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|
(in-package "ACL2")
#|
reductions.lisp
~~~~~~~~~~~~~~~
In this book, we use the conjunctive reduction and cone of influence reduction
compositionally to provide reduction algorithms for circuits.
|#
(local
(include-book "conjunction")
)
(include-book "cone-of-influence")
(defun ltl-semantics-for-circuit (C f)
(ltl-semantics f (create-kripke C)))
(defun ltl-semantics-for-circuits* (list)
(if (endp list) T
(and (ltl-semantics-for-circuit (second (first list))
(first (first list)))
(ltl-semantics-for-circuits* (rest list)))))
(defun reduce-problem-conjunction (f C)
(if (and (equal (len f) 3)
(equal (second f) '&))
(append (reduce-problem-conjunction (first f) C)
(reduce-problem-conjunction (third f) C))
(list (list f C))))
(defun reduce-problem-cone (f C)
(let ((vars (create-restricted-var-set f)))
(cone-of-influence-reduction C vars)))
(defun reduce-problem-cone* (list)
(if (endp list) nil
(cons (list (first (first list))
(reduce-problem-cone (first (first list)) (second (first list))))
(reduce-problem-cone* (rest list)))))
(defun compositional-reduction (C f)
(let ((list (reduce-problem-conjunction f C)))
(reduce-problem-cone* list)))
;; OK, so let us dispatch the obligations for conjunction first.
(local
(in-theory (disable ltl-semantics create-kripke ltl-formulap))
)
(local
(defthm ltl-semantics*-append-reduction
(equal (ltl-semantics-for-circuits* (append x y))
(and (ltl-semantics-for-circuits* x)
(ltl-semantics-for-circuits* y))))
)
(local
(defthm conjunction-produces-correct-list
(implies (ltl-formulap f)
(equal (ltl-semantics-for-circuits*
(reduce-problem-conjunction f C))
(ltl-semantics-for-circuit C f)))
:otf-flg t
:hints (("Goal"
:induct (reduce-problem-conjunction f C)
:do-not-induct t
:in-theory (enable ltl-formulap)
:do-not '(eliminate-destructors generalize))))
)
;; To work with reduce-problems-cone, we need to assume that the variables in f
;; are subsets of the variables in cone of influence reduction. We show that
;; assuming that the variables are subsets of variables of the circuit. We need
;; to show though that the variables of cone will be a superset of vars if we
;; start with a collection of vars that are subset of the variables of the
;; circuit.
(local
(encapsulate
()
(defthm not-memberp-union-reduction
(implies (and (not (memberp e x))
(not (memberp e y)))
(not (memberp e (set-union x y))))
:hints (("Goal"
:in-theory (enable set-union))))
(local
(defthm uniquep-set-union-reduction
(implies (and (uniquep x)
(uniquep y))
(uniquep (set-union x y)))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(in-theory (disable consistent-equation-record-p))
)
(local
(defthm consistent-equation-record-p-expanded
(implies (and (consistent-equation-record-p vars equations)
(uniquep vars)
(memberp v vars)
(memberp equation (<- equations v)))
(subset (find-variables equation)
vars))
:hints (("Goal"
:use consistent-equation-record-p-necc)))
)
(local
(in-theory (disable consistent-equation-record-p-necc))
)
(local
(defthm set-union-subset-reduction
(implies (and (subset x z)
(subset y z))
(subset (set-union x y) z))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm find-variables*-subset-of-variables
(implies (and (consistent-equation-record-p variables equations)
(uniquep variables)
(memberp v variables)
(subset equation-list (<- equations v)))
(subset (find-variables* equation-list)
variables))
:hints (("Goal"
:in-theory (disable find-variables)
:induct (find-variables* equation-list)
:do-not '(eliminate-destructors generalize)
:do-not-induct t)))
)
(local
(defthm find-variables*-is-subset-concretized
(implies (and (consistent-equation-record-p variables equations)
(memberp v variables)
(uniquep variables))
(subset (find-variables* (<- equations v)) variables)))
)
(local
(in-theory (disable find-variables*-subset-of-variables))
)
(local
(defthm find-variables-1-pass-is-subset
(implies (and (consistent-equation-record-p variables equations)
(subset vars variables)
(uniquep variables))
(subset (find-all-variables-1-pass vars equations)
variables)))
)
(local
(defthm memberp-union-reduction-1
(implies (memberp e x)
(memberp e (set-union y x)))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm memberp-find-all-variables-reduction
(implies (and (consistent-equation-record-p variables equations)
(subset vars variables)
(memberp v vars))
(memberp v (find-all-variables vars variables equations)))
:otf-flg t
:hints (("Goal"
:induct (find-all-variables vars variables equations)
:do-not '(eliminate-destructors generalize)
:do-not-induct t)))
)
(local
(defthm find-all-variables-produces-subset
(implies (and (consistent-equation-record-p variables equations)
(subset vars variables)
(subset vars-prime vars))
(subset vars-prime (find-all-variables vars variables equations))))
)
(local
(defthm set-intersect-is-subset
(implies (and (subset vars variables)
(subset vars vars-prime))
(subset vars (set-intersect vars-prime variables))))
)
(local
(defthm memberp-remove-reduction
(equal (memberp e (remove-duplicate-occurrences variables))
(memberp e variables)))
)
(local
(defthm remove-duplicates-is-subset
(implies (subset vars variables)
(subset vars (remove-duplicate-occurrences variables))))
)
(local
(defthm cone-variables-are-subset
(implies (and (consistent-equation-record-p variables equations)
(subset vars variables))
(subset vars (find-all-variables
(set-intersect
(remove-duplicate-occurrences vars)
variables)
variables equations)))
:hints (("Goal"
:do-not-induct t
:in-theory (disable find-all-variables-produces-subset)
:use ((:instance find-all-variables-produces-subset
(vars-prime vars)
(vars (set-intersect
(remove-duplicate-occurrences vars)
variables)))))))
)
(local
(defthm circuitp-to-cone-variables
(implies (and (circuitp C)
(subset vars (variables C)))
(subset vars (cone-variables vars C))))
)
(local
(in-theory (disable circuitp cone-variables cone-of-influence-reduction))
)
(defthm cone-of-influence-reduction-for-specific
(implies (and (circuitp C)
(ltl-formulap f)
(subset (create-restricted-var-set f)
(variables C)))
(equal (ltl-semantics-for-circuit (cone-of-influence-reduction
C (create-restricted-var-set
f))
f)
(ltl-semantics-for-circuit C f)))
:hints (("Goal"
:do-not-induct t
:in-theory (disable cone-of-influence-reduction-is-sound-generalized)
:use ((:instance cone-of-influence-reduction-is-sound-generalized
(interesting-vars (create-restricted-var-set f))
(vars (create-restricted-var-set f)))))))
)
)
(local
(in-theory (disable ltl-semantics-for-circuit create-restricted-var-set
cone-of-influence-reduction
circuitp ltl-formulap))
)
(local
(defthm reduce-problem-cone-reduction
(implies (and (circuitp C)
(ltl-formulap f)
(subset (create-restricted-var-set f) (variables C)))
(equal (ltl-semantics-for-circuit (reduce-problem-cone f C)
f)
(ltl-semantics-for-circuit C f))))
)
(local
(in-theory (disable reduce-problem-cone))
)
(local
(defun well-formed-problems-p (list)
(if (endp list) T
(and (ltl-formulap (first (first list)))
(circuitp (second (first list)))
(subset (create-restricted-var-set (first (first list)))
(variables (second (first list))))
(well-formed-problems-p (rest list)))))
)
(local
(defthm reduce-problem-cone*-reduction
(implies (well-formed-problems-p list)
(equal (ltl-semantics-for-circuits* (reduce-problem-cone* list))
(ltl-semantics-for-circuits* list)))
:otf-flg t
:hints (("Goal"
:in-theory (enable reduce-problem-cone*)
:do-not '(eliminate-destructors generalize))))
)
(local
(defthm subset-member-reduction
(implies (and (subset (set-union x y) z)
(memberp e x))
(memberp e z))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm subset-member-reduction-2
(implies (and (subset (set-union x y) z)
(memberp e y))
(memberp e z))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm set-union-subset-reduction
(implies (subset (set-union x y) z)
(subset x z))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm set-union-subset-reduction-2
(implies (subset (set-union x y) z)
(subset y z))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm conjunction-has-variables-subset-1
(implies (and (ltl-formulap f)
(equal (len f) 3)
(subset (create-restricted-var-set f) vars))
(subset (create-restricted-var-set (first f)) vars))
:hints (("Goal"
:do-not '(eliminate-destructors generalize)
:do-not-induct t
:in-theory (enable create-restricted-var-set ltl-formulap)
:expand (create-restricted-var-set f))))
)
(local
(defthm conjunction-has-variables-subset-2
(implies (and (ltl-formulap f)
(equal (len f) 3)
(subset (create-restricted-var-set f) vars))
(subset (create-restricted-var-set (third f)) vars))
:hints (("Goal"
:do-not '(eliminate-destructors generalize)
:do-not-induct t
:in-theory (enable create-restricted-var-set ltl-formulap)
:expand (create-restricted-var-set f))))
)
(local
(defthm well-formed-append-reduction
(implies (and (force (well-formed-problems-p first))
(force (well-formed-problems-p second)))
(well-formed-problems-p (append first second))))
)
(local
(defthm conjunction-produces-well-formed-problems
(implies (and (circuitp C)
(ltl-formulap f)
(subset (create-restricted-var-set f) (variables C)))
(well-formed-problems-p (reduce-problem-conjunction f C)))
:hints (("Goal"
:do-not-induct t
:do-not '(eliminate-destructors generalize)
:induct (reduce-problem-conjunction f C))))
)
(DEFTHM compositional-reduction-is-sound
(implies (and (circuitp C)
(ltl-formulap f)
(subset (create-restricted-var-set f) (variables C)))
(equal (ltl-semantics-for-circuits* (compositional-reduction C f))
(ltl-semantics-for-circuit C f))))
|
