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Substitution

Vl-initial-subst

Substitute into a vl-initial-p.

Signature
(vl-initial-subst x sigma) → new-x
Arguments: x — Guard (vl-initial-p x).; sigma — Guard (vl-sigma-p sigma).
Returns: new-x — Type (vl-initial-p new-x).

Definitions and Theorems

Function: vl-initial-subst

(defun vl-initial-subst (x sigma)
  (declare (xargs :guard (and (vl-initial-p x)
                              (vl-sigma-p sigma))))
  (declare (ignorable x sigma))
  (let ((__function__ 'vl-initial-subst))
    (declare (ignorable __function__))
    (change-vl-initial x
                       :stmt (vl-stmt-subst (vl-initial->stmt x)
                                            sigma))))

Theorem: vl-initial-p-of-vl-initial-subst

(defthm vl-initial-p-of-vl-initial-subst
  (b* ((new-x (vl-initial-subst x sigma)))
    (vl-initial-p new-x))
  :rule-classes :rewrite)

Theorem: vl-initial-subst-of-vl-initial-fix-x

(defthm vl-initial-subst-of-vl-initial-fix-x
  (equal (vl-initial-subst (vl-initial-fix x)
                           sigma)
         (vl-initial-subst x sigma)))

Theorem: vl-initial-subst-vl-initial-equiv-congruence-on-x

(defthm vl-initial-subst-vl-initial-equiv-congruence-on-x
  (implies (vl-initial-equiv x x-equiv)
           (equal (vl-initial-subst x sigma)
                  (vl-initial-subst x-equiv sigma)))
  :rule-classes :congruence)

Theorem: vl-initial-subst-of-vl-sigma-fix-sigma

(defthm vl-initial-subst-of-vl-sigma-fix-sigma
  (equal (vl-initial-subst x (vl-sigma-fix sigma))
         (vl-initial-subst x sigma)))

Theorem: vl-initial-subst-vl-sigma-equiv-congruence-on-sigma

(defthm vl-initial-subst-vl-sigma-equiv-congruence-on-sigma
  (implies (vl-sigma-equiv sigma sigma-equiv)
           (equal (vl-initial-subst x sigma)
                  (vl-initial-subst x sigma-equiv)))
  :rule-classes :congruence)