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