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Proof-tree

Proof-tree-equiv

Basic equivalence relation for proof-tree structures.

Definitions and Theorems

Function: proof-tree-equiv$inline

(defun proof-tree-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (proof-treep acl2::x)
                              (proof-treep acl2::y))))
  (equal (proof-tree-fix acl2::x)
         (proof-tree-fix acl2::y)))

Theorem: proof-tree-equiv-is-an-equivalence

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

Theorem: proof-tree-equiv-implies-equal-proof-tree-fix-1

(defthm proof-tree-equiv-implies-equal-proof-tree-fix-1
  (implies (proof-tree-equiv acl2::x x-equiv)
           (equal (proof-tree-fix acl2::x)
                  (proof-tree-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: proof-tree-fix-under-proof-tree-equiv

(defthm proof-tree-fix-under-proof-tree-equiv
  (proof-tree-equiv (proof-tree-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-proof-tree-fix-1-forward-to-proof-tree-equiv

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

Theorem: equal-of-proof-tree-fix-2-forward-to-proof-tree-equiv

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

Theorem: proof-tree-equiv-of-proof-tree-fix-1-forward

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

Theorem: proof-tree-equiv-of-proof-tree-fix-2-forward

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