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Expr-grade

Expr-grade-equiv

Basic equivalence relation for expr-grade structures.

Definitions and Theorems

Function: expr-grade-equiv$inline

(defun expr-grade-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (expr-gradep acl2::x)
                              (expr-gradep acl2::y))))
  (equal (expr-grade-fix acl2::x)
         (expr-grade-fix acl2::y)))

Theorem: expr-grade-equiv-is-an-equivalence

(defthm expr-grade-equiv-is-an-equivalence
  (and (booleanp (expr-grade-equiv x y))
       (expr-grade-equiv x x)
       (implies (expr-grade-equiv x y)
                (expr-grade-equiv y x))
       (implies (and (expr-grade-equiv x y)
                     (expr-grade-equiv y z))
                (expr-grade-equiv x z)))
  :rule-classes (:equivalence))

Theorem: expr-grade-equiv-implies-equal-expr-grade-fix-1

(defthm expr-grade-equiv-implies-equal-expr-grade-fix-1
  (implies (expr-grade-equiv acl2::x x-equiv)
           (equal (expr-grade-fix acl2::x)
                  (expr-grade-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: expr-grade-fix-under-expr-grade-equiv

(defthm expr-grade-fix-under-expr-grade-equiv
  (expr-grade-equiv (expr-grade-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-expr-grade-fix-1-forward-to-expr-grade-equiv

(defthm equal-of-expr-grade-fix-1-forward-to-expr-grade-equiv
  (implies (equal (expr-grade-fix acl2::x) acl2::y)
           (expr-grade-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-expr-grade-fix-2-forward-to-expr-grade-equiv

(defthm equal-of-expr-grade-fix-2-forward-to-expr-grade-equiv
  (implies (equal acl2::x (expr-grade-fix acl2::y))
           (expr-grade-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: expr-grade-equiv-of-expr-grade-fix-1-forward

(defthm expr-grade-equiv-of-expr-grade-fix-1-forward
  (implies (expr-grade-equiv (expr-grade-fix acl2::x)
                             acl2::y)
           (expr-grade-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: expr-grade-equiv-of-expr-grade-fix-2-forward

(defthm expr-grade-equiv-of-expr-grade-fix-2-forward
  (implies (expr-grade-equiv acl2::x (expr-grade-fix acl2::y))
           (expr-grade-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)