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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section {* Exponentiation *}
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theory Power
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imports Num Equiv_Relations
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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_Suc0_right [simp]:
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  "a ^ Suc 0 = 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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lemma power_commuting_commutes:
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  assumes "x * y = y * x"
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  shows "x ^ n * y = y * x ^n"
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proof (induct n)
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  case (Suc n)
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  have "x ^ Suc n * y = x ^ n * y * x"
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    by (subst power_Suc2) (simp add: assms ac_simps)
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  also have "\<dots> = y * x ^ Suc n"
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    unfolding Suc power_Suc2
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    by (simp add: ac_simps)
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  finally show ?case .
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qed simp
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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 [field_simps]:
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  "(a * b) ^ n = (a ^ n) * (b ^ n)"
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  by (induct n) (simp_all add: ac_simps)
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end
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text{*Extract constant factors from powers*}
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declare power_mult_distrib [where a = "numeral w" for w, simp]
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declare power_mult_distrib [where b = "numeral w" for w, simp]
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lemma power_add_numeral [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
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  by (simp add: power_add [symmetric])
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lemma power_add_numeral2 [simp]:
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  fixes a :: "'a :: monoid_mult"
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  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents: 59867
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   146
  by (simp add: mult.assoc [symmetric])
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents: 59867
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   147
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
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   148
lemma power_mult_numeral [simp]:
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paulson <lp15@cam.ac.uk>
parents: 59867
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   149
  fixes a :: "'a :: monoid_mult"
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents: 59867
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   150
  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents: 59867
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   151
  by (simp only: numeral_mult power_mult)
91477b3a2d6b Tidying. Improved simplification for numerals, esp in exponents.
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   152
47191
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context semiring_numeral
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   154
begin
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   155
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   156
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
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huffman
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   157
  by (simp only: sqr_conv_mult numeral_mult)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
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diff changeset
   158
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   159
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
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   160
  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
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diff changeset
   161
    numeral_sqr numeral_mult power_add power_one_right)
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   162
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   163
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
ebd8c46d156b bootstrap Num.thy before Power.thy;
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  by (rule numeral_pow [symmetric])
ebd8c46d156b bootstrap Num.thy before Power.thy;
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ebd8c46d156b bootstrap Num.thy before Power.thy;
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   166
end
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context semiring_1
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   169
begin
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   170
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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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   174
59009
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   175
lemma zero_power:
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haftmann
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   176
  "0 < n \<Longrightarrow> 0 ^ n = 0"
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haftmann
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  by (cases n) simp_all
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haftmann
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   178
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
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lemma power_zero_numeral [simp]:
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  "0 ^ numeral k = 0"
47209
4893907fe872 add constant pred_numeral k = numeral k - (1::nat);
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   181
  by (simp add: numeral_eq_Suc)
47191
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lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
47192
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  by (rule power_zero_numeral)
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   185
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a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
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lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
47192
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   187
  by (rule power_one)
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   188
30996
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   189
end
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   190
648d02b124d8 cleaned up Power theory
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   191
context comm_semiring_1
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   192
begin
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   193
648d02b124d8 cleaned up Power theory
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   194
text {* The divides relation *}
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   195
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   196
lemma le_imp_power_dvd:
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   197
  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
648d02b124d8 cleaned up Power theory
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parents: 30960
diff changeset
   198
proof
648d02b124d8 cleaned up Power theory
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   199
  have "a ^ n = a ^ (m + (n - m))"
648d02b124d8 cleaned up Power theory
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parents: 30960
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   200
    using `m \<le> n` by simp
648d02b124d8 cleaned up Power theory
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parents: 30960
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   201
  also have "\<dots> = a ^ m * a ^ (n - m)"
648d02b124d8 cleaned up Power theory
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parents: 30960
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   202
    by (rule power_add)
648d02b124d8 cleaned up Power theory
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   203
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
648d02b124d8 cleaned up Power theory
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parents: 30960
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   204
qed
648d02b124d8 cleaned up Power theory
haftmann
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   205
648d02b124d8 cleaned up Power theory
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   206
lemma power_le_dvd:
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   207
  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
648d02b124d8 cleaned up Power theory
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   208
  by (rule dvd_trans [OF le_imp_power_dvd])
648d02b124d8 cleaned up Power theory
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parents: 30960
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   209
648d02b124d8 cleaned up Power theory
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   210
lemma dvd_power_same:
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   211
  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
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   212
  by (induct n) (auto simp add: mult_dvd_mono)
648d02b124d8 cleaned up Power theory
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parents: 30960
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   213
648d02b124d8 cleaned up Power theory
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   214
lemma dvd_power_le:
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   215
  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
648d02b124d8 cleaned up Power theory
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parents: 30960
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   216
  by (rule power_le_dvd [OF dvd_power_same])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   217
30996
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lemma dvd_power [simp]:
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  assumes "n > (0::nat) \<or> x = 1"
648d02b124d8 cleaned up Power theory
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parents: 30960
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   220
  shows "x dvd (x ^ n)"
648d02b124d8 cleaned up Power theory
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   221
using assms proof
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   222
  assume "0 < n"
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   223
  then have "x ^ n = x ^ Suc (n - 1)" by simp
648d02b124d8 cleaned up Power theory
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parents: 30960
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   224
  then show "x dvd (x ^ n)" by simp
648d02b124d8 cleaned up Power theory
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   225
next
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   226
  assume "x = 1"
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   227
  then show "x dvd (x ^ n)" by simp
648d02b124d8 cleaned up Power theory
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   228
qed
648d02b124d8 cleaned up Power theory
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   229
648d02b124d8 cleaned up Power theory
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   230
end
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   231
648d02b124d8 cleaned up Power theory
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   232
context ring_1
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   233
begin
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   234
648d02b124d8 cleaned up Power theory
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   235
lemma power_minus:
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   236
  "(- a) ^ n = (- 1) ^ n * a ^ n"
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   237
proof (induct n)
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   238
  case 0 show ?case by simp
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   239
next
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   240
  case (Suc n) then show ?case
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
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   241
    by (simp del: power_Suc add: power_Suc2 mult.assoc)
30996
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   242
qed
648d02b124d8 cleaned up Power theory
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diff changeset
   243
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
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   244
lemma power_minus_Bit0:
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   245
  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   246
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   247
    power_one_right mult_minus_left mult_minus_right minus_minus)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
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   248
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
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   249
lemma power_minus_Bit1:
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   250
  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
47220
52426c62b5d0 replace lemmas eval_nat_numeral with a simpler reformulation
huffman
parents: 47209
diff changeset
   251
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   252
47192
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   253
lemma power2_minus [simp]:
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wenzelm
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   254
  "(- a)\<^sup>2 = a\<^sup>2"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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   255
  by (rule power_minus_Bit0)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   256
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   257
lemma power_minus1_even [simp]:
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
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diff changeset
   258
  "(- 1) ^ (2*n) = 1"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   259
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   260
  case 0 show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   261
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   262
  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
huffman
parents: 47191
diff changeset
   263
qed
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   264
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   265
lemma power_minus1_odd:
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58157
diff changeset
   266
  "(- 1) ^ Suc (2*n) = -1"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   267
  by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   268
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   269
lemma power_minus_even [simp]:
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diff changeset
   270
  "(-a) ^ (2*n) = a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   271
  by (simp add: power_minus [of a])
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   272
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   273
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   274
58787
af9eb5e566dd eliminated redundancies;
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   275
lemma power_eq_0_nat_iff [simp]:
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   276
  fixes m n :: nat
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
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   277
  shows "m ^ n = 0 \<longleftrightarrow> m = 0 \<and> n > 0"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   278
  by (induct n) auto
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   279
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   280
context ring_1_no_zero_divisors
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   281
begin
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   282
58787
af9eb5e566dd eliminated redundancies;
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   283
lemma power_eq_0_iff [simp]:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   284
  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   285
  by (induct n) auto
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   286
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
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diff changeset
   287
lemma field_power_not_zero:
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huffman
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diff changeset
   288
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   289
  by (induct n) auto
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   290
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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parents: 47191
diff changeset
   291
lemma zero_eq_power2 [simp]:
53015
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wenzelm
parents: 52435
diff changeset
   292
  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   293
  unfolding power2_eq_square by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   294
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   295
lemma power2_eq_1_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
diff changeset
   296
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   297
  unfolding power2_eq_square by (rule square_eq_1_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   298
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   299
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   300
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context idom
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   302
begin
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parents: 47191
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   303
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52435
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   304
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   305
  unfolding power2_eq_square by (rule square_eq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   306
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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   307
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   308
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   309
context normalization_semidom
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haftmann
parents: 60155
diff changeset
   310
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   311
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   312
lemma normalize_power:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   313
  "normalize (a ^ n) = normalize a ^ n"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   314
  by (induct n) (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   315
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   316
lemma unit_factor_power:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   317
  "unit_factor (a ^ n) = unit_factor a ^ n"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   318
  by (induct n) (simp_all add: unit_factor_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   319
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   320
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60155
diff changeset
   321
47192
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   322
context division_ring
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diff changeset
   323
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   324
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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diff changeset
   325
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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parents: 47191
diff changeset
   326
lemma nonzero_power_inverse:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
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parents: 47191
diff changeset
   327
  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   328
  by (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   329
    (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
huffman
parents: 47191
diff changeset
   330
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   331
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   332
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   333
context field
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   334
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   335
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   336
lemma nonzero_power_divide:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   337
  "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
huffman
parents: 47191
diff changeset
   338
  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
huffman
parents: 47191
diff changeset
   339
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59009
diff changeset
   340
declare nonzero_power_divide [where b = "numeral w" for w, simp]
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59009
diff changeset
   341
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   342
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   343
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   344
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   345
subsection {* Exponentiation on ordered types *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   346
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   347
context linordered_ring (* TODO: move *)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   348
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   349
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   350
lemma sum_squares_ge_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   351
  "0 \<le> x * x + y * y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   352
  by (intro add_nonneg_nonneg zero_le_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   353
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   354
lemma not_sum_squares_lt_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   355
  "\<not> x * x + y * y < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   356
  by (simp add: not_less sum_squares_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   357
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   358
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   359
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   360
context linordered_semidom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   361
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   362
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   363
lemma zero_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   364
  "0 < a \<Longrightarrow> 0 < a ^ n"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   365
  by (induct n) simp_all
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   366
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   367
lemma zero_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   368
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56481
diff changeset
   369
  by (induct n) simp_all
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   370
47241
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   371
lemma power_mono:
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   372
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   373
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   374
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   375
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   376
  using power_mono [of 1 a n] by simp
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   377
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   378
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   379
  using power_mono [of a 1 n] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   380
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   381
lemma power_gt1_lemma:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   382
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   383
  shows "1 < a * a ^ n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   384
proof -
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   385
  from gt1 have "0 \<le> a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   386
    by (fact order_trans [OF zero_le_one less_imp_le])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   387
  have "1 * 1 < a * 1" using gt1 by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   388
  also have "\<dots> \<le> a * a ^ n" using gt1
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   389
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   390
        zero_le_one order_refl)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   391
  finally show ?thesis by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   392
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   393
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   394
lemma power_gt1:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   395
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   396
  by (simp add: power_gt1_lemma)
24376
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
   397
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   398
lemma one_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   399
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   400
  by (cases n) (simp_all add: power_gt1_lemma)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   401
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   402
lemma power_le_imp_le_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   403
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   404
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   405
proof (induct m arbitrary: n)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   406
  case 0
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   407
  show ?case by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   408
next
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   409
  case (Suc m)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   410
  show ?case
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   411
  proof (cases n)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   412
    case 0
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   413
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   414
    with gt1 show ?thesis
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   415
      by (force simp only: power_gt1_lemma
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   416
          not_less [symmetric])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   417
  next
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   418
    case (Suc n)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   419
    with Suc.prems Suc.hyps show ?thesis
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   420
      by (force dest: mult_left_le_imp_le
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   421
          simp add: less_trans [OF zero_less_one gt1])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   422
  qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   423
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   424
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   425
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
   426
lemma power_inject_exp [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   427
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   428
  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
   429
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   430
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
   431
natural numbers.*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   432
lemma power_less_imp_less_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   433
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   434
  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
   435
    power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   436
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   437
lemma power_strict_mono [rule_format]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   438
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   439
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   440
   (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
   441
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   442
text{*Lemma for @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   443
lemma power_Suc_less:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   444
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   445
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   446
    (auto simp add: mult_strict_left_mono)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   447
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   448
lemma power_strict_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   449
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   450
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   451
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   452
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   453
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   454
  apply (auto simp add: power_Suc_less less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   455
  apply (subgoal_tac "a * a^N < 1 * a^n")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   456
  apply simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   457
  apply (rule mult_strict_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   458
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   459
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   460
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   461
text{*Proof resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   462
lemma power_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   463
  "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
   464
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   465
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   466
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   467
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   468
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   469
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   470
  apply (rule mult_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   471
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   472
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   473
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   474
lemma power_Suc_less_one:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   475
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   476
  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
   477
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   478
text{*Proof again resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   479
lemma power_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   480
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   481
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   482
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   483
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   484
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   485
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   486
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   487
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   488
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   489
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   490
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   491
text{*Lemma for @{text power_strict_increasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   492
lemma power_less_power_Suc:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   493
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   494
  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
   495
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   496
lemma power_strict_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   497
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   498
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   499
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   500
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   501
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   502
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   503
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   504
  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
   505
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   506
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   507
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   508
lemma power_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   509
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   510
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   511
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   512
lemma power_strict_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   513
  "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
   514
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   515
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   516
lemma power_le_imp_le_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   517
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   518
    and ynonneg: "0 \<le> b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   519
  shows "a \<le> b"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   520
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   521
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   522
  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
   523
  then have "b ^ Suc n < a ^ Suc n"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39438
diff changeset
   524
    by (simp only: assms power_strict_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   525
  from le and this show False
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   526
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   527
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   528
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   529
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   530
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   531
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   532
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   533
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   534
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   535
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   536
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   537
  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
   538
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   539
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   540
lemma power_inject_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   541
  "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
   542
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
   543
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   544
lemma power_eq_imp_eq_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   545
  "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
   546
  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
   547
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   548
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
   549
  "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
   550
  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
   551
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   552
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
   553
  "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
   554
  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
   555
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   556
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
   557
  "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
   558
  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
   559
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   560
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   561
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   562
context linordered_ring_strict
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   563
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   564
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   565
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
   566
  "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
   567
  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
   568
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   569
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
   570
  "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
   571
  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
   572
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   573
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
   574
  "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
   575
  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
   576
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   577
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   578
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   579
context linordered_idom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   580
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   581
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   582
lemma power_abs:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   583
  "abs (a ^ n) = abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   584
  by (induct n) (auto simp add: abs_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   585
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   586
lemma abs_power_minus [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   587
  "abs ((-a) ^ n) = abs (a ^ n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   588
  by (simp add: power_abs)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   589
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53076
diff changeset
   590
lemma zero_less_power_abs_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   591
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   592
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   593
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   594
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   595
  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
   596
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   597
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   598
lemma zero_le_power_abs [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   599
  "0 \<le> abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   600
  by (rule zero_le_power [OF abs_ge_zero])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   601
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   602
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
   603
  "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
   604
  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
   605
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   606
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
   607
  "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
   608
  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
   609
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   610
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
   611
  "\<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
   612
  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
   613
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   614
lemma power2_less_eq_zero_iff [simp]:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   615
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   616
  by (simp add: le_less)
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   617
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   618
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
   619
  "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
   620
  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
   621
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   622
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
   623
  "(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
   624
  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
   625
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   626
lemma odd_power_less_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   627
  "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
   628
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   629
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   630
  then show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   631
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   632
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   633
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   634
    by (simp add: ac_simps power_add power2_eq_square)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   635
  thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   636
    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
   637
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   638
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   639
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
   640
  "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
   641
  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
   642
    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
   643
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   644
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
   645
  "0 \<le> a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   646
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   647
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   648
    show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   649
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   650
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   651
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   652
      by (simp add: ac_simps power_add power2_eq_square)
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   653
    thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   654
      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
   655
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   656
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   657
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
   658
  "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
   659
  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
   660
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   661
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
   662
  "\<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
   663
  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
   664
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   665
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
   666
  "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
   667
  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
   668
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   669
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
   670
  "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
   671
  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
   672
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   673
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
   674
  "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
   675
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   676
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   677
lemma abs_le_square_iff:
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   678
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   679
proof
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   680
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   681
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   682
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   683
next
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   684
  assume "x\<^sup>2 \<le> y\<^sup>2"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   685
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   686
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   687
qed
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   688
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   689
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   690
  using abs_le_square_iff [of x 1]
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   691
  by simp
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   692
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   693
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   694
  by (auto simp add: abs_if power2_eq_1_iff)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   695
  
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   696
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   697
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   698
  by (auto simp add: le_less)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   699
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   700
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   701
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   702
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   703
subsection {* Miscellaneous rules *}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   704
55718
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55096
diff changeset
   705
lemma self_le_power:
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55096
diff changeset
   706
  fixes x::"'a::linordered_semidom" 
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55096
diff changeset
   707
  shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
55811
aa1acc25126b load Metis a little later
traytel
parents: 55718
diff changeset
   708
  using power_increasing[of 1 n x] power_one_right[of x] by auto
55718
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55096
diff changeset
   709
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   710
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
   711
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   712
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   713
lemma (in comm_semiring_1) power2_sum:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   714
  "(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
   715
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   716
58787
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   717
lemma (in comm_ring_1) power2_diff:
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   718
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
af9eb5e566dd eliminated redundancies;
haftmann
parents: 58656
diff changeset
   719
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   720
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   721
lemma power_0_Suc [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   722
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   723
  by simp
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   724
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   725
text{*It looks plausible as a simprule, but its effect can be strange.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   726
lemma power_0_left:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   727
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   728
  by (induct n) simp_all
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   729
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   730
lemma (in field) power_diff:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   731
  assumes nz: "a \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   732
  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
   733
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   734
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   735
text{*Perhaps these should be simprules.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   736
lemma power_inverse:
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59865
diff changeset
   737
  fixes a :: "'a::division_ring"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   738
  shows "inverse (a ^ n) = inverse a ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   739
apply (cases "a = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   740
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   741
apply (simp add: nonzero_power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   742
done (* TODO: reorient or rename to inverse_power *)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   743
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   744
lemma power_one_over:
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59865
diff changeset
   745
  "1 / (a::'a::{field, power}) ^ n =  (1 / a) ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   746
  by (simp add: divide_inverse) (rule power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   747
56481
47500d0881f9 add divide_simps
hoelzl
parents: 56480
diff changeset
   748
lemma power_divide [field_simps, divide_simps]:
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59865
diff changeset
   749
  "(a / b) ^ n = (a::'a::field) ^ n / b ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   750
apply (cases "b = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   751
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   752
apply (rule nonzero_power_divide)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   753
apply assumption
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   754
done
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   755
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   756
text {* Simprules for comparisons where common factors can be cancelled. *}
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   757
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   758
lemmas zero_compare_simps =
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   759
    add_strict_increasing add_strict_increasing2 add_increasing
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   760
    zero_le_mult_iff zero_le_divide_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   761
    zero_less_mult_iff zero_less_divide_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   762
    mult_le_0_iff divide_le_0_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   763
    mult_less_0_iff divide_less_0_iff 
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   764
    zero_le_power2 power2_less_0
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47241
diff changeset
   765
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   766
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   767
subsection {* Exponentiation for the Natural Numbers *}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   768
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   769
lemma nat_one_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   770
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   771
  by (rule one_le_power [of i n, unfolded One_nat_def])
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   772
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   773
lemma nat_zero_less_power_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   774
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   775
  by (induct n) auto
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   776
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   777
lemma nat_power_eq_Suc_0_iff [simp]: 
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   778
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   779
  by (induct m) auto
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   780
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   781
lemma power_Suc_0 [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   782
  "Suc 0 ^ n = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   783
  by simp
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   784
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   785
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
   786
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
   787
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   788
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   789
  assumes nonneg: "0 < (i\<Colon>nat)"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   790
  assumes less: "i ^ m < i ^ n"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   791
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   792
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   793
  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
   794
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   795
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   796
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   797
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   798
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   799
lemma power_dvd_imp_le:
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   800
  "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
   801
  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
   802
  apply (erule dvd_imp_le, simp)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   803
  done
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   804
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   805
lemma power2_nat_le_eq_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   806
  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
   807
  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
   808
  by (auto intro: power2_le_imp_le power_mono)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   809
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   810
lemma power2_nat_le_imp_le:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   811
  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
   812
  assumes "m\<^sup>2 \<le> n"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   813
  shows "m \<le> n"
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   814
proof (cases m)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   815
  case 0 then show ?thesis by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   816
next
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   817
  case (Suc k)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   818
  show ?thesis
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   819
  proof (rule ccontr)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   820
    assume "\<not> m \<le> n"
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   821
    then have "n < m" by simp
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   822
    with assms Suc show False
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   823
      by (auto simp add: algebra_simps) (simp add: power2_eq_square)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   824
  qed
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54147
diff changeset
   825
qed
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49824
diff changeset
   826
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   827
subsubsection {* Cardinality of the Powerset *}
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   828
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   829
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   830
  unfolding UNIV_bool by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   831
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   832
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   833
proof (induct rule: finite_induct)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   834
  case empty 
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   835
    show ?case by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   836
next
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   837
  case (insert x A)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   838
  then have "inj_on (insert x) (Pow A)" 
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   839
    unfolding inj_on_def by (blast elim!: equalityE)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   840
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A" 
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   841
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   842
  then show ?case using insert
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   843
    apply (simp add: Pow_insert)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   844
    apply (subst card_Un_disjoint, auto)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   845
    done
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   846
qed
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   847
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   848
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   849
subsubsection {* Generalized sum over a set *}
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   850
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   851
lemma setsum_zero_power [simp]:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   852
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   853
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   854
apply (cases "finite A")
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   855
  by (induction A rule: finite_induct) auto
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   856
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   857
lemma setsum_zero_power' [simp]:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   858
  fixes c :: "nat \<Rightarrow> 'a::field"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   859
  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   860
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   861
  by auto
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   862
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   863
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   864
subsubsection {* Generalized product over a set *}
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   865
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   866
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   867
apply (erule finite_induct)
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   868
apply auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   869
done
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   870
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   871
lemma setprod_power_distrib:
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   872
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   873
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   874
proof (cases "finite A") 
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   875
  case True then show ?thesis 
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   876
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   877
next
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   878
  case False then show ?thesis 
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   879
    by simp
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   880
qed
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   881
58437
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   882
lemma power_setsum:
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   883
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   884
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
8d124c73c37a added lemmas
haftmann
parents: 58410
diff changeset
   885
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   886
lemma setprod_gen_delta:
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   887
  assumes fS: "finite S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   888
  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   889
proof-
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   890
  let ?f = "(\<lambda>k. if k=a then b k else c)"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   891
  {assume a: "a \<notin> S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   892
    hence "\<forall> k\<in> S. ?f k = c" by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   893
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   894
  moreover 
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   895
  {assume a: "a \<in> S"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   896
    let ?A = "S - {a}"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   897
    let ?B = "{a}"
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   898
    have eq: "S = ?A \<union> ?B" using a by blast 
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   899
    have dj: "?A \<inter> ?B = {}" by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   900
    from fS have fAB: "finite ?A" "finite ?B" by auto  
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   901
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   902
      apply (rule setprod.cong) by auto
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   903
    have cA: "card ?A = card S - 1" using fS a by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   904
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   905
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   906
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   907
      by simp
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   908
    then have ?thesis using a cA
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56544
diff changeset
   909
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
55096
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   910
  ultimately show ?thesis by blast
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   911
qed
916b2ac758f4 removed theory dependency of BNF_LFP on Datatype
traytel
parents: 54489
diff changeset
   912
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   913
subsection {* Code generator tweak *}
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   914
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
   915
lemma power_power_power [code]:
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   916
  "power = power.power (1::'a::{power}) (op *)"
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   917
  unfolding power_def power.power_def ..
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   918
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   919
declare power.power.simps [code]
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   920
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51263
diff changeset
   921
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
   922
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   923
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   924
end
49824
c26665a197dc msetprod based directly on Multiset.fold;
haftmann
parents: 47255
diff changeset
   925