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Variable-substitution

Variable-substitutionp

Recognizer for variable-substitution.

Signature
(variable-substitutionp x) → *

Definitions and Theorems

Function: variable-substitutionp

(defun variable-substitutionp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (null x)
    (and (consp (car x))
         (identifierp (caar x))
         (expressionp (cdar x))
         (or (null (cdr x))
             (and (consp (cdr x))
                  (consp (cadr x))
                  (acl2::fast-<< (caar x) (caadr x))
                  (variable-substitutionp (cdr x)))))))

Theorem: booleanp-of-variable-substitutionp

(defthm booleanp-of-variable-substitutionp
  (booleanp (variable-substitutionp x)))

Theorem: mapp-when-variable-substitutionp

(defthm mapp-when-variable-substitutionp
  (implies (variable-substitutionp x)
           (omap::mapp x))
  :rule-classes (:rewrite :forward-chaining))

Theorem: variable-substitutionp-of-tail

(defthm variable-substitutionp-of-tail
  (implies (variable-substitutionp x)
           (variable-substitutionp (omap::tail x))))

Theorem: identifierp-of-head-key-when-variable-substitutionp

(defthm identifierp-of-head-key-when-variable-substitutionp
  (implies (and (variable-substitutionp x)
                (not (omap::emptyp x)))
           (identifierp (mv-nth 0 (omap::head x)))))

Theorem: expressionp-of-head-val-when-variable-substitutionp

(defthm expressionp-of-head-val-when-variable-substitutionp
  (implies (and (variable-substitutionp x)
                (not (omap::emptyp x)))
           (expressionp (mv-nth 1 (omap::head x)))))

Theorem: variable-substitutionp-of-update

(defthm variable-substitutionp-of-update
  (implies (and (variable-substitutionp x)
                (identifierp k)
                (expressionp v))
           (variable-substitutionp (omap::update k v x))))

Theorem: variable-substitutionp-of-update*

(defthm variable-substitutionp-of-update*
  (implies (and (variable-substitutionp x)
                (variable-substitutionp y))
           (variable-substitutionp (omap::update* x y))))

Theorem: variable-substitutionp-of-delete

(defthm variable-substitutionp-of-delete
  (implies (variable-substitutionp x)
           (variable-substitutionp (omap::delete k x))))

Theorem: variable-substitutionp-of-delete*

(defthm variable-substitutionp-of-delete*
  (implies (variable-substitutionp x)
           (variable-substitutionp (omap::delete* k x))))

Theorem: identifierp-when-assoc-variable-substitutionp-binds-free-x

(defthm identifierp-when-assoc-variable-substitutionp-binds-free-x
  (implies (and (omap::assoc k x)
                (variable-substitutionp x))
           (identifierp k)))

Theorem: identifierp-of-car-of-assoc-variable-substitutionp

(defthm identifierp-of-car-of-assoc-variable-substitutionp
  (implies (and (variable-substitutionp x)
                (omap::assoc k x))
           (identifierp (car (omap::assoc k x)))))

Theorem: expressionp-of-cdr-of-assoc-variable-substitutionp

(defthm expressionp-of-cdr-of-assoc-variable-substitutionp
  (implies (and (variable-substitutionp x)
                (omap::assoc k x))
           (expressionp (cdr (omap::assoc k x)))))

Theorem: expressionp-of-lookup-when-variable-substitutionp

(defthm expressionp-of-lookup-when-variable-substitutionp
  (implies (and (variable-substitutionp x)
                (omap::assoc k x))
           (expressionp (omap::lookup k x))))