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Sig-table

Sig-table-equiv

Basic equivalence relation for sig-table structures.

Definitions and Theorems

Function: sig-table-equiv$inline

(defun sig-table-equiv$inline (x y)
  (declare (xargs :guard (and (sig-table-p x) (sig-table-p y))))
  (equal (sig-table-fix x)
         (sig-table-fix y)))

Theorem: sig-table-equiv-is-an-equivalence

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

Theorem: sig-table-equiv-implies-equal-sig-table-fix-1

(defthm sig-table-equiv-implies-equal-sig-table-fix-1
  (implies (sig-table-equiv x x-equiv)
           (equal (sig-table-fix x)
                  (sig-table-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sig-table-fix-under-sig-table-equiv

(defthm sig-table-fix-under-sig-table-equiv
  (sig-table-equiv (sig-table-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sig-table-fix-1-forward-to-sig-table-equiv

(defthm equal-of-sig-table-fix-1-forward-to-sig-table-equiv
  (implies (equal (sig-table-fix x) y)
           (sig-table-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sig-table-fix-2-forward-to-sig-table-equiv

(defthm equal-of-sig-table-fix-2-forward-to-sig-table-equiv
  (implies (equal x (sig-table-fix y))
           (sig-table-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sig-table-equiv-of-sig-table-fix-1-forward

(defthm sig-table-equiv-of-sig-table-fix-1-forward
  (implies (sig-table-equiv (sig-table-fix x) y)
           (sig-table-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sig-table-equiv-of-sig-table-fix-2-forward

(defthm sig-table-equiv-of-sig-table-fix-2-forward
  (implies (sig-table-equiv x (sig-table-fix y))
           (sig-table-equiv x y))
  :rule-classes :forward-chaining)