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• Sexpr-equivs

Key-and-env-equiv

(key-and-env-equiv x y) is an extension of 4v-env-equiv that additionally requires the keys match between the two alists.

This is a universal equivalence, introduced using def-universal-equiv.

Function: key-and-env-equiv

(defun key-and-env-equiv (x y)
  (and (keys-equiv x y)
       (4v-env-equiv x y)))

Definitions and Theorems

Function: key-and-env-equiv

(defun key-and-env-equiv (x y)
  (and (keys-equiv x y)
       (4v-env-equiv x y)))

Theorem: key-and-env-equiv-is-an-equivalence

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

Theorem: key-and-env-equiv-refines-4v-env-equiv

(defthm key-and-env-equiv-refines-4v-env-equiv
  (implies (key-and-env-equiv x y)
           (4v-env-equiv x y))
  :rule-classes (:refinement))

Theorem: key-and-env-equiv-implies-key-and-env-equiv-append-1

(defthm key-and-env-equiv-implies-key-and-env-equiv-append-1
  (implies (key-and-env-equiv a a-equiv)
           (key-and-env-equiv (append a b)
                              (append a-equiv b)))
  :rule-classes (:congruence))

Theorem: key-and-env-equiv-implies-key-and-env-equiv-append-2

(defthm key-and-env-equiv-implies-key-and-env-equiv-append-2
  (implies (key-and-env-equiv b b-equiv)
           (key-and-env-equiv (append a b)
                              (append a b-equiv)))
  :rule-classes (:congruence))

Theorem: 4v-env-equiv-to-key-and-env-equiv

(defthm 4v-env-equiv-to-key-and-env-equiv
  (implies (keys-equiv a b)
           (equal (4v-env-equiv a b)
                  (key-and-env-equiv a b))))