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• M-assumption-n-output-comb-transform

M-assumption-n-output-comb-transform-equiv

Basic equivalence relation for m-assumption-n-output-comb-transform structures.

Definitions and Theorems

Function: m-assumption-n-output-comb-transform-equiv$inline

(defun
 m-assumption-n-output-comb-transform-equiv$inline
 (x acl2::y)
 (declare
  (xargs
     :guard (and (m-assumption-n-output-comb-transform-p x)
                 (m-assumption-n-output-comb-transform-p acl2::y))))
 (equal (m-assumption-n-output-comb-transform-fix x)
        (m-assumption-n-output-comb-transform-fix acl2::y)))

Theorem: m-assumption-n-output-comb-transform-equiv-is-an-equivalence

(defthm
 m-assumption-n-output-comb-transform-equiv-is-an-equivalence
 (and
     (booleanp (m-assumption-n-output-comb-transform-equiv x y))
     (m-assumption-n-output-comb-transform-equiv x x)
     (implies (m-assumption-n-output-comb-transform-equiv x y)
              (m-assumption-n-output-comb-transform-equiv y x))
     (implies (and (m-assumption-n-output-comb-transform-equiv x y)
                   (m-assumption-n-output-comb-transform-equiv y z))
              (m-assumption-n-output-comb-transform-equiv x z)))
 :rule-classes (:equivalence))

Theorem: m-assumption-n-output-comb-transform-equiv-implies-equal-m-assumption-n-output-comb-transform-fix-1

(defthm
 m-assumption-n-output-comb-transform-equiv-implies-equal-m-assumption-n-output-comb-transform-fix-1
 (implies
      (m-assumption-n-output-comb-transform-equiv x x-equiv)
      (equal (m-assumption-n-output-comb-transform-fix x)
             (m-assumption-n-output-comb-transform-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: m-assumption-n-output-comb-transform-fix-under-m-assumption-n-output-comb-transform-equiv

(defthm
 m-assumption-n-output-comb-transform-fix-under-m-assumption-n-output-comb-transform-equiv
 (m-assumption-n-output-comb-transform-equiv
      (m-assumption-n-output-comb-transform-fix x)
      x)
 :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-m-assumption-n-output-comb-transform-fix-1-forward-to-m-assumption-n-output-comb-transform-equiv

(defthm
 equal-of-m-assumption-n-output-comb-transform-fix-1-forward-to-m-assumption-n-output-comb-transform-equiv
 (implies (equal (m-assumption-n-output-comb-transform-fix x)
                 acl2::y)
          (m-assumption-n-output-comb-transform-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-m-assumption-n-output-comb-transform-fix-2-forward-to-m-assumption-n-output-comb-transform-equiv

(defthm
 equal-of-m-assumption-n-output-comb-transform-fix-2-forward-to-m-assumption-n-output-comb-transform-equiv
 (implies (equal x
                 (m-assumption-n-output-comb-transform-fix acl2::y))
          (m-assumption-n-output-comb-transform-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: m-assumption-n-output-comb-transform-equiv-of-m-assumption-n-output-comb-transform-fix-1-forward

(defthm
 m-assumption-n-output-comb-transform-equiv-of-m-assumption-n-output-comb-transform-fix-1-forward
 (implies (m-assumption-n-output-comb-transform-equiv
               (m-assumption-n-output-comb-transform-fix x)
               acl2::y)
          (m-assumption-n-output-comb-transform-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: m-assumption-n-output-comb-transform-equiv-of-m-assumption-n-output-comb-transform-fix-2-forward

(defthm
 m-assumption-n-output-comb-transform-equiv-of-m-assumption-n-output-comb-transform-fix-2-forward
 (implies (m-assumption-n-output-comb-transform-equiv
               x
               (m-assumption-n-output-comb-transform-fix acl2::y))
          (m-assumption-n-output-comb-transform-equiv x acl2::y))
 :rule-classes :forward-chaining)