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• Rename

Ident-ident-subst

Signature

(ident-ident-subst ident alist) → new-ident

Arguments

ident — Guard (identp ident).

alist — Guard (ident-ident-alistp alist).

Returns

new-ident — Type (identp new-ident).

Definitions and Theorems

Function: ident-ident-subst

(defun ident-ident-subst (ident alist)
  (declare (xargs :guard (and (identp ident)
                              (ident-ident-alistp alist))))
  (let ((__function__ 'ident-ident-subst))
    (declare (ignorable __function__))
    (b* ((lookup (assoc-equal (ident-fix ident)
                              (ident-ident-alist-fix alist))))
      (ident-fix (if lookup (cdr lookup) ident)))))

Theorem: identp-of-ident-ident-subst

(defthm identp-of-ident-ident-subst
  (b* ((new-ident (ident-ident-subst ident alist)))
    (identp new-ident))
  :rule-classes :rewrite)

Theorem: ident-ident-subst-of-ident-fix-ident

(defthm ident-ident-subst-of-ident-fix-ident
  (equal (ident-ident-subst (ident-fix ident)
                            alist)
         (ident-ident-subst ident alist)))

Theorem: ident-ident-subst-ident-equiv-congruence-on-ident

(defthm ident-ident-subst-ident-equiv-congruence-on-ident
  (implies (c$::ident-equiv ident ident-equiv)
           (equal (ident-ident-subst ident alist)
                  (ident-ident-subst ident-equiv alist)))
  :rule-classes :congruence)

Theorem: ident-ident-subst-of-ident-ident-alist-fix-alist

(defthm ident-ident-subst-of-ident-ident-alist-fix-alist
  (equal (ident-ident-subst ident (ident-ident-alist-fix alist))
         (ident-ident-subst ident alist)))

Theorem: ident-ident-subst-ident-ident-alist-equiv-congruence-on-alist

(defthm
      ident-ident-subst-ident-ident-alist-equiv-congruence-on-alist
  (implies (ident-ident-alist-equiv alist alist-equiv)
           (equal (ident-ident-subst ident alist)
                  (ident-ident-subst ident alist-equiv)))
  :rule-classes :congruence)