author | hoelzl |
Mon, 03 Dec 2012 18:19:08 +0100 | |
changeset 50328 | 25b1e8686ce0 |
parent 49824 | c26665a197dc |
child 51263 | 31e786e0e6a7 |
permissions | -rw-r--r-- |
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0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
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(* Title: HOL/Power.thy |
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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New theory "Power" of exponentiation (and binomial coefficients)
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Copyright 1997 University of Cambridge |
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New theory "Power" of exponentiation (and binomial coefficients)
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*) |
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New theory "Power" of exponentiation (and binomial coefficients)
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header {* Exponentiation *} |
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theory Power |
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imports Num |
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begin |
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subsection {* Powers for Arbitrary Monoids *} |
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||
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class power = one + times |
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begin |
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|
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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" ("(_\<twosuperior>)" [1000] 999) where |
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"x\<twosuperior> \<equiv> x ^ 2" |
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notation (latex output) |
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power2 ("(_\<twosuperior>)" [1000] 999) |
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notation (HTML output) |
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power2 ("(_\<twosuperior>)" [1000] 999) |
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end |
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context monoid_mult |
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begin |
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subclass power . |
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lemma power_one [simp]: |
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"1 ^ n = 1" |
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by (induct n) simp_all |
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lemma power_one_right [simp]: |
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"a ^ 1 = a" |
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by simp |
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lemma power_commutes: |
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"a ^ n * a = a * a ^ n" |
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by (induct n) (simp_all add: mult_assoc) |
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lemma power_Suc2: |
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"a ^ Suc n = a ^ n * a" |
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by (simp add: power_commutes) |
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lemma power_add: |
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"a ^ (m + n) = a ^ m * a ^ n" |
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by (induct m) (simp_all add: algebra_simps) |
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lemma power_mult: |
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"a ^ (m * n) = (a ^ m) ^ n" |
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by (induct n) (simp_all add: power_add) |
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lemma power2_eq_square: "a\<twosuperior> = 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) ^ 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) ^ 2" |
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by (simp add: power_even_eq) |
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lemma power_numeral_even: |
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"z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)" |
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unfolding numeral_Bit0 power_add Let_def .. |
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lemma power_numeral_odd: |
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"z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)" |
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unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right |
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unfolding power_Suc power_add Let_def mult_assoc .. |
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lemma funpow_times_power: |
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"(times x ^^ f x) = times (x ^ f x)" |
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proof (induct "f x" arbitrary: f) |
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case 0 then show ?case by (simp add: fun_eq_iff) |
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next |
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case (Suc n) |
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def g \<equiv> "\<lambda>x. f x - 1" |
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with Suc have "n = g x" by simp |
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with Suc have "times x ^^ g x = times (x ^ g x)" by simp |
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moreover from Suc g_def have "f x = g x + 1" by simp |
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ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc) |
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qed |
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end |
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context comm_monoid_mult |
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begin |
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lemma power_mult_distrib: |
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"(a * b) ^ n = (a ^ n) * (b ^ n)" |
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by (induct n) (simp_all add: mult_ac) |
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end |
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context semiring_numeral |
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begin |
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lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k" |
|
121 |
by (simp only: sqr_conv_mult numeral_mult) |
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lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l" |
|
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by (induct l, simp_all only: numeral_class.numeral.simps pow.simps |
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numeral_sqr numeral_mult power_add power_one_right) |
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lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)" |
|
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by (rule numeral_pow [symmetric]) |
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||
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end |
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context semiring_1 |
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begin |
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||
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lemma of_nat_power: |
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"of_nat (m ^ n) = of_nat m ^ n" |
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by (induct n) (simp_all add: of_nat_mult) |
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lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0" |
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by (simp add: numeral_eq_Suc) |
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lemma zero_power2: "0\<twosuperior> = 0" (* delete? *) |
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by (rule power_zero_numeral) |
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lemma one_power2: "1\<twosuperior> = 1" (* delete? *) |
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by (rule power_one) |
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end |
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context comm_semiring_1 |
|
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begin |
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||
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text {* The divides relation *} |
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lemma le_imp_power_dvd: |
|
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assumes "m \<le> n" shows "a ^ m dvd a ^ n" |
|
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proof |
|
158 |
have "a ^ n = a ^ (m + (n - m))" |
|
159 |
using `m \<le> n` by simp |
|
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also have "\<dots> = a ^ m * a ^ (n - m)" |
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by (rule power_add) |
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finally show "a ^ n = a ^ m * a ^ (n - m)" . |
|
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qed |
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||
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lemma power_le_dvd: |
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"a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b" |
|
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by (rule dvd_trans [OF le_imp_power_dvd]) |
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lemma dvd_power_same: |
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170 |
"x dvd y \<Longrightarrow> x ^ n dvd y ^ n" |
|
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by (induct n) (auto simp add: mult_dvd_mono) |
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||
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lemma dvd_power_le: |
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"x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m" |
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by (rule power_le_dvd [OF dvd_power_same]) |
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lemma dvd_power [simp]: |
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assumes "n > (0::nat) \<or> x = 1" |
|
179 |
shows "x dvd (x ^ n)" |
|
180 |
using assms proof |
|
181 |
assume "0 < n" |
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182 |
then have "x ^ n = x ^ Suc (n - 1)" by simp |
|
183 |
then show "x dvd (x ^ n)" by simp |
|
184 |
next |
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185 |
assume "x = 1" |
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186 |
then show "x dvd (x ^ n)" by simp |
|
187 |
qed |
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||
189 |
end |
|
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||
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context ring_1 |
|
192 |
begin |
|
193 |
||
194 |
lemma power_minus: |
|
195 |
"(- a) ^ n = (- 1) ^ n * a ^ n" |
|
196 |
proof (induct n) |
|
197 |
case 0 show ?case by simp |
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198 |
next |
|
199 |
case (Suc n) then show ?case |
|
200 |
by (simp del: power_Suc add: power_Suc2 mult_assoc) |
|
201 |
qed |
|
202 |
||
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lemma power_minus_Bit0: |
204 |
"(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)" |
|
205 |
by (induct k, simp_all only: numeral_class.numeral.simps power_add |
|
206 |
power_one_right mult_minus_left mult_minus_right minus_minus) |
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||
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lemma power_minus_Bit1: |
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209 |
"(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))" |
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by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left) |
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|
212 |
lemma power_neg_numeral_Bit0 [simp]: |
|
213 |
"neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))" |
|
214 |
by (simp only: neg_numeral_def power_minus_Bit0 power_numeral) |
|
215 |
||
216 |
lemma power_neg_numeral_Bit1 [simp]: |
|
217 |
"neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))" |
|
218 |
by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps) |
|
219 |
||
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lemma power2_minus [simp]: |
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"(- a)\<twosuperior> = a\<twosuperior>" |
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by (rule power_minus_Bit0) |
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lemma power_minus1_even [simp]: |
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"-1 ^ (2*n) = 1" |
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proof (induct n) |
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227 |
case 0 show ?case by simp |
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228 |
next |
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229 |
case (Suc n) then show ?case by (simp add: power_add power2_eq_square) |
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230 |
qed |
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231 |
|
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232 |
lemma power_minus1_odd: |
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233 |
"-1 ^ Suc (2*n) = -1" |
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234 |
by simp |
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235 |
|
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236 |
lemma power_minus_even [simp]: |
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237 |
"(-a) ^ (2*n) = a ^ (2*n)" |
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238 |
by (simp add: power_minus [of a]) |
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239 |
|
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240 |
end |
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241 |
|
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242 |
context ring_1_no_zero_divisors |
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243 |
begin |
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244 |
|
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245 |
lemma field_power_not_zero: |
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246 |
"a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0" |
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247 |
by (induct n) auto |
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248 |
|
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249 |
lemma zero_eq_power2 [simp]: |
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250 |
"a\<twosuperior> = 0 \<longleftrightarrow> a = 0" |
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251 |
unfolding power2_eq_square by simp |
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252 |
|
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253 |
lemma power2_eq_1_iff: |
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254 |
"a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1" |
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255 |
unfolding power2_eq_square by (rule square_eq_1_iff) |
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256 |
|
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257 |
end |
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258 |
|
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259 |
context idom |
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260 |
begin |
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261 |
|
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262 |
lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y" |
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263 |
unfolding power2_eq_square by (rule square_eq_iff) |
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264 |
|
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265 |
end |
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266 |
|
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267 |
context division_ring |
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268 |
begin |
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269 |
|
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270 |
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *} |
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271 |
lemma nonzero_power_inverse: |
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272 |
"a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n" |
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273 |
by (induct n) |
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274 |
(simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero) |
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275 |
|
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276 |
end |
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277 |
|
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278 |
context field |
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279 |
begin |
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280 |
|
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281 |
lemma nonzero_power_divide: |
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282 |
"b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n" |
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283 |
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse) |
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284 |
|
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285 |
end |
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286 |
|
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287 |
|
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288 |
subsection {* Exponentiation on ordered types *} |
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289 |
|
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290 |
context linordered_ring (* TODO: move *) |
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291 |
begin |
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292 |
|
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293 |
lemma sum_squares_ge_zero: |
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294 |
"0 \<le> x * x + y * y" |
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295 |
by (intro add_nonneg_nonneg zero_le_square) |
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296 |
|
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297 |
lemma not_sum_squares_lt_zero: |
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298 |
"\<not> x * x + y * y < 0" |
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299 |
by (simp add: not_less sum_squares_ge_zero) |
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300 |
|
30996 | 301 |
end |
302 |
||
35028
108662d50512
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303 |
context linordered_semidom |
30996 | 304 |
begin |
305 |
||
306 |
lemma zero_less_power [simp]: |
|
307 |
"0 < a \<Longrightarrow> 0 < a ^ n" |
|
308 |
by (induct n) (simp_all add: mult_pos_pos) |
|
309 |
||
310 |
lemma zero_le_power [simp]: |
|
311 |
"0 \<le> a \<Longrightarrow> 0 \<le> a ^ n" |
|
312 |
by (induct n) (simp_all add: mult_nonneg_nonneg) |
|
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313 |
|
47241 | 314 |
lemma power_mono: |
315 |
"a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n" |
|
316 |
by (induct n) (auto intro: mult_mono order_trans [of 0 a b]) |
|
317 |
||
318 |
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n" |
|
319 |
using power_mono [of 1 a n] by simp |
|
320 |
||
321 |
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1" |
|
322 |
using power_mono [of a 1 n] by simp |
|
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323 |
|
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324 |
lemma power_gt1_lemma: |
30996 | 325 |
assumes gt1: "1 < a" |
326 |
shows "1 < a * a ^ n" |
|
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327 |
proof - |
30996 | 328 |
from gt1 have "0 \<le> a" |
329 |
by (fact order_trans [OF zero_le_one less_imp_le]) |
|
330 |
have "1 * 1 < a * 1" using gt1 by simp |
|
331 |
also have "\<dots> \<le> a * a ^ n" using gt1 |
|
332 |
by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le |
|
14577 | 333 |
zero_le_one order_refl) |
334 |
finally show ?thesis by simp |
|
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335 |
qed |
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336 |
|
30996 | 337 |
lemma power_gt1: |
338 |
"1 < a \<Longrightarrow> 1 < a ^ Suc n" |
|
339 |
by (simp add: power_gt1_lemma) |
|
24376 | 340 |
|
30996 | 341 |
lemma one_less_power [simp]: |
342 |
"1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n" |
|
343 |
by (cases n) (simp_all add: power_gt1_lemma) |
|
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344 |
|
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345 |
lemma power_le_imp_le_exp: |
30996 | 346 |
assumes gt1: "1 < a" |
347 |
shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n" |
|
348 |
proof (induct m arbitrary: n) |
|
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349 |
case 0 |
14577 | 350 |
show ?case by simp |
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351 |
next |
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352 |
case (Suc m) |
14577 | 353 |
show ?case |
354 |
proof (cases n) |
|
355 |
case 0 |
|
30996 | 356 |
with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp |
14577 | 357 |
with gt1 show ?thesis |
358 |
by (force simp only: power_gt1_lemma |
|
30996 | 359 |
not_less [symmetric]) |
14577 | 360 |
next |
361 |
case (Suc n) |
|
30996 | 362 |
with Suc.prems Suc.hyps show ?thesis |
14577 | 363 |
by (force dest: mult_left_le_imp_le |
30996 | 364 |
simp add: less_trans [OF zero_less_one gt1]) |
14577 | 365 |
qed |
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366 |
qed |
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367 |
|
14577 | 368 |
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*} |
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369 |
lemma power_inject_exp [simp]: |
30996 | 370 |
"1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n" |
14577 | 371 |
by (force simp add: order_antisym power_le_imp_le_exp) |
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372 |
|
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373 |
text{*Can relax the first premise to @{term "0<a"} in the case of the |
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374 |
natural numbers.*} |
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375 |
lemma power_less_imp_less_exp: |
30996 | 376 |
"1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n" |
377 |
by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] |
|
378 |
power_le_imp_le_exp) |
|
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379 |
|
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380 |
lemma power_strict_mono [rule_format]: |
30996 | 381 |
"a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n" |
382 |
by (induct n) |
|
383 |
(auto simp add: mult_strict_mono le_less_trans [of 0 a b]) |
|
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384 |
|
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385 |
text{*Lemma for @{text power_strict_decreasing}*} |
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|
386 |
lemma power_Suc_less: |
30996 | 387 |
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n" |
388 |
by (induct n) |
|
389 |
(auto simp add: mult_strict_left_mono) |
|
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changeset
|
390 |
|
30996 | 391 |
lemma power_strict_decreasing [rule_format]: |
392 |
"n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n" |
|
393 |
proof (induct N) |
|
394 |
case 0 then show ?case by simp |
|
395 |
next |
|
396 |
case (Suc N) then show ?case |
|
397 |
apply (auto simp add: power_Suc_less less_Suc_eq) |
|
398 |
apply (subgoal_tac "a * a^N < 1 * a^n") |
|
399 |
apply simp |
|
400 |
apply (rule mult_strict_mono) apply auto |
|
401 |
done |
|
402 |
qed |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
403 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
404 |
text{*Proof resembles that of @{text power_strict_decreasing}*} |
30996 | 405 |
lemma power_decreasing [rule_format]: |
406 |
"n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n" |
|
407 |
proof (induct N) |
|
408 |
case 0 then show ?case by simp |
|
409 |
next |
|
410 |
case (Suc N) then show ?case |
|
411 |
apply (auto simp add: le_Suc_eq) |
|
412 |
apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp) |
|
413 |
apply (rule mult_mono) apply auto |
|
414 |
done |
|
415 |
qed |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
416 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
417 |
lemma power_Suc_less_one: |
30996 | 418 |
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1" |
419 |
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
|
420 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
421 |
text{*Proof again resembles that of @{text power_strict_decreasing}*} |
30996 | 422 |
lemma power_increasing [rule_format]: |
423 |
"n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N" |
|
424 |
proof (induct N) |
|
425 |
case 0 then show ?case by simp |
|
426 |
next |
|
427 |
case (Suc N) then show ?case |
|
428 |
apply (auto simp add: le_Suc_eq) |
|
429 |
apply (subgoal_tac "1 * a^n \<le> a * a^N", simp) |
|
430 |
apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one]) |
|
431 |
done |
|
432 |
qed |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
433 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
434 |
text{*Lemma for @{text power_strict_increasing}*} |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
435 |
lemma power_less_power_Suc: |
30996 | 436 |
"1 < a \<Longrightarrow> a ^ n < a * a ^ n" |
437 |
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
|
438 |
|
30996 | 439 |
lemma power_strict_increasing [rule_format]: |
440 |
"n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N" |
|
441 |
proof (induct N) |
|
442 |
case 0 then show ?case by simp |
|
443 |
next |
|
444 |
case (Suc N) then show ?case |
|
445 |
apply (auto simp add: power_less_power_Suc less_Suc_eq) |
|
446 |
apply (subgoal_tac "1 * a^n < a * a^N", simp) |
|
447 |
apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le) |
|
448 |
done |
|
449 |
qed |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
450 |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
451 |
lemma power_increasing_iff [simp]: |
30996 | 452 |
"1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y" |
453 |
by (blast intro: power_le_imp_le_exp power_increasing less_imp_le) |
|
15066 | 454 |
|
455 |
lemma power_strict_increasing_iff [simp]: |
|
30996 | 456 |
"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
|
457 |
by (blast intro: power_less_imp_less_exp power_strict_increasing) |
15066 | 458 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
459 |
lemma power_le_imp_le_base: |
30996 | 460 |
assumes le: "a ^ Suc n \<le> b ^ Suc n" |
461 |
and ynonneg: "0 \<le> b" |
|
462 |
shows "a \<le> b" |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
463 |
proof (rule ccontr) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
464 |
assume "~ a \<le> b" |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
465 |
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
|
466 |
then have "b ^ Suc n < a ^ Suc n" |
41550 | 467 |
by (simp only: assms power_strict_mono) |
30996 | 468 |
from le and this show False |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
469 |
by (simp add: linorder_not_less [symmetric]) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset
|
470 |
qed |
14577 | 471 |
|
22853 | 472 |
lemma power_less_imp_less_base: |
473 |
assumes less: "a ^ n < b ^ n" |
|
474 |
assumes nonneg: "0 \<le> b" |
|
475 |
shows "a < b" |
|
476 |
proof (rule contrapos_pp [OF less]) |
|
477 |
assume "~ a < b" |
|
478 |
hence "b \<le> a" by (simp only: linorder_not_less) |
|
479 |
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) |
|
30996 | 480 |
thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less) |
22853 | 481 |
qed |
482 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
483 |
lemma power_inject_base: |
30996 | 484 |
"a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b" |
485 |
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
|
486 |
|
22955 | 487 |
lemma power_eq_imp_eq_base: |
30996 | 488 |
"a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b" |
489 |
by (cases n) (simp_all del: power_Suc, rule power_inject_base) |
|
22955 | 490 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
491 |
lemma power2_le_imp_le: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
492 |
"x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
493 |
unfolding numeral_2_eq_2 by (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
|
494 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
495 |
lemma power2_less_imp_less: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
496 |
"x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
497 |
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
|
498 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
499 |
lemma power2_eq_imp_eq: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
500 |
"x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
501 |
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
|
502 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
503 |
end |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
504 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
505 |
context linordered_ring_strict |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
506 |
begin |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
507 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
508 |
lemma sum_squares_eq_zero_iff: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
509 |
"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
|
510 |
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
|
511 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
512 |
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
|
513 |
"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
|
514 |
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
|
515 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
516 |
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
|
517 |
"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
|
518 |
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
|
519 |
|
30996 | 520 |
end |
521 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33364
diff
changeset
|
522 |
context linordered_idom |
30996 | 523 |
begin |
29978
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset
|
524 |
|
30996 | 525 |
lemma power_abs: |
526 |
"abs (a ^ n) = abs a ^ n" |
|
527 |
by (induct n) (auto simp add: abs_mult) |
|
528 |
||
529 |
lemma abs_power_minus [simp]: |
|
530 |
"abs ((-a) ^ n) = abs (a ^ n)" |
|
35216 | 531 |
by (simp add: power_abs) |
30996 | 532 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35216
diff
changeset
|
533 |
lemma zero_less_power_abs_iff [simp, no_atp]: |
30996 | 534 |
"0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0" |
535 |
proof (induct n) |
|
536 |
case 0 show ?case by simp |
|
537 |
next |
|
538 |
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
|
539 |
qed |
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset
|
540 |
|
30996 | 541 |
lemma zero_le_power_abs [simp]: |
542 |
"0 \<le> abs a ^ n" |
|
543 |
by (rule zero_le_power [OF abs_ge_zero]) |
|
544 |
||
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
545 |
lemma zero_le_power2 [simp]: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
546 |
"0 \<le> a\<twosuperior>" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
547 |
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
|
548 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
549 |
lemma zero_less_power2 [simp]: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
550 |
"0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
551 |
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
|
552 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
553 |
lemma power2_less_0 [simp]: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
554 |
"\<not> a\<twosuperior> < 0" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
555 |
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
|
556 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
557 |
lemma abs_power2 [simp]: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
558 |
"abs (a\<twosuperior>) = a\<twosuperior>" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
559 |
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
|
560 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
561 |
lemma power2_abs [simp]: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
562 |
"(abs a)\<twosuperior> = a\<twosuperior>" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
563 |
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
|
564 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
565 |
lemma odd_power_less_zero: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
566 |
"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
|
567 |
proof (induct n) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
568 |
case 0 |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
569 |
then show ?case by simp |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
570 |
next |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
571 |
case (Suc n) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
572 |
have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
573 |
by (simp add: mult_ac power_add power2_eq_square) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
574 |
thus ?case |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
575 |
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
|
576 |
qed |
30996 | 577 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
578 |
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
|
579 |
"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
|
580 |
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
|
581 |
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
|
582 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
583 |
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
|
584 |
"0 \<le> a ^ (2*n)" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
585 |
proof (induct n) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
586 |
case 0 |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
587 |
show ?case by simp |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
588 |
next |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
589 |
case (Suc n) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
590 |
have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
591 |
by (simp add: mult_ac power_add power2_eq_square) |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
592 |
thus ?case |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
593 |
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
|
594 |
qed |
30996 | 595 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
596 |
lemma sum_power2_ge_zero: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
597 |
"0 \<le> x\<twosuperior> + y\<twosuperior>" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
598 |
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
|
599 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
600 |
lemma not_sum_power2_lt_zero: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
601 |
"\<not> x\<twosuperior> + y\<twosuperior> < 0" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
602 |
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
|
603 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
604 |
lemma sum_power2_eq_zero_iff: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
605 |
"x\<twosuperior> + y\<twosuperior> = 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
|
606 |
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
|
607 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
608 |
lemma sum_power2_le_zero_iff: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
609 |
"x\<twosuperior> + y\<twosuperior> \<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
|
610 |
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
|
611 |
|
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
612 |
lemma sum_power2_gt_zero_iff: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
613 |
"0 < x\<twosuperior> + y\<twosuperior> \<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
|
614 |
unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff) |
30996 | 615 |
|
616 |
end |
|
617 |
||
29978
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset
|
618 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
619 |
subsection {* Miscellaneous rules *} |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
620 |
|
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
621 |
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
|
622 |
unfolding One_nat_def by (cases m) simp_all |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
623 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
624 |
lemma power2_sum: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
625 |
fixes x y :: "'a::comm_semiring_1" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
626 |
shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
627 |
by (simp add: algebra_simps power2_eq_square mult_2_right) |
30996 | 628 |
|
47192
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
629 |
lemma power2_diff: |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
630 |
fixes x y :: "'a::comm_ring_1" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
631 |
shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y" |
0c0501cb6da6
move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents:
47191
diff
changeset
|
632 |
by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute) |
30996 | 633 |
|
634 |
lemma power_0_Suc [simp]: |
|
635 |
"(0::'a::{power, semiring_0}) ^ Suc n = 0" |
|
636 |
by simp |
|
30313 | 637 |
|
30996 | 638 |
text{*It looks plausible as a simprule, but its effect can be strange.*} |
639 |
lemma power_0_left: |
|
640 |
"0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))" |
|
641 |
by (induct n) simp_all |
|
642 |
||
643 |
lemma power_eq_0_iff [simp]: |
|
644 |
"a ^ n = 0 \<longleftrightarrow> |
|
645 |
a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0" |
|
646 |
by (induct n) |
|
647 |
(auto simp add: no_zero_divisors elim: contrapos_pp) |
|
648 |
||
36409 | 649 |
lemma (in field) power_diff: |
30996 | 650 |
assumes nz: "a \<noteq> 0" |
651 |
shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n" |
|
36409 | 652 |
by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero) |
30313 | 653 |
|
30996 | 654 |
text{*Perhaps these should be simprules.*} |
655 |
lemma power_inverse: |
|
36409 | 656 |
fixes a :: "'a::division_ring_inverse_zero" |
657 |
shows "inverse (a ^ n) = inverse a ^ n" |
|
30996 | 658 |
apply (cases "a = 0") |
659 |
apply (simp add: power_0_left) |
|
660 |
apply (simp add: nonzero_power_inverse) |
|
661 |
done (* TODO: reorient or rename to inverse_power *) |
|
662 |
||
663 |
lemma power_one_over: |
|
36409 | 664 |
"1 / (a::'a::{field_inverse_zero, power}) ^ n = (1 / a) ^ n" |
30996 | 665 |
by (simp add: divide_inverse) (rule power_inverse) |
666 |
||
667 |
lemma power_divide: |
|
36409 | 668 |
"(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n" |
30996 | 669 |
apply (cases "b = 0") |
670 |
apply (simp add: power_0_left) |
|
671 |
apply (rule nonzero_power_divide) |
|
672 |
apply assumption |
|
30313 | 673 |
done |
674 |
||
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
675 |
text {* Simprules for comparisons where common factors can be cancelled. *} |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
676 |
|
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
677 |
lemmas zero_compare_simps = |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
678 |
add_strict_increasing add_strict_increasing2 add_increasing |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
679 |
zero_le_mult_iff zero_le_divide_iff |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
680 |
zero_less_mult_iff zero_less_divide_iff |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
681 |
mult_le_0_iff divide_le_0_iff |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
682 |
mult_less_0_iff divide_less_0_iff |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
683 |
zero_le_power2 power2_less_0 |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47241
diff
changeset
|
684 |
|
30313 | 685 |
|
30960 | 686 |
subsection {* Exponentiation for the Natural Numbers *} |
14577 | 687 |
|
30996 | 688 |
lemma nat_one_le_power [simp]: |
689 |
"Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n" |
|
690 |
by (rule one_le_power [of i n, unfolded One_nat_def]) |
|
23305 | 691 |
|
30996 | 692 |
lemma nat_zero_less_power_iff [simp]: |
693 |
"x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0" |
|
694 |
by (induct n) auto |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
695 |
|
30056 | 696 |
lemma nat_power_eq_Suc_0_iff [simp]: |
30996 | 697 |
"x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0" |
698 |
by (induct m) auto |
|
30056 | 699 |
|
30996 | 700 |
lemma power_Suc_0 [simp]: |
701 |
"Suc 0 ^ n = Suc 0" |
|
702 |
by simp |
|
30056 | 703 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
704 |
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
|
705 |
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
|
706 |
@{term "m=1"} and @{term "n=0"}.*} |
21413 | 707 |
lemma nat_power_less_imp_less: |
708 |
assumes nonneg: "0 < (i\<Colon>nat)" |
|
30996 | 709 |
assumes less: "i ^ m < i ^ n" |
21413 | 710 |
shows "m < n" |
711 |
proof (cases "i = 1") |
|
712 |
case True with less power_one [where 'a = nat] show ?thesis by simp |
|
713 |
next |
|
714 |
case False with nonneg have "1 < i" by auto |
|
715 |
from power_strict_increasing_iff [OF this] less show ?thesis .. |
|
716 |
qed |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset
|
717 |
|
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
31998
diff
changeset
|
718 |
lemma power_dvd_imp_le: |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
31998
diff
changeset
|
719 |
"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
|
720 |
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
|
721 |
apply (erule dvd_imp_le, simp) |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
31998
diff
changeset
|
722 |
done |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
31998
diff
changeset
|
723 |
|
31155
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
724 |
|
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
725 |
subsection {* Code generator tweak *} |
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
726 |
|
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
|
727 |
lemma power_power_power [code]: |
31155
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
728 |
"power = power.power (1::'a::{power}) (op *)" |
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
729 |
unfolding power_def power.power_def .. |
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
730 |
|
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
731 |
declare power.power.simps [code] |
92d8ff6af82c
monomorphic code generation for power operations
haftmann
parents:
31021
diff
changeset
|
732 |
|
33364 | 733 |
code_modulename SML |
734 |
Power Arith |
|
735 |
||
736 |
code_modulename OCaml |
|
737 |
Power Arith |
|
738 |
||
739 |
code_modulename Haskell |
|
740 |
Power Arith |
|
741 |
||
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset
|
742 |
end |
49824 | 743 |