Fixing function for funtab+vartab+tagenv structures.
(funtab+vartab+tagenv-fix x) → new-x
Function:
(defun funtab+vartab+tagenv-fix$inline (x) (declare (xargs :guard (funtab+vartab+tagenv-p x))) (let ((__function__ 'funtab+vartab+tagenv-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((funs (fun-table-fix (cdr (std::da-nth 0 x)))) (vars (var-table-fix (cdr (std::da-nth 1 x)))) (tags (tag-env-fix (cdr (std::da-nth 2 x))))) (list (cons 'funs funs) (cons 'vars vars) (cons 'tags tags))) :exec x)))
Theorem:
(defthm funtab+vartab+tagenv-p-of-funtab+vartab+tagenv-fix (b* ((new-x (funtab+vartab+tagenv-fix$inline x))) (funtab+vartab+tagenv-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm funtab+vartab+tagenv-fix-when-funtab+vartab+tagenv-p (implies (funtab+vartab+tagenv-p x) (equal (funtab+vartab+tagenv-fix x) x)))
Function:
(defun funtab+vartab+tagenv-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (funtab+vartab+tagenv-p acl2::x) (funtab+vartab+tagenv-p acl2::y)))) (equal (funtab+vartab+tagenv-fix acl2::x) (funtab+vartab+tagenv-fix acl2::y)))
Theorem:
(defthm funtab+vartab+tagenv-equiv-is-an-equivalence (and (booleanp (funtab+vartab+tagenv-equiv x y)) (funtab+vartab+tagenv-equiv x x) (implies (funtab+vartab+tagenv-equiv x y) (funtab+vartab+tagenv-equiv y x)) (implies (and (funtab+vartab+tagenv-equiv x y) (funtab+vartab+tagenv-equiv y z)) (funtab+vartab+tagenv-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm funtab+vartab+tagenv-equiv-implies-equal-funtab+vartab+tagenv-fix-1 (implies (funtab+vartab+tagenv-equiv acl2::x x-equiv) (equal (funtab+vartab+tagenv-fix acl2::x) (funtab+vartab+tagenv-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm funtab+vartab+tagenv-fix-under-funtab+vartab+tagenv-equiv (funtab+vartab+tagenv-equiv (funtab+vartab+tagenv-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-funtab+vartab+tagenv-fix-1-forward-to-funtab+vartab+tagenv-equiv (implies (equal (funtab+vartab+tagenv-fix acl2::x) acl2::y) (funtab+vartab+tagenv-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-funtab+vartab+tagenv-fix-2-forward-to-funtab+vartab+tagenv-equiv (implies (equal acl2::x (funtab+vartab+tagenv-fix acl2::y)) (funtab+vartab+tagenv-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm funtab+vartab+tagenv-equiv-of-funtab+vartab+tagenv-fix-1-forward (implies (funtab+vartab+tagenv-equiv (funtab+vartab+tagenv-fix acl2::x) acl2::y) (funtab+vartab+tagenv-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm funtab+vartab+tagenv-equiv-of-funtab+vartab+tagenv-fix-2-forward (implies (funtab+vartab+tagenv-equiv acl2::x (funtab+vartab+tagenv-fix acl2::y)) (funtab+vartab+tagenv-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)