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