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• Expt

Basic-expt-type-rules

Rules about when expt produces integers, positive numbers, etc.

Definitions and Theorems

Theorem: expt-type-prescription-rationalp

(defthm expt-type-prescription-rationalp
        (implies (rationalp r)
                 (rationalp (expt r i)))
        :rule-classes (:type-prescription :generalize))

Theorem: expt-type-prescription-positive

(defthm expt-type-prescription-positive
        (implies (and (< 0 r) (real/rationalp r))
                 (< 0 (expt r i)))
        :rule-classes (:type-prescription :generalize))

Theorem: expt-type-prescription-nonzero

(defthm expt-type-prescription-nonzero
        (implies (and (fc (acl2-numberp r))
                      (not (equal r 0)))
                 (not (equal 0 (expt r i))))
        :rule-classes (:type-prescription :generalize))

Theorem: expt-type-prescription-integerp

(defthm expt-type-prescription-integerp
        (implies (and (<= 0 i) (integerp r))
                 (integerp (expt r i)))
        :rule-classes (:type-prescription :generalize))