Basic equivalence relation for pseudo-term-alist structures.
Function:
(defun pseudo-term-alist-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (pseudo-term-alistp acl2::x) (pseudo-term-alistp acl2::y)))) (equal (pseudo-term-alist-fix acl2::x) (pseudo-term-alist-fix acl2::y)))
Theorem:
(defthm pseudo-term-alist-equiv-is-an-equivalence (and (booleanp (pseudo-term-alist-equiv x y)) (pseudo-term-alist-equiv x x) (implies (pseudo-term-alist-equiv x y) (pseudo-term-alist-equiv y x)) (implies (and (pseudo-term-alist-equiv x y) (pseudo-term-alist-equiv y z)) (pseudo-term-alist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm pseudo-term-alist-equiv-implies-equal-pseudo-term-alist-fix-1 (implies (pseudo-term-alist-equiv acl2::x x-equiv) (equal (pseudo-term-alist-fix acl2::x) (pseudo-term-alist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm pseudo-term-alist-fix-under-pseudo-term-alist-equiv (pseudo-term-alist-equiv (pseudo-term-alist-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-pseudo-term-alist-fix-1-forward-to-pseudo-term-alist-equiv (implies (equal (pseudo-term-alist-fix acl2::x) acl2::y) (pseudo-term-alist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-pseudo-term-alist-fix-2-forward-to-pseudo-term-alist-equiv (implies (equal acl2::x (pseudo-term-alist-fix acl2::y)) (pseudo-term-alist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm pseudo-term-alist-equiv-of-pseudo-term-alist-fix-1-forward (implies (pseudo-term-alist-equiv (pseudo-term-alist-fix acl2::x) acl2::y) (pseudo-term-alist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm pseudo-term-alist-equiv-of-pseudo-term-alist-fix-2-forward (implies (pseudo-term-alist-equiv acl2::x (pseudo-term-alist-fix acl2::y)) (pseudo-term-alist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)