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Lambda-binding

Lambda-binding-equiv

Basic equivalence relation for lambda-binding structures.

Definitions and Theorems

Function: lambda-binding-equiv$inline

(defun lambda-binding-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (lambda-binding-p acl2::x)
                              (lambda-binding-p acl2::y))))
  (equal (lambda-binding-fix acl2::x)
         (lambda-binding-fix acl2::y)))

Theorem: lambda-binding-equiv-is-an-equivalence

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

Theorem: lambda-binding-equiv-implies-equal-lambda-binding-fix-1

(defthm lambda-binding-equiv-implies-equal-lambda-binding-fix-1
  (implies (lambda-binding-equiv acl2::x x-equiv)
           (equal (lambda-binding-fix acl2::x)
                  (lambda-binding-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: lambda-binding-fix-under-lambda-binding-equiv

(defthm lambda-binding-fix-under-lambda-binding-equiv
  (lambda-binding-equiv (lambda-binding-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-lambda-binding-fix-1-forward-to-lambda-binding-equiv

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

Theorem: equal-of-lambda-binding-fix-2-forward-to-lambda-binding-equiv

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

Theorem: lambda-binding-equiv-of-lambda-binding-fix-1-forward

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

Theorem: lambda-binding-equiv-of-lambda-binding-fix-2-forward

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