))))))
(local (defthm obvious-inv-cancellation
(implies (and (pred x)
(pred y))
(equiv (op (op x (inv x)) y) y))
:rule-classes nil))
(defthm inv-cancellation-on-right
(implies (and (pred x)
(pred y))
(equiv (op x (op y (inv x)))
y))
:hints (("Goal"
:use obvious-inv-cancellation
:in-theory (disable right-inverse-of-inv-for-op))))
(defthm inv-cancellation-on-left
(implies (and (pred x)
(pred y))
(equiv (op x (op (inv x) y))
y))))
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