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Inequalities-of-exponents

Rules for resolving inequalities between exponents.

Definitions and Theorems

Theorem: expt->-1

(defthm expt->-1
  (implies (and (< 1 r)
                (< 0 i)
                (fc (real/rationalp r))
                (fc (integerp i)))
           (< 1 (expt r i)))
  :rule-classes :linear)

Theorem: expt-is-increasing-for-base>1

(defthm expt-is-increasing-for-base>1
  (implies (and (< 1 r)
                (< i j)
                (fc (real/rationalp r))
                (fc (integerp i))
                (fc (integerp j)))
           (< (expt r i) (expt r j)))
  :rule-classes (:rewrite :linear))

Theorem: expt-is-decreasing-for-pos-base<1

(defthm expt-is-decreasing-for-pos-base<1
  (implies (and (< 0 r)
                (< r 1)
                (< i j)
                (fc (real/rationalp r))
                (fc (integerp i))
                (fc (integerp j)))
           (< (expt r j) (expt r i))))

Theorem: expt-is-weakly-increasing-for-base>1

(defthm expt-is-weakly-increasing-for-base>1
  (implies (and (< 1 r)
                (<= i j)
                (fc (real/rationalp r))
                (fc (integerp i))
                (fc (integerp j)))
           (<= (expt r i) (expt r j))))

Theorem: expt-is-weakly-decreasing-for-pos-base<1

(defthm expt-is-weakly-decreasing-for-pos-base<1
  (implies (and (< 0 r)
                (< r 1)
                (<= i j)
                (fc (real/rationalp r))
                (fc (integerp i))
                (fc (integerp j)))
           (<= (expt r j) (expt r i))))