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(in-package "ACL2")
;; Soundness of the skolem step functions step-sk/step-ex,
;; which operate on the leftmost existential quantifier.
;; Recall that step-sk does a skolemization, and step-ex
;; does the corresponding extension of an interpretation.
(include-book "sk-useless")
(include-book "sk-step")
;;-------------------------
(defthm not-validator-not-feval-d
(implies (and (domain-term-list (fringe dom))
(not (validator f x dom i)))
(not (feval-d (list 'exists x f) dom i))))
(defthm not-validator-not-feval
(implies (and (not (validator f x dom i))
(subsetp-equal (fringe dom) (fringe (domain i)))
(member-equal e (fringe dom)))
(not (feval (subst-free f x e) i))))
(defthm not-validator-not-feval-2
(implies (and (not (validator f x (domain i) i))
(member-equal e (fringe (domain i))))
(not (feval (subst-free f x e) i))))
(defthm validator-feval
(implies (validator f x dom i)
(feval (subst-free f x (validator f x dom i)) i)))
(defthm validator-feval-exists
(implies (and (variable-term x)
(validator f x dom i))
(feval-d (list 'exists x f) dom i)))
;;------------------------------------
;; not validator case
(encapsulate
nil
(local (include-book "instance"))
(defthm ground-term-eval ;; redundant event from instance book
(implies (and (variable-term x)
(domain-term-list (fringe dom))
(not (vars-in-term-list (list tm))))
(equal (xeval (subst-free f x (eval-term tm i)) dom i)
(xeval (subst-free f x tm) dom i))))
)
(defthm ground-term-feval
(implies (and (variable-term x)
(not (vars-in-term-list (list tm))))
(equal (feval (subst-free f x (eval-term tm i)) i)
(feval (subst-free f x tm) i)))
:hints (("Goal"
:in-theory (enable xeval-feval))))
(defthm not-validator-case
(implies (and (variable-term x)
(domain-term-list vals)
(not (validator f x (domain i) i))
(not (member-equal fsym (funcs-in-formula f))))
(not (feval (subst-free f x (cons fsym vals))
(list* (domain i)
(relations i)
(list (cons fsym (len vals))
(cons vals 0))
(functions i)))))
:hints (("Goal"
:do-not generalize
:in-theory (disable eval-term-list ground-term-feval)
:use ((:instance ground-term-feval
(tm (cons fsym vals))
(i (list* (domain i)
(relations i)
(list (cons fsym (len vals))
(cons vals 0))
(functions i)))
))))
:otf-flg t)
;;------------------------------------
;; validator case
(defthm validator-in-domain
(implies (validator f x dom i)
(member-equal (validator f x dom i) (fringe dom))))
(defthm validator-is-domain-term
(implies (and (validator f x dom i)
(domain-term-list (fringe dom)))
(domain-term (validator f x dom i))))
(defthm eval-term-with-inserted-tuple
(implies (and (domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(function-symbol fsym)
(member-equal val (fringe (domain i))))
(equal (car (eval-term-list
(list (cons fsym vals))
(list* (domain i)
(relations i)
(list (cons fsym (len vals))
(cons vals val))
(functions i))))
val))
:hints (("Goal"
:in-theory (enable fapply domain))))
(defthm validator-case
(implies (and (variable-term x)
(function-symbol fsym)
(not (member-equal x (quantified-vars f)))
(setp (quantified-vars f))
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f)))
(validator f x (domain i) i))
(equal (feval (subst-free f x (cons fsym vals))
(list* (domain i)
(relations i)
(list (cons fsym (len vals))
(cons vals
(validator f x
(domain i)
i)))
(functions i)))
(feval-d (list 'exists x f)
(domain i)
i)))
:hints (("Goal"
:do-not-induct t
:in-theory (disable eval-term-list ground-term-feval)
:use ((:instance ground-term-feval
(tm (cons fsym vals))
(i (list* (domain i)
(relations i)
(list (cons fsym (len vals))
(cons vals (validator f x
(domain i)
i)))
(functions i)))
)))))
;;---------------------------------
;; append case
(include-book "sk-xbuild")
(defthm xeval-append-a-x
(implies
(and (variable-term f3)
(function-symbol fsym)
(step-sk-arity f5 (+ 1 (len vals)))
(not (member-equal f3 (quantified-vars f5)))
(setp (quantified-vars f5))
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f5)))
(domain-term-list (append (fringe dom1) (fringe dom2)))
(setp (append (fringe dom1) (fringe dom2)))
(subsetp-equal (append (fringe dom1) (fringe dom2))
(fringe (domain i))))
(equal (feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom2
(list* (domain i)
(relations i)
(cons (cons fsym
(step-sk-arity f5 (+ 1 (len vals))))
(append (build-sk-d (list 'all f3 f5)
vals dom1 i)
(build-sk-d (list 'all f3 f5)
vals dom2 i)))
(functions i)))
(feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom2
(list* (domain i)
(relations i)
(cons (cons fsym
(step-sk-arity f5 (+ 1 (len vals))))
(build-sk-d (list 'all f3 f5)
vals dom2 i))
(functions i)))))
:hints (("Goal"
:use ((:instance xeval-append-a
(y f3) (f f5) (dm dom1) (dom dom2)
(func2 (build-sk-d (list 'all f3 f5)
vals dom2 i))))
:in-theory (disable XEVAL-APPEND-A)
:do-not-induct t
)))
(defthm xeval-append-b-x1
(implies
(and (variable-term f3)
(function-symbol fsym)
(step-sk-arity f5 (+ 1 (len vals)))
(not (member-equal f3 (quantified-vars f5)))
(setp (quantified-vars f5))
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f5)))
(domain-term-list (append (fringe dom1) (fringe dom2)))
(setp (append (fringe dom1) (fringe dom2)))
(subsetp-equal (append (fringe dom1) (fringe dom2))
(fringe (domain i)))
(feval-d (list 'all f3 f5) dom1 i)
)
(equal (feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom1
(list* (domain i)
(relations i)
(cons (cons fsym (step-sk-arity f5 (+ 1 (len vals))))
(append (build-sk-d (list 'all f3 f5)
vals dom1 i)
(build-sk-d (list 'all f3 f5)
vals dom2 i)))
(functions i)))
(feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom1
(list* (domain i)
(relations i)
(cons (cons fsym (step-sk-arity f5 (+ 1 (len vals))))
(build-sk-d (list 'all f3 f5)
vals dom1 i))
(functions i)))))
:hints (("Goal"
:use ((:instance xeval-append-b
(y f3) (f f5) (dm dom2) (dom dom1)
(func1 (build-sk-d (list 'all f3 f5)
vals dom1 i))))
:in-theory (disable XEVAL-APPEND-B)
:do-not-induct t
)))
(defthm xeval-append-b-x2
(implies
(and (variable-term f3)
(function-symbol fsym)
(step-sk-arity f5 (+ 1 (len vals)))
(not (feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom1
(list* (domain i)
(relations i)
(cons (cons fsym
(step-sk-arity f5 (+ 1 (len vals))))
(build-sk-d (list 'all f3 f5)
vals dom1 i))
(functions i))))
(not (member-equal f3 (quantified-vars f5)))
(setp (quantified-vars f5))
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f5)))
(domain-term-list (append (fringe dom1) (fringe dom2)))
(setp (append (fringe dom1) (fringe dom2)))
(subsetp-equal (append (fringe dom1) (fringe dom2))
(fringe (domain i)))
(not (feval-d (list 'all f3 f5) dom1 i))
)
(not (feval-d (list 'all f3 (step-sk f5 (append vals (list f3)) fsym))
dom1
(list* (domain i)
(relations i)
(cons (cons fsym (step-sk-arity f5 (+ 1 (len vals))))
(append (build-sk-d (list 'all f3 f5)
vals dom1 i)
(build-sk-d (list 'all f3 f5)
vals dom2 i)))
(functions i)))))
:hints (("Goal"
:use ((:instance xeval-append-b
(y f3) (f f5) (dm dom2) (dom dom1)
(func1 (build-sk-d (list 'all f3 f5)
vals dom1 i))))
:in-theory (disable XEVAL-APPEND-B)
:do-not-induct t
)))
;;----------------------------------------------
;; Here it is, the heart of the soundness of Skolemization.
;; We split it into 2 cases: with and without a skolemizable existential quant.
(defthm step-sk-e-fsound-flg
(implies (and (setp (quantified-vars f))
(step-sk-arity f (len vals)) ;; there is a skolemizable exists
(function-symbol fsym)
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f))))
(if flg
(equal (feval (step-sk f vals fsym) (step-ex f fsym vals i))
(feval f i))
(implies (and (wfall f)
(domain-term-list (fringe dom))
(setp (fringe dom))
(subsetp-equal (fringe dom) (fringe (domain i))))
(equal (feval-d (step-sk f vals fsym ) dom
(step-ex-d f fsym vals dom i))
(feval-d f dom i)))))
:hints (("Goal"
:do-not generalize
:induct (build-sk-i flg f vals dom i))
)
:rule-classes nil)
(in-theory (disable step-ex))
(defthm step-sk-e-fsound-vals
(implies (and (setp (quantified-vars f))
(step-sk-arity f (len vals)) ;; there is a skolemizable exists
(function-symbol fsym)
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f)))
)
(equal (feval (step-sk f vals fsym) (step-ex f fsym vals i))
(feval f i)))
:hints (("Goal"
:by (:instance step-sk-e-fsound-flg (flg t)))))
(defthm step-sk-without-exists-2
(implies (and (natp n)
(not (step-sk-arity f n)))
(equal (step-sk f vars fsym) f))
:hints (("goal"
:induct (step-sk-sym-i2 f vars n))))
(defthm step-sk-x-fsound-vals
(implies (and (setp (quantified-vars f))
(not (step-sk-arity f (len vals))) ;; no skolemizable exists
(function-symbol fsym)
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f)))
)
(equal (feval (step-sk f vals fsym) (step-ex f fsym vals i))
(feval f i)))
:hints (("Goal"
:in-theory (enable step-ex)
:use ((:instance step-sk-without-exists-2
(n (len vals))
(vars vals))))))
(defthm step-sk-fsound-vals
(implies (and (setp (quantified-vars f))
(function-symbol fsym)
(domain-term-list vals)
(subsetp-equal vals (fringe (domain i)))
(not (member-equal fsym (funcs-in-formula f)))
)
(equal (feval (step-sk f vals fsym) (step-ex f fsym vals i))
(feval f i)))
:hints (("Goal"
:use ((:instance step-sk-e-fsound-vals)
(:instance step-sk-x-fsound-vals)))))
(defthm step-sk-fsound
(implies (and (setp (quantified-vars f))
(function-symbol fsym)
(not (member-equal fsym (funcs-in-formula f))))
(equal (feval (step-sk f nil fsym) (step-ex f fsym nil i))
(feval f i))))
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