(congruence-rule-table-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.
(congruence-rule-table-fix x) → fty::newx
Note that in the execution this is just an inline identity function.
Function:
(defun congruence-rule-table-fix$inline (x) (declare (xargs :guard (congruence-rule-table-p x))) (let ((__function__ 'congruence-rule-table-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) nil (let ((rest (congruence-rule-table-fix (cdr x)))) (if (and (consp (car x)) (pseudo-fnsym-p (caar x))) (let ((fty::first-key (caar x)) (fty::first-val (congruence-rulelist-fix (cdar x)))) (cons (cons fty::first-key fty::first-val) rest)) rest))) :exec x)))
Theorem:
(defthm congruence-rule-table-p-of-congruence-rule-table-fix (b* ((fty::newx (congruence-rule-table-fix$inline x))) (congruence-rule-table-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm congruence-rule-table-fix-when-congruence-rule-table-p (implies (congruence-rule-table-p x) (equal (congruence-rule-table-fix x) x)))
Function:
(defun congruence-rule-table-equiv$inline (x y) (declare (xargs :guard (and (congruence-rule-table-p x) (congruence-rule-table-p y)))) (equal (congruence-rule-table-fix x) (congruence-rule-table-fix y)))
Theorem:
(defthm congruence-rule-table-equiv-is-an-equivalence (and (booleanp (congruence-rule-table-equiv x y)) (congruence-rule-table-equiv x x) (implies (congruence-rule-table-equiv x y) (congruence-rule-table-equiv y x)) (implies (and (congruence-rule-table-equiv x y) (congruence-rule-table-equiv y z)) (congruence-rule-table-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm congruence-rule-table-equiv-implies-equal-congruence-rule-table-fix-1 (implies (congruence-rule-table-equiv x x-equiv) (equal (congruence-rule-table-fix x) (congruence-rule-table-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm congruence-rule-table-fix-under-congruence-rule-table-equiv (congruence-rule-table-equiv (congruence-rule-table-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-congruence-rule-table-fix-1-forward-to-congruence-rule-table-equiv (implies (equal (congruence-rule-table-fix x) y) (congruence-rule-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-congruence-rule-table-fix-2-forward-to-congruence-rule-table-equiv (implies (equal x (congruence-rule-table-fix y)) (congruence-rule-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm congruence-rule-table-equiv-of-congruence-rule-table-fix-1-forward (implies (congruence-rule-table-equiv (congruence-rule-table-fix x) y) (congruence-rule-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm congruence-rule-table-equiv-of-congruence-rule-table-fix-2-forward (implies (congruence-rule-table-equiv x (congruence-rule-table-fix y)) (congruence-rule-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm cons-of-congruence-rulelist-fix-v-under-congruence-rule-table-equiv (congruence-rule-table-equiv (cons (cons k (congruence-rulelist-fix v)) x) (cons (cons k v) x)))
Theorem:
(defthm cons-congruence-rulelist-equiv-congruence-on-v-under-congruence-rule-table-equiv (implies (congruence-rulelist-equiv v v-equiv) (congruence-rule-table-equiv (cons (cons k v) x) (cons (cons k v-equiv) x))) :rule-classes :congruence)
Theorem:
(defthm cons-of-congruence-rule-table-fix-y-under-congruence-rule-table-equiv (congruence-rule-table-equiv (cons x (congruence-rule-table-fix y)) (cons x y)))
Theorem:
(defthm cons-congruence-rule-table-equiv-congruence-on-y-under-congruence-rule-table-equiv (implies (congruence-rule-table-equiv y y-equiv) (congruence-rule-table-equiv (cons x y) (cons x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm congruence-rule-table-fix-of-acons (equal (congruence-rule-table-fix (cons (cons a b) x)) (let ((rest (congruence-rule-table-fix x))) (if (and (pseudo-fnsym-p a)) (let ((fty::first-key a) (fty::first-val (congruence-rulelist-fix b))) (cons (cons fty::first-key fty::first-val) rest)) rest))))
Theorem:
(defthm hons-assoc-equal-of-congruence-rule-table-fix (equal (hons-assoc-equal k (congruence-rule-table-fix x)) (let ((fty::pair (hons-assoc-equal k x))) (and (pseudo-fnsym-p k) fty::pair (cons k (congruence-rulelist-fix (cdr fty::pair)))))))
Theorem:
(defthm congruence-rule-table-fix-of-append (equal (congruence-rule-table-fix (append std::a std::b)) (append (congruence-rule-table-fix std::a) (congruence-rule-table-fix std::b))))
Theorem:
(defthm consp-car-of-congruence-rule-table-fix (equal (consp (car (congruence-rule-table-fix x))) (consp (congruence-rule-table-fix x))))