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Modscope

Modscope-equiv

Basic equivalence relation for modscope structures.

Definitions and Theorems

Function: modscope-equiv$inline

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

Theorem: modscope-equiv-is-an-equivalence

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

Theorem: modscope-equiv-implies-equal-modscope-fix-1

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

Theorem: modscope-fix-under-modscope-equiv

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

Theorem: equal-of-modscope-fix-1-forward-to-modscope-equiv

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

Theorem: equal-of-modscope-fix-2-forward-to-modscope-equiv

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

Theorem: modscope-equiv-of-modscope-fix-1-forward

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

Theorem: modscope-equiv-of-modscope-fix-2-forward

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