src/HOL/Power.thy
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(*  Title:      HOL/Power.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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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 Num
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begin
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subsection {* Powers for Arbitrary Monoids *}
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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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text {* Special syntax for squares. *}
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abbreviation (xsymbols)
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  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
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  "x\<^sup>2 \<equiv> x ^ 2"
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notation (latex output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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notation (HTML output)
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  power2  ("(_\<^sup>2)" [1000] 999)
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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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lemma power2_eq_square: "a\<^sup>2 = a * a"
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  by (simp add: numeral_2_eq_2)
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lemma power3_eq_cube: "a ^ 3 = a * a * a"
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  by (simp add: numeral_3_eq_3 mult_assoc)
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lemma power_even_eq:
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  "a ^ (2 * n) = (a ^ n)\<^sup>2"
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  by (subst mult_commute) (simp add: power_mult)
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lemma power_odd_eq:
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  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
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  by (simp add: power_even_eq)
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lemma power_numeral_even:
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  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
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  unfolding numeral_Bit0 power_add Let_def ..
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lemma power_numeral_odd:
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  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
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  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
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  unfolding power_Suc power_add Let_def mult_assoc ..
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lemma funpow_times_power:
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  "(times x ^^ f x) = times (x ^ f x)"
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proof (induct "f x" arbitrary: f)
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  case 0 then show ?case by (simp add: fun_eq_iff)
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next
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  case (Suc n)
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  def g \<equiv> "\<lambda>x. f x - 1"
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  with Suc have "n = g x" by simp
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  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
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  moreover from Suc g_def have "f x = g x + 1" by simp
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  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc)
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qed
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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_numeral
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begin
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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  by (simp only: sqr_conv_mult numeral_mult)
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
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  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
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    numeral_sqr numeral_mult power_add power_one_right)
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
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  by (rule numeral_pow [symmetric])
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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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lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
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  by (simp add: numeral_eq_Suc)
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
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  by (rule power_zero_numeral)
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
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  by (rule power_one)
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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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   157
proof
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   158
  have "a ^ n = a ^ (m + (n - m))"
648d02b124d8 cleaned up Power theory
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   159
    using `m \<le> n` by simp
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   160
  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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   164
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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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   168
648d02b124d8 cleaned up Power theory
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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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   181
  assume "0 < n"
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   182
  then have "x ^ n = x ^ Suc (n - 1)" by simp
648d02b124d8 cleaned up Power theory
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   183
  then show "x dvd (x ^ n)" by simp
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   184
next
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   185
  assume "x = 1"
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   186
  then show "x dvd (x ^ n)" by simp
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   187
qed
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   188
648d02b124d8 cleaned up Power theory
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end
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context ring_1
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begin
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   193
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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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   200
    by (simp del: power_Suc add: power_Suc2 mult_assoc)
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   201
qed
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47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
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lemma power_minus_Bit0:
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  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   205
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
ebd8c46d156b bootstrap Num.thy before Power.thy;
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    power_one_right mult_minus_left mult_minus_right minus_minus)
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   207
ebd8c46d156b bootstrap Num.thy before Power.thy;
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lemma power_minus_Bit1:
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  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
47220
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  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
47191
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lemma power2_minus [simp]:
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  "(- a)\<^sup>2 = a\<^sup>2"
47192
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  by (rule power_minus_Bit0)
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   215
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma power_minus1_even [simp]:
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  "-1 ^ (2*n) = 1"
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proof (induct n)
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  case 0 show ?case by simp
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next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   222
qed
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   223
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma power_minus1_odd:
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  "-1 ^ Suc (2*n) = -1"
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  by simp
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   227
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lemma power_minus_even [simp]:
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  "(-a) ^ (2*n) = a ^ (2*n)"
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  by (simp add: power_minus [of a])
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   231
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   232
end
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   233
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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context ring_1_no_zero_divisors
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begin
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   236
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma field_power_not_zero:
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  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
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  by (induct n) auto
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   240
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma zero_eq_power2 [simp]:
53015
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  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
47192
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   243
  unfolding power2_eq_square by simp
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   244
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lemma power2_eq_1_iff:
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   246
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
47192
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  unfolding power2_eq_square by (rule square_eq_1_iff)
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   248
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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end
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   250
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   251
context idom
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begin
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lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
47192
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   255
  unfolding power2_eq_square by (rule square_eq_iff)
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   256
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   257
end
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   258
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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context division_ring
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begin
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0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
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lemma nonzero_power_inverse:
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  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   265
  by (induct n)
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   266
    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   267
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   268
end
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   269
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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context field
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begin
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   272
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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lemma nonzero_power_divide:
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  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   276
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   277
end
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0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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subsection {* Exponentiation on ordered types *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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context linordered_ring (* TODO: move *)
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   283
begin
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   284
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   285
lemma sum_squares_ge_zero:
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   286
  "0 \<le> x * x + y * y"
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   287
  by (intro add_nonneg_nonneg zero_le_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   288
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   289
lemma not_sum_squares_lt_zero:
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   290
  "\<not> x * x + y * y < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   291
  by (simp add: not_less sum_squares_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   292
30996
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   293
end
648d02b124d8 cleaned up Power theory
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   294
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
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   295
context linordered_semidom
30996
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   296
begin
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   297
648d02b124d8 cleaned up Power theory
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   298
lemma zero_less_power [simp]:
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   299
  "0 < a \<Longrightarrow> 0 < a ^ n"
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   300
  by (induct n) (simp_all add: mult_pos_pos)
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diff changeset
   301
648d02b124d8 cleaned up Power theory
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   302
lemma zero_le_power [simp]:
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   303
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
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   304
  by (induct n) (simp_all add: mult_nonneg_nonneg)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   305
47241
243b33052e34 add lemma power_le_one
huffman
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   306
lemma power_mono:
243b33052e34 add lemma power_le_one
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diff changeset
   307
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
243b33052e34 add lemma power_le_one
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   308
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
243b33052e34 add lemma power_le_one
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diff changeset
   309
243b33052e34 add lemma power_le_one
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   310
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
243b33052e34 add lemma power_le_one
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diff changeset
   311
  using power_mono [of 1 a n] by simp
243b33052e34 add lemma power_le_one
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parents: 47220
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   312
243b33052e34 add lemma power_le_one
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   313
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
243b33052e34 add lemma power_le_one
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parents: 47220
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   314
  using power_mono [of a 1 n] by simp
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   315
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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   316
lemma power_gt1_lemma:
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   317
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
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diff changeset
   318
  shows "1 < a * a ^ n"
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diff changeset
   319
proof -
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   320
  from gt1 have "0 \<le> a"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   321
    by (fact order_trans [OF zero_le_one less_imp_le])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   322
  have "1 * 1 < a * 1" using gt1 by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   323
  also have "\<dots> \<le> a * a ^ n" using gt1
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   324
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
14577
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parents: 14438
diff changeset
   325
        zero_le_one order_refl)
dbb95b825244 tuned document;
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parents: 14438
diff changeset
   326
  finally show ?thesis by simp
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   327
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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diff changeset
   328
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   329
lemma power_gt1:
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parents: 30960
diff changeset
   330
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   331
  by (simp add: power_gt1_lemma)
24376
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
   332
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parents: 30960
diff changeset
   333
lemma one_less_power [simp]:
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parents: 30960
diff changeset
   334
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   335
  by (cases n) (simp_all add: power_gt1_lemma)
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diff changeset
   336
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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   337
lemma power_le_imp_le_exp:
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   338
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   339
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
648d02b124d8 cleaned up Power theory
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diff changeset
   340
proof (induct m arbitrary: n)
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   341
  case 0
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   342
  show ?case by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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   343
next
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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diff changeset
   344
  case (Suc m)
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   345
  show ?case
dbb95b825244 tuned document;
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   346
  proof (cases n)
dbb95b825244 tuned document;
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parents: 14438
diff changeset
   347
    case 0
30996
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haftmann
parents: 30960
diff changeset
   348
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
14577
dbb95b825244 tuned document;
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parents: 14438
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   349
    with gt1 show ?thesis
dbb95b825244 tuned document;
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parents: 14438
diff changeset
   350
      by (force simp only: power_gt1_lemma
30996
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haftmann
parents: 30960
diff changeset
   351
          not_less [symmetric])
14577
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   352
  next
dbb95b825244 tuned document;
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   353
    case (Suc n)
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parents: 30960
diff changeset
   354
    with Suc.prems Suc.hyps show ?thesis
14577
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parents: 14438
diff changeset
   355
      by (force dest: mult_left_le_imp_le
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haftmann
parents: 30960
diff changeset
   356
          simp add: less_trans [OF zero_less_one gt1])
14577
dbb95b825244 tuned document;
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parents: 14438
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   357
  qed
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   358
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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parents: 8844
diff changeset
   359
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dbb95b825244 tuned document;
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diff changeset
   360
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   361
lemma power_inject_exp [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   362
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   363
  by (force simp add: order_antisym power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   364
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   365
text{*Can relax the first premise to @{term "0<a"} in the case of the
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   366
natural numbers.*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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parents: 8844
diff changeset
   367
lemma power_less_imp_less_exp:
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parents: 30960
diff changeset
   368
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   369
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   370
    power_le_imp_le_exp)
14348
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parents: 8844
diff changeset
   371
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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parents: 8844
diff changeset
   372
lemma power_strict_mono [rule_format]:
30996
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haftmann
parents: 30960
diff changeset
   373
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   374
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   375
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   376
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   377
text{*Lemma for @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   378
lemma power_Suc_less:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   379
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   380
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   381
    (auto simp add: mult_strict_left_mono)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   382
30996
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haftmann
parents: 30960
diff changeset
   383
lemma power_strict_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   384
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   385
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   386
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   387
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   388
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   389
  apply (auto simp add: power_Suc_less less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   390
  apply (subgoal_tac "a * a^N < 1 * a^n")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   391
  apply simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   392
  apply (rule mult_strict_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   393
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   394
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   395
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   396
text{*Proof resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   397
lemma power_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   398
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   399
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   400
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   401
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   402
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   403
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   404
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   405
  apply (rule mult_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   406
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   407
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   408
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   409
lemma power_Suc_less_one:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   410
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   411
  using power_strict_decreasing [of 0 "Suc n" a] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   412
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   413
text{*Proof again resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   414
lemma power_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   415
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   416
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   417
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   418
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   419
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   420
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   421
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   422
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   423
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   424
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   425
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   426
text{*Lemma for @{text power_strict_increasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   427
lemma power_less_power_Suc:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   428
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   429
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   430
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   431
lemma power_strict_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   432
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   433
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   434
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   435
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   436
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   437
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   438
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   439
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   440
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   441
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   442
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   443
lemma power_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   444
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   445
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   446
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   447
lemma power_strict_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   448
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   449
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   450
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   451
lemma power_le_imp_le_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   452
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   453
    and ynonneg: "0 \<le> b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   454
  shows "a \<le> b"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   455
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   456
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   457
  then have "b < a" by (simp only: linorder_not_le)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   458
  then have "b ^ Suc n < a ^ Suc n"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39438
diff changeset
   459
    by (simp only: assms power_strict_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   460
  from le and this show False
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   461
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   462
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   463
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   464
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   465
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   466
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   467
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   468
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   469
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   470
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   471
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   472
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   473
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   474
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   475
lemma power_inject_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   476
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   477
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   478
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   479
lemma power_eq_imp_eq_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   480
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   481
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   482
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   483
lemma power2_le_imp_le:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   484
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   485
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   486
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   487
lemma power2_less_imp_less:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   488
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   489
  by (rule power_less_imp_less_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   490
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   491
lemma power2_eq_imp_eq:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   492
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   493
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   494
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   495
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   496
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   497
context linordered_ring_strict
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   498
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   499
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   500
lemma sum_squares_eq_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   501
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   502
  by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   503
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   504
lemma sum_squares_le_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   505
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   506
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   507
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   508
lemma sum_squares_gt_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   509
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   510
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   511
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   512
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   513
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   514
context linordered_idom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   515
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   516
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   517
lemma power_abs:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   518
  "abs (a ^ n) = abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   519
  by (induct n) (auto simp add: abs_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   520
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   521
lemma abs_power_minus [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   522
  "abs ((-a) ^ n) = abs (a ^ n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   523
  by (simp add: power_abs)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   524
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53076
diff changeset
   525
lemma zero_less_power_abs_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   526
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   527
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   528
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   529
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   530
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   531
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   532
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   533
lemma zero_le_power_abs [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   534
  "0 \<le> abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   535
  by (rule zero_le_power [OF abs_ge_zero])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   536
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   537
lemma zero_le_power2 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   538
  "0 \<le> a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   539
  by (simp add: power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   540
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   541
lemma zero_less_power2 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   542
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   543
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   544
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   545
lemma power2_less_0 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   546
  "\<not> a\<^sup>2 < 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   547
  by (force simp add: power2_eq_square mult_less_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   548
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   549
lemma abs_power2 [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   550
  "abs (a\<^sup>2) = a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   551
  by (simp add: power2_eq_square abs_mult abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   552
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   553
lemma power2_abs [simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   554
  "(abs a)\<^sup>2 = a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   555
  by (simp add: power2_eq_square abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   556
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   557
lemma odd_power_less_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   558
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   559
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   560
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   561
  then show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   562
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   563
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   564
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   565
    by (simp add: mult_ac power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   566
  thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   567
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   568
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   569
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   570
lemma odd_0_le_power_imp_0_le:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   571
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   572
  using odd_power_less_zero [of a n]
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   573
    by (force simp add: linorder_not_less [symmetric]) 
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   574
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   575
lemma zero_le_even_power'[simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   576
  "0 \<le> a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   577
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   578
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   579
    show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   580
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   581
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   582
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   583
      by (simp add: mult_ac power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   584
    thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   585
      by (simp add: Suc zero_le_mult_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   586
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   587
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   588
lemma sum_power2_ge_zero:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   589
  "0 \<le> x\<^sup>2 + y\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   590
  by (intro add_nonneg_nonneg zero_le_power2)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   591
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   592
lemma not_sum_power2_lt_zero:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   593
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   594
  unfolding not_less by (rule sum_power2_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   595
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   596
lemma sum_power2_eq_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   597
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   598
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   599
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   600
lemma sum_power2_le_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   601
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   602
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   603
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   604
lemma sum_power2_gt_zero_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   605
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   606
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   607
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   608
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   609
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   610
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   611
subsection {* Miscellaneous rules *}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   612
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   613
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   614
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   615
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   616
lemma power2_sum:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   617
  fixes x y :: "'a::comm_semiring_1"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   618
  shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   619
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   620
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   621
lemma power2_diff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   622
  fixes x y :: "'a::comm_ring_1"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   623
  shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   624
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   625
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   626
lemma power_0_Suc [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   627
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   628
  by simp
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   629
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   630
text{*It looks plausible as a simprule, but its effect can be strange.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   631
lemma power_0_left:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   632
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   633
  by (induct n) simp_all
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   634
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   635
lemma power_eq_0_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   636
  "a ^ n = 0 \<longleftrightarrow>
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   637
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   638
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   639
    (auto simp add: no_zero_divisors elim: contrapos_pp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   640
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   641
lemma (in field) power_diff:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   642
  assumes nz: "a \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   643
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   644
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   645
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   646
text{*Perhaps these should be simprules.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   647
lemma power_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   648
  fixes a :: "'a::division_ring_inverse_zero"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   649
  shows "inverse (a ^ n) = inverse a ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   650
apply (cases "a = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   651
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   652
apply (simp add: nonzero_power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   653
done (* TODO: reorient or rename to inverse_power *)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   654
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   655
lemma power_one_over:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   656
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   657
  by (simp add: divide_inverse) (rule power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   658
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   659
lemma power_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   660
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   661
apply (cases "b = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   662
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   663
apply (rule nonzero_power_divide)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   664
apply assumption
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   665
done
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   666
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   667
text {* Simprules for comparisons where common factors can be cancelled. *}
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   668
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   669
lemmas zero_compare_simps =
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   670
    add_strict_increasing add_strict_increasing2 add_increasing
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   671
    zero_le_mult_iff zero_le_divide_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   672
    zero_less_mult_iff zero_less_divide_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   673
    mult_le_0_iff divide_le_0_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   674
    mult_less_0_iff divide_less_0_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   675
    zero_le_power2 power2_less_0
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   676
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   677
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   678
subsection {* Exponentiation for the Natural Numbers *}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   679
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   680
lemma nat_one_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   681
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   682
  by (rule one_le_power [of i n, unfolded One_nat_def])
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   683
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   684
lemma nat_zero_less_power_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   685
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   686
  by (induct n) auto
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   687
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   688
lemma nat_power_eq_Suc_0_iff [simp]: 
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   689
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   690
  by (induct m) auto
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   691
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   692
lemma power_Suc_0 [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   693
  "Suc 0 ^ n = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   694
  by simp
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   695
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   696
text{*Valid for the naturals, but what if @{text"0<i<1"}?
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   697
Premises cannot be weakened: consider the case where @{term "i=0"},
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   698
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   699
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   700
  assumes nonneg: "0 < (i\<Colon>nat)"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   701
  assumes less: "i ^ m < i ^ n"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   702
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   703
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   704
  case True with less power_one [where 'a = nat] show ?thesis by simp
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   705
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   706
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   707
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   708
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   709
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   710
lemma power_dvd_imp_le:
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   711
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   712
  apply (rule power_le_imp_le_exp, assumption)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   713
  apply (erule dvd_imp_le, simp)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   714
  done
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   715
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   716
lemma power2_nat_le_eq_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   717
  fixes m n :: nat
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   718
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   719
  by (auto intro: power2_le_imp_le power_mono)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   720
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   721
lemma power2_nat_le_imp_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   722
  fixes m n :: nat
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   723
  assumes "m\<^sup>2 \<le> n"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   724
  shows "m \<le> n"
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   725
proof (cases m)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   726
  case 0 then show ?thesis by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   727
next
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   728
  case (Suc k)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   729
  show ?thesis
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   730
  proof (rule ccontr)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   731
    assume "\<not> m \<le> n"
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   732
    then have "n < m" by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   733
    with assms Suc show False
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   734
      by (auto simp add: algebra_simps) (simp add: power2_eq_square)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   735
  qed
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   736
qed
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   737
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   738
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   739
subsection {* Code generator tweak *}
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   740
45231
d85a2fdc586c replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
bulwahn
parents: 41550
diff changeset
   741
lemma power_power_power [code]:
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   742
  "power = power.power (1::'a::{power}) (op *)"
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   743
  unfolding power_def power.power_def ..
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   744
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   745
declare power.power.simps [code]
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   746
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51263
diff changeset
   747
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51263
diff changeset
   748
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   749
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   750
end
49824
c26665a197dc msetprod based directly on Multiset.fold;
haftmann
parents: 47255
diff changeset
   751