author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35216 | 7641e8d831d2 |
child 35828 | 46cfc4b8112e |
permissions | -rw-r--r-- |
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New theory "Power" of exponentiation (and binomial coefficients)
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(* Title: HOL/Power.thy |
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New theory "Power" of exponentiation (and binomial coefficients)
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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New theory "Power" of exponentiation (and binomial coefficients)
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Copyright 1997 University of Cambridge |
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*) |
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header {* Exponentiation *} |
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theory Power |
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imports Nat |
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begin |
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subsection {* Powers for Arbitrary Monoids *} |
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||
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class power = one + times |
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begin |
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primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where |
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power_0: "a ^ 0 = 1" |
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| power_Suc: "a ^ Suc n = a * a ^ n" |
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notation (latex output) |
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power ("(_\<^bsup>_\<^esup>)" [1000] 1000) |
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notation (HTML output) |
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power ("(_\<^bsup>_\<^esup>)" [1000] 1000) |
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||
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end |
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context monoid_mult |
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begin |
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subclass power .. |
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lemma power_one [simp]: |
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"1 ^ n = 1" |
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by (induct n) simp_all |
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lemma power_one_right [simp]: |
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"a ^ 1 = a" |
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by simp |
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lemma power_commutes: |
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"a ^ n * a = a * a ^ n" |
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by (induct n) (simp_all add: mult_assoc) |
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lemma power_Suc2: |
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"a ^ Suc n = a ^ n * a" |
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by (simp add: power_commutes) |
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lemma power_add: |
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"a ^ (m + n) = a ^ m * a ^ n" |
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by (induct m) (simp_all add: algebra_simps) |
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lemma power_mult: |
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"a ^ (m * n) = (a ^ m) ^ n" |
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by (induct n) (simp_all add: power_add) |
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end |
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context comm_monoid_mult |
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begin |
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lemma power_mult_distrib: |
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"(a * b) ^ n = (a ^ n) * (b ^ n)" |
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by (induct n) (simp_all add: mult_ac) |
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end |
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context semiring_1 |
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begin |
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lemma of_nat_power: |
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"of_nat (m ^ n) = of_nat m ^ n" |
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by (induct n) (simp_all add: of_nat_mult) |
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end |
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context comm_semiring_1 |
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begin |
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text {* The divides relation *} |
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lemma le_imp_power_dvd: |
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assumes "m \<le> n" shows "a ^ m dvd a ^ n" |
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proof |
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have "a ^ n = a ^ (m + (n - m))" |
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using `m \<le> n` by simp |
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also have "\<dots> = a ^ m * a ^ (n - m)" |
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by (rule power_add) |
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finally show "a ^ n = a ^ m * a ^ (n - m)" . |
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qed |
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lemma power_le_dvd: |
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"a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b" |
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by (rule dvd_trans [OF le_imp_power_dvd]) |
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lemma dvd_power_same: |
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"x dvd y \<Longrightarrow> x ^ n dvd y ^ n" |
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by (induct n) (auto simp add: mult_dvd_mono) |
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lemma dvd_power_le: |
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"x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m" |
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by (rule power_le_dvd [OF dvd_power_same]) |
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lemma dvd_power [simp]: |
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assumes "n > (0::nat) \<or> x = 1" |
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shows "x dvd (x ^ n)" |
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using assms proof |
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assume "0 < n" |
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then have "x ^ n = x ^ Suc (n - 1)" by simp |
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then show "x dvd (x ^ n)" by simp |
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next |
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assume "x = 1" |
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then show "x dvd (x ^ n)" by simp |
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qed |
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end |
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context ring_1 |
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begin |
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lemma power_minus: |
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"(- a) ^ n = (- 1) ^ n * a ^ n" |
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proof (induct n) |
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case 0 show ?case by simp |
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next |
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case (Suc n) then show ?case |
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by (simp del: power_Suc add: power_Suc2 mult_assoc) |
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qed |
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end |
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context linordered_semidom |
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begin |
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lemma zero_less_power [simp]: |
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"0 < a \<Longrightarrow> 0 < a ^ n" |
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by (induct n) (simp_all add: mult_pos_pos) |
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lemma zero_le_power [simp]: |
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"0 \<le> a \<Longrightarrow> 0 \<le> a ^ n" |
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by (induct n) (simp_all add: mult_nonneg_nonneg) |
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lemma one_le_power[simp]: |
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"1 \<le> a \<Longrightarrow> 1 \<le> a ^ n" |
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apply (induct n) |
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apply simp_all |
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apply (rule order_trans [OF _ mult_mono [of 1 _ 1]]) |
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apply (simp_all add: order_trans [OF zero_le_one]) |
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done |
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lemma power_gt1_lemma: |
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assumes gt1: "1 < a" |
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shows "1 < a * a ^ n" |
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proof - |
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from gt1 have "0 \<le> a" |
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by (fact order_trans [OF zero_le_one less_imp_le]) |
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have "1 * 1 < a * 1" using gt1 by simp |
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also have "\<dots> \<le> a * a ^ n" using gt1 |
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by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le |
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zero_le_one order_refl) |
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finally show ?thesis by simp |
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qed |
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lemma power_gt1: |
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"1 < a \<Longrightarrow> 1 < a ^ Suc n" |
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by (simp add: power_gt1_lemma) |
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lemma one_less_power [simp]: |
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"1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n" |
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by (cases n) (simp_all add: power_gt1_lemma) |
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lemma power_le_imp_le_exp: |
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assumes gt1: "1 < a" |
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shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n" |
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proof (induct m arbitrary: n) |
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case 0 |
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show ?case by simp |
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next |
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case (Suc m) |
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show ?case |
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proof (cases n) |
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case 0 |
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with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp |
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with gt1 show ?thesis |
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by (force simp only: power_gt1_lemma |
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not_less [symmetric]) |
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next |
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case (Suc n) |
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with Suc.prems Suc.hyps show ?thesis |
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by (force dest: mult_left_le_imp_le |
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simp add: less_trans [OF zero_less_one gt1]) |
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qed |
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qed |
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text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*} |
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lemma power_inject_exp [simp]: |
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"1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n" |
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by (force simp add: order_antisym power_le_imp_le_exp) |
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text{*Can relax the first premise to @{term "0<a"} in the case of the |
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natural numbers.*} |
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lemma power_less_imp_less_exp: |
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"1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n" |
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by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] |
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power_le_imp_le_exp) |
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lemma power_mono: |
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"a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n" |
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by (induct n) |
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(auto intro: mult_mono order_trans [of 0 a b]) |
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lemma power_strict_mono [rule_format]: |
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"a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n" |
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by (induct n) |
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(auto simp add: mult_strict_mono le_less_trans [of 0 a b]) |
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text{*Lemma for @{text power_strict_decreasing}*} |
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lemma power_Suc_less: |
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"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n" |
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by (induct n) |
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(auto simp add: mult_strict_left_mono) |
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lemma power_strict_decreasing [rule_format]: |
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"n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n" |
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proof (induct N) |
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case 0 then show ?case by simp |
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next |
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case (Suc N) then show ?case |
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apply (auto simp add: power_Suc_less less_Suc_eq) |
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apply (subgoal_tac "a * a^N < 1 * a^n") |
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apply simp |
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apply (rule mult_strict_mono) apply auto |
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done |
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qed |
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text{*Proof resembles that of @{text power_strict_decreasing}*} |
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lemma power_decreasing [rule_format]: |
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"n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n" |
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proof (induct N) |
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case 0 then show ?case by simp |
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next |
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case (Suc N) then show ?case |
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apply (auto simp add: le_Suc_eq) |
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apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp) |
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apply (rule mult_mono) apply auto |
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done |
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qed |
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lemma power_Suc_less_one: |
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"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1" |
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using power_strict_decreasing [of 0 "Suc n" a] by simp |
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text{*Proof again resembles that of @{text power_strict_decreasing}*} |
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lemma power_increasing [rule_format]: |
256 |
"n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N" |
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proof (induct N) |
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case 0 then show ?case by simp |
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next |
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case (Suc N) then show ?case |
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apply (auto simp add: le_Suc_eq) |
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apply (subgoal_tac "1 * a^n \<le> a * a^N", simp) |
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apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one]) |
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done |
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qed |
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text{*Lemma for @{text power_strict_increasing}*} |
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lemma power_less_power_Suc: |
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"1 < a \<Longrightarrow> a ^ n < a * a ^ n" |
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by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one]) |
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271 |
|
30996 | 272 |
lemma power_strict_increasing [rule_format]: |
273 |
"n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N" |
|
274 |
proof (induct N) |
|
275 |
case 0 then show ?case by simp |
|
276 |
next |
|
277 |
case (Suc N) then show ?case |
|
278 |
apply (auto simp add: power_less_power_Suc less_Suc_eq) |
|
279 |
apply (subgoal_tac "1 * a^n < a * a^N", simp) |
|
280 |
apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le) |
|
281 |
done |
|
282 |
qed |
|
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283 |
|
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284 |
lemma power_increasing_iff [simp]: |
30996 | 285 |
"1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y" |
286 |
by (blast intro: power_le_imp_le_exp power_increasing less_imp_le) |
|
15066 | 287 |
|
288 |
lemma power_strict_increasing_iff [simp]: |
|
30996 | 289 |
"1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y" |
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290 |
by (blast intro: power_less_imp_less_exp power_strict_increasing) |
15066 | 291 |
|
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292 |
lemma power_le_imp_le_base: |
30996 | 293 |
assumes le: "a ^ Suc n \<le> b ^ Suc n" |
294 |
and ynonneg: "0 \<le> b" |
|
295 |
shows "a \<le> b" |
|
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296 |
proof (rule ccontr) |
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assume "~ a \<le> b" |
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then have "b < a" by (simp only: linorder_not_le) |
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299 |
then have "b ^ Suc n < a ^ Suc n" |
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|
300 |
by (simp only: prems power_strict_mono) |
30996 | 301 |
from le and this show False |
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by (simp add: linorder_not_less [symmetric]) |
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303 |
qed |
14577 | 304 |
|
22853 | 305 |
lemma power_less_imp_less_base: |
306 |
assumes less: "a ^ n < b ^ n" |
|
307 |
assumes nonneg: "0 \<le> b" |
|
308 |
shows "a < b" |
|
309 |
proof (rule contrapos_pp [OF less]) |
|
310 |
assume "~ a < b" |
|
311 |
hence "b \<le> a" by (simp only: linorder_not_less) |
|
312 |
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) |
|
30996 | 313 |
thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less) |
22853 | 314 |
qed |
315 |
||
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316 |
lemma power_inject_base: |
30996 | 317 |
"a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b" |
318 |
by (blast intro: power_le_imp_le_base antisym eq_refl sym) |
|
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319 |
|
22955 | 320 |
lemma power_eq_imp_eq_base: |
30996 | 321 |
"a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b" |
322 |
by (cases n) (simp_all del: power_Suc, rule power_inject_base) |
|
22955 | 323 |
|
30996 | 324 |
end |
325 |
||
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326 |
context linordered_idom |
30996 | 327 |
begin |
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328 |
|
30996 | 329 |
lemma power_abs: |
330 |
"abs (a ^ n) = abs a ^ n" |
|
331 |
by (induct n) (auto simp add: abs_mult) |
|
332 |
||
333 |
lemma abs_power_minus [simp]: |
|
334 |
"abs ((-a) ^ n) = abs (a ^ n)" |
|
35216 | 335 |
by (simp add: power_abs) |
30996 | 336 |
|
337 |
lemma zero_less_power_abs_iff [simp, noatp]: |
|
338 |
"0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0" |
|
339 |
proof (induct n) |
|
340 |
case 0 show ?case by simp |
|
341 |
next |
|
342 |
case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff) |
|
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343 |
qed |
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|
344 |
|
30996 | 345 |
lemma zero_le_power_abs [simp]: |
346 |
"0 \<le> abs a ^ n" |
|
347 |
by (rule zero_le_power [OF abs_ge_zero]) |
|
348 |
||
349 |
end |
|
350 |
||
351 |
context ring_1_no_zero_divisors |
|
352 |
begin |
|
353 |
||
354 |
lemma field_power_not_zero: |
|
355 |
"a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0" |
|
356 |
by (induct n) auto |
|
357 |
||
358 |
end |
|
359 |
||
360 |
context division_ring |
|
361 |
begin |
|
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362 |
|
30997 | 363 |
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *} |
30996 | 364 |
lemma nonzero_power_inverse: |
365 |
"a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n" |
|
366 |
by (induct n) |
|
367 |
(simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero) |
|
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368 |
|
30996 | 369 |
end |
370 |
||
371 |
context field |
|
372 |
begin |
|
373 |
||
374 |
lemma nonzero_power_divide: |
|
375 |
"b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n" |
|
376 |
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse) |
|
377 |
||
378 |
end |
|
379 |
||
380 |
lemma power_0_Suc [simp]: |
|
381 |
"(0::'a::{power, semiring_0}) ^ Suc n = 0" |
|
382 |
by simp |
|
30313 | 383 |
|
30996 | 384 |
text{*It looks plausible as a simprule, but its effect can be strange.*} |
385 |
lemma power_0_left: |
|
386 |
"0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))" |
|
387 |
by (induct n) simp_all |
|
388 |
||
389 |
lemma power_eq_0_iff [simp]: |
|
390 |
"a ^ n = 0 \<longleftrightarrow> |
|
391 |
a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0" |
|
392 |
by (induct n) |
|
393 |
(auto simp add: no_zero_divisors elim: contrapos_pp) |
|
394 |
||
395 |
lemma power_diff: |
|
396 |
fixes a :: "'a::field" |
|
397 |
assumes nz: "a \<noteq> 0" |
|
398 |
shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n" |
|
399 |
by (induct m n rule: diff_induct) (simp_all add: nz) |
|
30313 | 400 |
|
30996 | 401 |
text{*Perhaps these should be simprules.*} |
402 |
lemma power_inverse: |
|
403 |
fixes a :: "'a::{division_ring,division_by_zero,power}" |
|
404 |
shows "inverse (a ^ n) = (inverse a) ^ n" |
|
405 |
apply (cases "a = 0") |
|
406 |
apply (simp add: power_0_left) |
|
407 |
apply (simp add: nonzero_power_inverse) |
|
408 |
done (* TODO: reorient or rename to inverse_power *) |
|
409 |
||
410 |
lemma power_one_over: |
|
411 |
"1 / (a::'a::{field,division_by_zero, power}) ^ n = (1 / a) ^ n" |
|
412 |
by (simp add: divide_inverse) (rule power_inverse) |
|
413 |
||
414 |
lemma power_divide: |
|
415 |
"(a / b) ^ n = (a::'a::{field,division_by_zero}) ^ n / b ^ n" |
|
416 |
apply (cases "b = 0") |
|
417 |
apply (simp add: power_0_left) |
|
418 |
apply (rule nonzero_power_divide) |
|
419 |
apply assumption |
|
30313 | 420 |
done |
421 |
||
422 |
||
30960 | 423 |
subsection {* Exponentiation for the Natural Numbers *} |
14577 | 424 |
|
30996 | 425 |
lemma nat_one_le_power [simp]: |
426 |
"Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n" |
|
427 |
by (rule one_le_power [of i n, unfolded One_nat_def]) |
|
23305 | 428 |
|
30996 | 429 |
lemma nat_zero_less_power_iff [simp]: |
430 |
"x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0" |
|
431 |
by (induct n) auto |
|
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432 |
|
30056 | 433 |
lemma nat_power_eq_Suc_0_iff [simp]: |
30996 | 434 |
"x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0" |
435 |
by (induct m) auto |
|
30056 | 436 |
|
30996 | 437 |
lemma power_Suc_0 [simp]: |
438 |
"Suc 0 ^ n = Suc 0" |
|
439 |
by simp |
|
30056 | 440 |
|
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441 |
text{*Valid for the naturals, but what if @{text"0<i<1"}? |
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442 |
Premises cannot be weakened: consider the case where @{term "i=0"}, |
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|
443 |
@{term "m=1"} and @{term "n=0"}.*} |
21413 | 444 |
lemma nat_power_less_imp_less: |
445 |
assumes nonneg: "0 < (i\<Colon>nat)" |
|
30996 | 446 |
assumes less: "i ^ m < i ^ n" |
21413 | 447 |
shows "m < n" |
448 |
proof (cases "i = 1") |
|
449 |
case True with less power_one [where 'a = nat] show ?thesis by simp |
|
450 |
next |
|
451 |
case False with nonneg have "1 < i" by auto |
|
452 |
from power_strict_increasing_iff [OF this] less show ?thesis .. |
|
453 |
qed |
|
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|
454 |
|
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changeset
|
455 |
lemma power_dvd_imp_le: |
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changeset
|
456 |
"i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n" |
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changeset
|
457 |
apply (rule power_le_imp_le_exp, assumption) |
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haftmann
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changeset
|
458 |
apply (erule dvd_imp_le, simp) |
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haftmann
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changeset
|
459 |
done |
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|
460 |
|
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changeset
|
461 |
|
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changeset
|
462 |
subsection {* Code generator tweak *} |
92d8ff6af82c
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|
463 |
|
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|
464 |
lemma power_power_power [code, code_unfold, code_inline del]: |
31155
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|
465 |
"power = power.power (1::'a::{power}) (op *)" |
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|
466 |
unfolding power_def power.power_def .. |
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|
467 |
|
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|
468 |
declare power.power.simps [code] |
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changeset
|
469 |
|
33364 | 470 |
code_modulename SML |
471 |
Power Arith |
|
472 |
||
473 |
code_modulename OCaml |
|
474 |
Power Arith |
|
475 |
||
476 |
code_modulename Haskell |
|
477 |
Power Arith |
|
478 |
||
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paulson
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diff
changeset
|
479 |
end |