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