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• Boolean-operations

Boolean-eq

Equality == on booleans [JLS14:4.2.5] [JLS14:15.21.2].

Signature

(boolean-eq operand-left operand-right) → result

Arguments

operand-left — Guard (boolean-valuep operand-left).

operand-right — Guard (boolean-valuep operand-right).

Returns

result — Type (boolean-valuep result).

Definitions and Theorems

Function: boolean-eq

(defun boolean-eq (operand-left operand-right)
  (declare (xargs :guard (and (boolean-valuep operand-left)
                              (boolean-valuep operand-right))))
  (b* ((x (boolean-value->bool operand-left))
       (y (boolean-value->bool operand-right)))
    (boolean-value (equal x y))))

Theorem: boolean-valuep-of-boolean-eq

(defthm boolean-valuep-of-boolean-eq
  (b* ((result (boolean-eq operand-left operand-right)))
    (boolean-valuep result))
  :rule-classes :rewrite)

Theorem: boolean-eq-commutative

(defthm boolean-eq-commutative
  (equal (boolean-eq x y)
         (boolean-eq y x)))

Theorem: boolean-eq-of-boolean-value-fix-operand-left

(defthm boolean-eq-of-boolean-value-fix-operand-left
  (equal (boolean-eq (boolean-value-fix operand-left)
                     operand-right)
         (boolean-eq operand-left operand-right)))

Theorem: boolean-eq-boolean-value-equiv-congruence-on-operand-left

(defthm boolean-eq-boolean-value-equiv-congruence-on-operand-left
  (implies (boolean-value-equiv operand-left operand-left-equiv)
           (equal (boolean-eq operand-left operand-right)
                  (boolean-eq operand-left-equiv operand-right)))
  :rule-classes :congruence)

Theorem: boolean-eq-of-boolean-value-fix-operand-right

(defthm boolean-eq-of-boolean-value-fix-operand-right
  (equal (boolean-eq operand-left
                     (boolean-value-fix operand-right))
         (boolean-eq operand-left operand-right)))

Theorem: boolean-eq-boolean-value-equiv-congruence-on-operand-right

(defthm boolean-eq-boolean-value-equiv-congruence-on-operand-right
  (implies (boolean-value-equiv operand-right operand-right-equiv)
           (equal (boolean-eq operand-left operand-right)
                  (boolean-eq operand-left operand-right-equiv)))
  :rule-classes :congruence)