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|
(in-package "ACL2")
#|
circuits.lisp
~~~~~~~~~~~~~
In this book, we discuss a procedure to construct Kripke Structures from
"circuit descriptions. A circuit in our world is a collection of variables, a
collection of equations, and a collection of equations. An equation is a
boolean evaluator of the current circuit valuaes producing the next state
function. We show that under certain "well-formed-ness constraints", our
procedure produces a valid model, in terms of the circuit-modelp predicate
defined earlier.
|#
(include-book "circuit-bisim")
;; A circuit is a collection of variables, equations and initial states. We
;; will add equations to the macros, and tell you what is a good circuit.
(defmacro equations (c) `(<- ,c :equations))
;; Now we define what it means for the equations to be consistent with the
;; variables of the circuit.
(defun find-variables (equation)
(cond ((and (atom equation) (not (booleanp equation)))
(list equation))
((and (equal (len equation) 3) (memberp (second equation) '(& +)))
(set-union (find-variables (first equation))
(find-variables (third equation))))
((and (equal (len equation) 2) (equal (first equation) '~))
(find-variables (second equation)))
(t nil)))
(defun-sk consistent-equation-record-p (vars equations)
(forall (v equation)
(implies (and (uniquep vars)
(memberp v vars)
(memberp equation (<- equations v)))
(subset (find-variables equation) vars))))
(defun cons-list-p (vars equations)
(if (endp vars) T
(and (consp (<- equations (first vars)))
(cons-list-p (rest vars) equations))))
;; OK, now let us define the function circuitp.
(defun circuitp (C)
(and (only-evaluations-p (initial-states C) (variables C))
(strict-evaluation-list-p (variables C) (initial-states C))
(uniquep (variables C))
(cons-list-p (variables C) (equations C))
(consistent-equation-record-p (variables C) (equations C))))
;; Now let us try to create a Kripke Structure from the circuit. We need to
;; show that under (circuitp C), the kripke structure we produce is a
;; circuit-model-p.
(defun assign-T (v states)
(if (endp states) nil
(cons (-> (first states) v T)
(assign-T v (rest states)))))
(defun assign-nil (v states)
(if (endp states) nil
(cons (-> (first states) v nil)
(assign-nil v (rest states)))))
;; Now we create all the states of the model.
(defun create-all-evaluations (vars states)
(if (endp vars) states
(let ((rec-states (create-all-evaluations (cdr vars) states)))
(append (assign-t (car vars) rec-states)
(assign-nil (car vars) rec-states)))))
;; Now let us create the label function.
(defun label-fn-of-st (st vars)
(if (endp vars) nil
(if (equal (<- st (first vars)) T)
(cons (first vars)
(label-fn-of-st st (rest vars)))
(label-fn-of-st st (rest vars)))))
(defun create-label-fn (states vars label)
(if (endp states) label
(create-label-fn (rest states) vars
(-> label (first states)
(label-fn-of-st (first states) vars)))))
;; And finally the transitions.
(defun apply-equation (equation st)
(cond ((atom equation) (if (booleanp equation)
equation
(<- st equation)))
((equal (len equation) 2)
(case (first equation)
(~ (not (apply-equation (second equation) st)))
(t nil)))
((equal (len equation) 3)
(case (second equation)
(& (and (apply-equation (first equation) st)
(apply-equation (third equation) st)))
(+ (or (apply-equation (first equation) st)
(apply-equation (third equation) st)))
(t nil)))
(t nil)))
(defun produce-next-state (vars st equations)
(if (endp vars) st
(-> (produce-next-state (rest vars) st equations)
(first vars)
(apply-equation (<- equations (first vars)) st))))
(defun consistent-p-equations (vars eqn equations)
(if (endp vars) T
(and (memberp (<- eqn (first vars)) (<- equations (first vars)))
(consistent-p-equations (rest vars) eqn equations))))
(defun-sk next-state-is-ok (p q vars equations)
(exists eqn (and (consistent-p-equations vars eqn equations)
(evaluation-eq q (produce-next-state vars p eqn) vars))))
(defun create-next-states-of-p (p states vars equations)
(if (endp states) nil
(if (next-state-is-ok p (first states) vars equations)
(cons (first states) (create-next-states-of-p
p (rest states) vars equations))
(create-next-states-of-p p (rest states) vars equations))))
(defun create-next-states (states states-prime vars equations)
(if (endp states) ()
(->
(create-next-states (rest states) states-prime vars equations)
(first states)
(create-next-states-of-p (first states) states-prime vars equations))))
(defun create-kripke (C)
(let ((vars (variables C))
(equations (equations C))
(initial-states (initial-states C)))
(let* ((states (create-all-evaluations vars (list ())))
(label-fn (create-label-fn (set-union initial-states states) vars ()))
(transition (create-next-states (set-union initial-states states)
(set-union initial-states states)
vars equations)))
(>_ :states (set-union initial-states states)
:initial-states initial-states
:label-fn label-fn
:transition transition
:variables vars))))
;; Since I have defined the Kripke model for a circuit, let us prove that it is
;; a circuit-model-p.
;; We start with the initial states.
;; The theorem that initial-states are subsets of states is trivial by
;; union. So there is nothing to prove.
(local
(defthm initial-states-are-subset-of-states
(subset (initial-states (create-kripke C)) (states (create-kripke C))))
)
;; END of proofs on initial-states.
;; OK, let us prove that create-label-fn is a valid label function.
(local
(defthm label-fn-is-subset
(subset (label-fn-of-st st vars) vars))
)
(local
(defthm label-fn-of-st-is-truth-p-label
(truthp-label (label-fn-of-st st vars) st))
)
(local
(defthm label-fn-of-st-is-all-truths-p-label
(all-truthsp-label (label-fn-of-st st vars) st vars))
)
(local
(defun abs-only-all-truths-p (states label vars)
(if (endp states) T
(and (all-truthsp-label (<- label (first states)) (first states) vars)
(abs-only-all-truths-p (rest states) label vars))))
)
(local
(defthm abs-concrete-only-all-truthsp-reduction
(equal (only-all-truths-p states m vars)
(abs-only-all-truths-p states (label-fn m) vars))
:hints (("Goal"
:in-theory (enable label-of))))
)
;; And now let us just prove abs-all-truthsp-label for the label-fn
(local
(defthm create-label-fn-does-not-mess-with-non-members
(implies (not (memberp s states))
(equal (<- (create-label-fn states vars label) s)
(<- label s))))
)
(local
(defthm create-label-fn-creates-an-all-truthsp-label
(implies (memberp s states)
(equal (<- (create-label-fn states vars label) s)
(label-fn-of-st s vars))))
)
(local
(defthm label-fn-is-abs-only--all-truthsp
(abs-only-all-truths-p states (create-label-fn states vars label) vars)
:hints (("Subgoal *1/3"
:cases ((memberp (car states) (cdr states)))
:do-not-induct t)))
)
(local
(defthm label-fn-is-only-all-truthsp
(only-all-truths-p (states (create-kripke C)) (create-kripke C)
(variables C)))
)
(local
(in-theory (disable abs-concrete-only-all-truthsp-reduction))
)
(local
(defun abs-label-subset-vars (states label vars)
(if (endp states) T
(and (subset (<- label (first states)) vars)
(abs-label-subset-vars (rest states) label vars))))
)
(local
(defthm abs-label-subset-vars-is-same-as-concrete
(equal (label-subset-vars states m vars)
(abs-label-subset-vars states (label-fn m) vars))
:hints (("Goal"
:in-theory (enable label-of))))
)
(local
(defthm create-label-fn-is-abs-label-subset-vars
(abs-label-subset-vars states (create-label-fn states vars label) vars)
:hints (("Subgoal *1/3"
:cases ((memberp (car states) (cdr states)))
:do-not-induct t)))
)
(local
(defthm label-fn-is-label-subset-vars
(label-subset-vars (states (create-kripke C)) (create-kripke C) (variables
C)))
)
(local
(in-theory (disable abs-label-subset-vars-is-same-as-concrete))
)
(local
(defun abs-only-truth-p (states label)
(if (endp states) T
(and (truthp-label (<- label (first states)) (first states))
(abs-only-truth-p (rest states) label))))
)
(local
(defthm only-truth-p-abs-reduction
(equal (only-truth-p states m)
(abs-only-truth-p states (label-fn m)))
:hints (("Goal"
:in-theory (enable label-of))))
)
(local
(defthm label-fn-is-abs-only-truth-p
(abs-only-truth-p states (create-label-fn states vars label))
:hints (("Subgoal *1/3"
:cases ((memberp (car states) (cdr states))))))
)
(local
(defthm label-fn-is-only-truth-p
(only-truth-p (states (create-kripke C)) (create-kripke C)))
)
(local
(in-theory (disable only-truth-p-abs-reduction))
)
;; END of proofs for label function.
;; Let us now work with the transition function.
(local
(defthm create-next-states-is-subset-of-states-aux
(implies (memberp q (create-next-states-of-p p states vars equations))
(memberp q states)))
)
(local
(defthm create-next-states-of-p-subset-helper
(implies (subset states-prime (create-next-states-of-p p states vars
equations))
(subset states-prime states)))
)
(local
(defthm create-next-states-is-subset-of-states
(subset (create-next-states-of-p p states vars equations)
states)
:hints (("Goal"
:use ((:instance create-next-states-of-p-subset-helper
(states-prime (create-next-states-of-p p states
vars equations)))))))
)
(local
(defthm not-memberp-next-states-reduction
(implies (not (memberp s states))
(equal (<- (create-next-states states states-prime vars equations)
s)
nil)))
)
(local
(defthm memberp-next-state-reduction
(implies (memberp s states)
(equal (<- (create-next-states states states-prime vars equations)
s)
(create-next-states-of-p s states-prime vars equations)))
:hints (("Subgoal *1/3"
:cases ((equal s (car states))))))
)
(local
(defthm transition-subset-p-for-next-state
(transition-subset-p states states-prime
(create-next-states states states-prime vars equations))
:hints (("Subgoal *1/2"
:cases ((memberp (car states) (cdr states))))))
)
(local
(defthm transition-subset-p-holds-for-kripke
(transition-subset-p (states (create-kripke C))
(states (create-kripke C))
(transition (create-kripke C))))
)
(local
(defthm next-states-in-states-concretized
(equal (next-states-in-states m states)
(transition-subset-p states (states m) (transition m)))
:hints (("Goal"
:in-theory (enable next-states-in-states))))
)
(local
(defthm next-states-in-states-holds-for-create-kripke
(next-states-in-states (create-kripke C) (states (create-kripke C))))
)
;; END of proofs for transition function.
;; BEGIN proofs for states
;; first states is a consp
(local
(defthm consp-states-for-consp-vars
(implies (consp states)
(consp (create-all-evaluations vars states))))
)
;; The following theorem is a hack. This theorem is known as a
;; type-prescription rule for append. Unfortunately, we need it as a rewrite
;; rule.
(local
(in-theory (enable set-union))
)
(local
(defthm consp-union-reduction
(implies (consp y)
(consp (set-union x y))))
)
(local
(defthm create-kripke-is-consp-states
(consp (states (create-kripke C))))
)
;; OK let us prove that everything is boolean with create-all-evaluations
(local
(defthm only-evaluations-p-union-reduction
(implies (and (only-evaluations-p init vars)
(only-evaluations-p states vars))
(only-evaluations-p (set-union init states) vars)))
)
;; OK that takes care of the set-union part. Now we only need to show the
;; create-all-evaluations produces only-evaluations-p
(local
(defun boolean-p-states (v states)
(if (endp states) T
(and (booleanp (<- (first states) v))
(boolean-p-states v (rest states)))))
)
(local
(defun boolean-list-p-states (vars states)
(if (endp vars) T
(and (boolean-p-states (first vars) states)
(boolean-list-p-states (rest vars) states))))
)
;; Now can we prove that boolean-p-states holds for create-all-evaluations?
(local
(defthm assign-t-produces-boolean-p
(boolean-p-states v (assign-T v states)))
)
(local
(defthm assign-nil-produces-boolean-p
(boolean-p-states v (assign-nil v states)))
)
(local
(defthm assign-T-remains-same-for-not-v
(implies (not (equal v v-prime))
(equal (boolean-p-states v (assign-T v-prime states))
(boolean-p-states v states))))
)
(local
(defthm assign-nil-remains-same-for-not-v
(implies (not (equal v v-prime))
(equal (boolean-p-states v (assign-nil v-prime states))
(boolean-p-states v states))))
)
(local
(defthm boolean-p-append-reduction
(equal (boolean-p-states v (append states states-prime))
(and (boolean-p-states v states)
(boolean-p-states v states-prime))))
)
(local
(defthm boolean-p-create-non-member-reduction
(implies (not (memberp v vars))
(equal (boolean-p-states v (create-all-evaluations vars states))
(boolean-p-states v states)))
:hints (("Goal"
:induct (create-all-evaluations vars states)
:do-not-induct t)))
)
(local
(defthm create-all-evaluations-for-member-is-boolean
(implies (memberp v vars)
(boolean-p-states v (create-all-evaluations vars states)))
:hints (("Goal"
:induct (create-all-evaluations vars states)
:do-not-induct t)
("Subgoal *1/2"
:cases ((equal v (car vars))))))
)
(local
(defthm create-all-evaluations-is-boolean-list-p-aux
(implies (subset vars vars-prime)
(boolean-list-p-states vars
(create-all-evaluations vars-prime states))))
)
(local
(defthm create-all-evaluations-is-boolean-list-p
(boolean-list-p-states vars (create-all-evaluations vars states)))
)
;; Can we prove that if we produce a boolean list then it is an evaluation?
(local
(defun evaluation-witness-variable (vars st)
(if (endp vars) nil
(if (not (booleanp (<- st (first vars))))
(first vars)
(evaluation-witness-variable (rest vars) st))))
)
(local
(defthm evaluation-p-from-witness
(implies (booleanp (<- st (evaluation-witness-variable vars st)))
(evaluation-p st vars)))
)
(local
(defthm boolean-list-p-to-boolean-vars
(implies (and (boolean-list-p-states vars states)
(memberp v vars))
(boolean-p-states v states)))
)
(local
(defthm boolean-p-states-implies-boolean-v
(implies (and (boolean-p-states v states)
(memberp st states))
(booleanp (<- st v))))
)
(local
(defthm boolean-p-states-to-evaluation-p
(implies (and (boolean-list-p-states vars states)
(memberp st states))
(evaluation-p st vars)))
)
(local
(defthm boolean-p-states-to-only-evaluation-p-aux
(implies (and (boolean-list-p-states vars states)
(subset states-prime states))
(only-evaluations-p states-prime vars)))
)
(local
(defthm boolean-p-states-to-only-evaluations-p
(implies (boolean-list-p-states vars states)
(only-evaluations-p states vars)))
)
(local
(defthm create-all-evaluations-is-only-evaluations-p
(only-evaluations-p (create-all-evaluations vars states) vars))
)
(local
(defthm create-kripke-is-only-evaluations-p
(implies (circuitp C)
(only-evaluations-p (states (create-kripke C)) (variables C))))
)
;; The final predicate is all-evaluations-p. This is tricky, since it is
;; defined using defun-sk. We try to create a witness for all-evaluations-p.
(local
(defun find-matching-states (st vars states)
(cond ((endp vars) states)
((equal (<- st (first vars)) T)
(assign-t (first vars)
(find-matching-states st (rest vars) states)))
(t (assign-nil (first vars)
(find-matching-states st (rest vars) states)))))
)
;; Let us first prove find-matching-states is a consp
(local
(defthm find-matching-states-is-consp
(implies (consp states)
(consp (find-matching-states st vars states))))
)
;; Now let us prove that for every member of find-matching-states it is
;; evaluation-eq to st.
(local
(defthm nth-member-reduction
(implies (and (< i (len x))
(consp x))
(memberp (nth i x) x)))
)
(local
(defthm nth-member-reduction-2
(implies (and (>= i (len x))
(integerp i))
(equal (nth i x) nil))
:hints (("Goal"
:in-theory (enable zp))))
)
(local
(defthm assign-nil-produces-nil-member
(implies (memberp q (assign-nil v states))
(equal (<- q v) nil)))
)
(local
(defthm assign-t-produces-t-member
(implies (memberp q (assign-t v states))
(equal (<- q v) t)))
)
(local
(defthm assign-nil-produces-nil
(implies (and (consp states)
(integerp i))
(not (<- (nth i (assign-nil v states)) v)))
:otf-flg t
:hints (("Goal"
:cases ((>= i (len (assign-nil v states))))
:do-not-induct t)
("Subgoal 2"
:in-theory (disable nth-member-reduction)
:use ((:instance nth-member-reduction
(x (assign-nil v states)))))))
)
(local
(defthm assign-t-has-same-len
(equal (len (assign-t v states))
(len states)))
)
(local
(defthm assign-nil-has-same-len
(equal (len (assign-nil v states))
(len states)))
)
(local
(defthm len-consp-reduction
(implies (and (equal (len x) (len y))
(consp x))
(consp y)))
)
(local
(defthm assign-t-produces-t
(implies (and (consp states)
(< i (len states))
(integerp i))
(equal (<- (nth i (assign-t v states)) v) t))
:otf-flg t
:hints (("Goal"
:in-theory (disable nth-member-reduction)
:use ((:instance nth-member-reduction
(x (assign-t v states)))))))
)
(local
(defthm assign-t-does-not-fuss
(implies (and (consp states)
(< i (len states))
(integerp i)
(not (equal v v-prime)))
(equal (<- (nth i (assign-t v states)) v-prime)
(<- (nth i states) v-prime))))
)
(local
(defthm assign-nil-does-not-fuss
(implies (and (consp states)
(< i (len states))
(integerp i)
(not (equal v v-prime)))
(equal (<- (nth i (assign-nil v states)) v-prime)
(<- (nth i states) v-prime))))
)
(local
(defthm len-of-find-matching-states-is-same
(equal (len (find-matching-states st vars states))
(len states)))
)
(local
(defthm find-matching-state-produces-equivalent-assignment
(implies (and (memberp v vars)
(consp states)
(integerp i)
(< i (len states))
(evaluation-p st vars))
(equal (<- (nth i (find-matching-states st vars states)) v)
(<- st v)))
:otf-flg t
:hints (("Goal"
:induct (find-matching-states st vars states)
:do-not '(eliminate-destructors generalize)
:do-not-induct t)
("Subgoal *1/3.1"
:cases ((equal v (car vars))))
("Subgoal *1/2.1"
:cases ((equal v (car vars))))))
)
(local
(defun falsifier-evaluation-eq (p q vars)
(if (endp vars) nil
(if (not (equal (<- p (first vars))
(<- q (first vars))))
(first vars)
(falsifier-evaluation-eq p q (rest vars)))))
)
(local
(defthm falsifier-means-evaluation-eq
(implies (equal (<- p (falsifier-evaluation-eq p q vars))
(<- q (falsifier-evaluation-eq p q vars)))
(evaluation-eq p q vars)))
)
(local
(defthm falsifier-not-member-to-evaluation-eq
(implies (not (memberp (falsifier-evaluation-eq p q vars) vars))
(evaluation-eq p q vars)))
)
(local
(defthm find-matching-states-evaluation-eq
(implies (and (consp states)
(integerp i)
(< i (len states))
(evaluation-p st vars))
(evaluation-eq (nth i (find-matching-states st vars states))
st vars))
:hints (("Goal"
:cases ((not (memberp
(falsifier-evaluation-eq
(nth i (find-matching-states st vars states))
st vars)
vars))))))
)
(local
(defthm find-matching-is-evaluation-eq-concretized
(implies (and (consp states)
(evaluation-p st vars))
(evaluation-eq (car (find-matching-states st vars states))
st vars))
:hints (("Goal"
:in-theory (disable find-matching-states-evaluation-eq)
:use ((:instance find-matching-states-evaluation-eq
(i 0))))))
)
(local
(defthm memberp-append-reduction
(equal (memberp a (append x y))
(or (memberp a x)
(memberp a y))))
)
(local
(defthm member-assign-t-reduction
(implies (memberp e x)
(memberp (-> e v t)
(assign-t v x))))
)
(local
(defthm assign-t-subset-reduction
(implies (subset x y)
(subset (assign-t v x)
(assign-t v y))))
)
(local
(defthm member-assign-nil-reduction
(implies (memberp e x)
(memberp (-> e v nil)
(assign-nil v x))))
)
(local
(defthm assign-nil-subset-reduction
(implies (subset x y)
(subset (assign-nil v x)
(assign-nil v y))))
)
(local
(defthm append-subset-reduction-1
(implies (subset x y)
(subset x (append y z))))
)
(local
(defthm append-subset-reduction-2
(implies (subset x y)
(subset x (append z y))))
)
(local
(defthm find-matching-subset-reduction
(subset (find-matching-states st vars states)
(create-all-evaluations vars states)))
)
(local
(defthm car-of-find-matching-is-member-of-all-evaluations
(implies (consp states)
(memberp (car (find-matching-states st vars states))
(create-all-evaluations vars states))))
)
(local
(defthm evaluation-eq-memberp-from-memberp
(implies (and (evaluation-eq p q vars)
(memberp q states))
(evaluation-eq-member-p p states vars)))
)
(local
(defthm evalaution-eq-symmetry-hack
(implies (and (evaluation-eq p q vars)
(memberp p states))
(evaluation-eq-member-p q states vars))
:hints (("Goal"
:in-theory (disable evaluation-eq evaluation-eq-member-p
evaluation-eq-memberp-from-memberp)
:use ((:instance evaluation-eq-memberp-from-memberp
(p q)
(q p))
(:instance evaluation-eq-is-symmetric)))))
)
(local
(in-theory (disable evaluation-eq-memberp-from-memberp))
)
(local
(defthm create-all-evaluations-is-evaluation-eq-memberp
(implies (and (evaluation-p st vars)
(consp states))
(evaluation-eq-member-p st (create-all-evaluations vars states)
vars))
:hints (("Goal"
:do-not '(eliminate-destructors generalize)
:do-not-induct t
:in-theory (disable evalaution-eq-symmetry-hack)
:use ((:instance evalaution-eq-symmetry-hack
(q st)
(states (create-all-evaluations vars states))
(p (car (find-matching-states st vars
states))))))))
)
(local
(defthm consp-states-to-all-evaluations-p
(implies (consp states)
(all-evaluations-p (create-all-evaluations vars states) vars))
:hints (("Goal"
:use ((:instance (:definition all-evaluations-p)
(states (create-all-evaluations vars states)))))))
)
(local
(defthm append-evaluation-eq-member-reduction
(implies (evaluation-eq-member-p st states vars)
(evaluation-eq-member-p st (set-union init states) vars)))
)
(local
(defthm all-evaluations-p-union-reduction
(implies (all-evaluations-p states vars)
(all-evaluations-p (set-union init states) vars))
:hints (("Goal"
:use ((:instance all-evaluations-p-necc)
(:instance (:definition all-evaluations-p)
(states (set-union init states)))))))
)
(local
(defthm create-kripke-is-all-evaluations-p
(all-evaluations-p (states (create-kripke C))
(variables c)))
)
(local
(defthm variables-of-create-kripke-are-original-vars
(equal (variables (create-kripke C))
(variables C)))
)
(local
(defthm strict-evaluations-list-to-evaluation
(implies (and (strict-evaluation-list-p vars states)
(memberp st states))
(strict-evaluation-p st vars)))
)
(local
(defthm strict-evaluations-append-reduction
(implies (and (strict-evaluation-list-p vars states)
(strict-evaluation-list-p vars states-prime))
(strict-evaluation-list-p vars (append states states-prime))))
)
(local
(defthm strict-evaluation-list-p-nth-reduction
(implies (and (strict-evaluation-list-p vars states)
(integerp i)
(< i (len states))
(consp states))
(strict-evaluation-p (nth i states) vars)))
)
(local
(defthm assign-t-strict-evaluations-reduction
(implies (and (strict-evaluation-list-p vars states)
(memberp v vars)
(consp states)
(integerp i)
(< i (len states))
(not (memberp v-prime vars)))
(not (<- (nth i (assign-t v states)) v-prime)))
:hints (("Goal"
:do-not-induct t
:in-theory (disable assign-t-does-not-fuss)
:use ((:instance assign-t-does-not-fuss)
(:instance strict-evaluation-p-necc
(v v-prime)
(st (nth i states)))))))
)
(local
(defthm assign-nil-strict-evaluations-reduction
(implies (and (strict-evaluation-list-p vars states)
(memberp v vars)
(consp states)
(integerp i)
(< i (len states))
(not (memberp v-prime vars)))
(not (<- (nth i (assign-nil v states)) v-prime)))
:hints (("Goal"
:do-not-induct t
:in-theory (disable assign-nil-does-not-fuss)
:use ((:instance assign-nil-does-not-fuss)
(:instance strict-evaluation-p-necc
(v v-prime)
(st (nth i states)))))))
)
(local
(defthm strict-evaluations-assign-t-reduction
(implies (and (integerp i)
(consp states)
(strict-evaluation-list-p vars states)
(memberp v vars)
(< i (len states)))
(strict-evaluation-p (nth i (assign-t v states)) vars)))
)
(local
(defthm strict-evaluations-assign-nil-reduction
(implies (and (integerp i)
(consp states)
(strict-evaluation-list-p vars states)
(memberp v vars)
(< i (len states)))
(strict-evaluation-p (nth i (assign-nil v states)) vars)))
)
(local
(defun find-index (st states)
(if (endp states) 0
(if (equal st (first states)) 0
(1+ (find-index st (rest states))))))
)
(local
(defthm find-index-is-memberp
(implies (memberp st states)
(equal (nth (find-index st states) states)
st)))
)
(local
(defthm find-index-returns-<-len
(implies (memberp st states)
(< (find-index st states) (len states)))
:rule-classes :linear)
)
(local
(defthm strict-evaluation-for-memberp-assign-t
(implies (and (consp states)
(strict-evaluation-list-p vars states)
(memberp v vars)
(memberp st (assign-t v states)))
(strict-evaluation-p st vars))
:hints (("Goal"
:do-not-induct t
:in-theory (disable assign-t-strict-evaluations-reduction
strict-evaluation-p)
:use ((:instance strict-evaluations-assign-t-reduction
(i (find-index st (assign-t v states))))))))
)
(local
(defthm strict-evaluation-for-memberp-assign-nil
(implies (and (consp states)
(strict-evaluation-list-p vars states)
(memberp v vars)
(memberp st (assign-nil v states)))
(strict-evaluation-p st vars))
:hints (("Goal"
:do-not-induct t
:in-theory (disable assign-nil-strict-evaluations-reduction
strict-evaluation-p)
:use ((:instance strict-evaluations-assign-nil-reduction
(i (find-index st (assign-nil v states))))))))
)
(local
(in-theory (disable strict-evaluation-p))
)
(local
(defthm strict-evaluations-for-assign-t
(implies (and (consp states)
(strict-evaluation-list-p vars states)
(memberp v vars))
(strict-evaluation-list-p vars (assign-t v states))))
)
(local
(defthm strict-evaluations-for-assign-nil
(implies (and (consp states)
(strict-evaluation-list-p vars states)
(memberp v vars))
(strict-evaluation-list-p vars (assign-nil v states))))
)
(local
(defun null-list-p (states)
(if (endp states) T
(and (null (first states))
(null-list-p (rest states)))))
)
(local
(defthm strict-evaluation-p-cons-reduction
(implies (strict-evaluation-p st vars)
(strict-evaluation-p (-> st v t) (cons v vars)))
:hints (("Goal"
:expand (strict-evaluation-p (-> st v t) (cons v vars)))))
)
(local
(defthm strict-evaluation-p-cons-reduction-2
(implies (strict-evaluation-p st vars)
(strict-evaluation-p (-> st v nil) (cons v vars)))
:hints (("Goal"
:expand (strict-evaluation-p (-> st v nil) (cons v vars)))))
)
(local
(defthm strict-evaluation-p-assign-reduction-t
(implies (strict-evaluation-list-p vars states)
(strict-evaluation-list-p (cons v vars) (assign-t v states))))
)
(local
(defthm strict-evaluation-p-assign-reduction-nil
(implies (strict-evaluation-list-p vars states)
(strict-evaluation-list-p (cons v vars) (assign-nil v states))))
)
(local
(defthm nil-is-strict-evaluation-p
(strict-evaluation-p nil vars)
:hints (("Goal"
:in-theory (enable strict-evaluation-p))))
)
(local
(defthm null-list-p-is-strict-evaluation-p
(implies (null-list-p states)
(strict-evaluation-list-p vars states)))
)
(local
(defthm create-evaluations-is-strict-evaluation-list-p
(implies (and (consp states)
(null-list-p states)
(uniquep vars))
(strict-evaluation-list-p
vars (create-all-evaluations vars states)))
:otf-flg t
:hints (("Goal"
:induct (create-all-evaluations vars states)
:do-not '(eliminate-destructors generalize)
:do-not-induct t)
("Subgoal *1/2"
:in-theory (disable strict-evaluation-p-assign-reduction-t
strict-evaluation-p-assign-reduction-nil)
:use ((:instance strict-evaluation-p-assign-reduction-t
(states (create-all-evaluations (cdr vars) states))
(vars (cdr vars))
(v (car vars)))
(:instance strict-evaluation-p-assign-reduction-nil
(states (create-all-evaluations (cdr vars) states))
(vars (cdr vars))
(v (car vars)))))))
)
(local
(defthm strict-evaluation-set-union-reduction
(implies (and (strict-evaluation-list-p vars init)
(strict-evaluation-list-p vars states))
(strict-evaluation-list-p vars (set-union init states)))
:hints (("Goal"
:in-theory (enable set-union))))
)
(local
(defthm strict-evaluation-list-p-holds
(implies (circuitp C)
(strict-evaluation-list-p (variables C) (states (create-kripke C)))))
)
(local
(in-theory (disable create-kripke))
)
(DEFTHM create-kripke-produces-circuit-model
(implies (circuitp C)
(circuit-modelp (create-kripke C))))
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